Linear motion topics 1 Linear Motion - multiple choice What is the cross product A x B of the vectors A = (1,0,1) and B = (-1,0,1)? (a) (0,0,2) (b) (0,-1,0) (c) (0,2,0) (d) (0,-2,0) (e) none of [a]-[d] The two vectors each have length √2 and lie in the xz-plane, making angles of +45o and -45o with respect to the z-axis, respectively. The magnitude of the cross product is then (√2)2 sin90o = 2. By the right-hand rule, the direction is in the -y direction. Hence, (0,2,0) Consider two vectors A = (2,2,0) and B(4,-1,0). If a vector C = (1,0,0) is added to B, what is the change in the cross-product A x B (i.e., the new value minus the old value). [a] (0,0,-2) [b] 2 [c] (0,0,2) [d] (-10,0,0) [e] none of [a]-[d] The difference in the cross products is Ax(B + C) - AxB = AxB + AxC - AxB = AxC. Thus, the difference is (0, 0, 0 - 1•2) = (0, 0, -2). Find the vector C in the cross product C = A x B, if A = (0,1,0) and B = (1,0,0). (a) (1,1,0) (b) (0,0,1) (c) (0,0, -1) (d) (1, -1,0) (e) (0,0,1) Either use the explicit Cartesian representation, or the following logic: Vectors A and B are unit vectors pointing along the y and x axes, respectively. Thus, the cross product must be perpendicular to the xy-plane, pointing along the +z or -z directions. From the right-hand rule, the direction must be negative. Lastly, the magnitude of C equals the product of the magnitudes of A and B (each of which equals 1) and the sine of the angle between them. Here, the sine equals 1, so the resulting vector is C = (0,0, -1). The displacement of an object for a round trip between two locations (a) is always greater than zero. (b) is always less than zero. (c) is zero. (d) is not zero. (e) can have any value. For a round trip, the displacement must vanish because the initial and final positions coincide. An object moves with constant speed in a straight line. Which of the following statements must be true? (a) No force acts on the object. (b) A single constant force acts on the object in the direction of motion. (c) The net force acting on the object depends on the value of the speed. (d) The net force acting on the object is zero. (e) The net force acting on the object cannot be determined. The acceleration is determined by the net force; hence, a vanishing acceleration implies that the net force vanishes, but not that there is no force. Linear motion topics 2 As a sky diver falls through the air, her terminal speed (a) depends on her mass. (b) depends on her body’s orientation. (c) depends on the local value of the acceleration due to gravity. (d) depends on the density of air. (e) is described by all of the above. All of these features apply to terminal velocity. A man riding in an elevator has an apparent weight greater than his actual weight. Which one of the following statements could be true? (a) The elevator moves upward with constant speed. (b) The elevator moves downward with constant speed. (c) The elevator moves upward with decreasing speed. (d) The elevator moves downward with decreasing speed. (e) The elevator moves downward with increasing speed. If his apparent weight is less than his real weight, then his acceleration must be upward. This could arise in two situations: v is upward and increasing or v is downward and decreasing. A moving box receives an impulse directed to the north. As a result, the box (a) has a velocity pointing north. (b) is north of the equator. (c) stops. (d) has a velocity towards the south. (e) accelerates towards the north. The direction of the acceleration must be the same as the net force, but the velocity may be in any direction. A lamp of mass m hangs from a spring scale that is attached to the ceiling of an elevator. When the elevator is stopped at the fortieth floor, the scale reads mg. What does it read while the elevator decends toward the ground floor at a constant speed? (a) More than mg. (b) Less than mg. (c) mg. (d) Zero. (e)This cannot be answered without knowing how fast the elevator is descending. If the speed of the lamp with respect to the elevator cage is constant, then there is no relative acceleration, and the scale must still read mg. Hence: (c). A falling abject experiences the drag force due to air resistance. Which statement is NOT true? (a) The drag force depends on the falling speed. (b) The faster the ball falls, the stronger the air resistance. (c) The mechanical energy of the object is conserved. (d) The speed of the object will reach a maximum value and then stop changing. (e) The net force acting on the object will eventually reach zero. Linear motion topics 3 The mechanical energy of the object must decrease because of the dissipative force. The displacement of an object during any time interval is always (a) greater than or equal to the distance it travels during the same time interval. (b) less than or equal to the distance it travels during the same time interval. (c) equal to the distance it travels during the same time interval. (d) greater than the distance it travels during the same time interval. (e) much greater than the distance it travels during the same time interval. Distance increases monotonically with time, with the result that displacement can never exceed distance. The change in the position of a boat is recorded every hour. Viewing the water's surface as an xy coordinate system, the change in the (x,y) coordinates of the boat over two hours is observed to be (0,2) and (1, -1) (quoted in km). What is the average speed of the boat divided by its average velocity? (a) 4.0 (b) 2.4 (c) 1.00 (d) 2.0 (e) 1.08 The total distance covered in two hours is 2 + (12 + [-1]2)1/2 = 2 + √2. The total displacement vector is (0,2) + (1, -1) = (1,1), which has a length of √2. Thus, speed / velocity = distance / displacement = (2 + √2) / √2 = 1 + √2 = 2.4. A jogger changes her speed several times during the first four minutes of her run: 50 m/min for the first 30 seconds, 70 m/min for the next 1.5 minutes, and 90 m/min for the last two minutes. What is her average speed in m/min? (a) 70 (b) 78 (c) 80 (d) 73 (e) 75 The total distance covered by the jogger is 0.5 x 50 + 1.5 x 70 + 2.0 x 90 = 310 m. Hence, the average speed is 310/4 = 78 m/min. An automobile is moving towards the east at 50 km/hr, and the wind is from the north at 50 km/hr. Which vector represents the wind velocity as observed by a passenger in the car, where the first component points to the east and the second component to the north? [a] (50, -50) [b] (-50, 50) [c] (50, 50) [d] (100, 0) [e] (-50, -50) Subtract the car velocity from the wind velocity to obtain the relative velocity: vwind - vcar = (0, -50) - (50, 0) = (-50, -50). An object moving along the x-axis has a position x = +3 m and is moving with a velocity of -4 m/s. The object is slowing down. The sign of its acceleration is (a) 0 (b) negative (c) positive (d) <0 until it stops, then >0 (e) impossible to tell from the data given. The initial velocity is negative, but tends to zero as the object stops. Thus, Δv > 0 and the acceleration must be positive. Starting at rest, an object falls a height h in time t. Assuming that the only force on the object is its gravitational attraction to the Earth, how far does the object fall in an elapsed time of 3t, starting from rest? (a) 3h (b) 9h (c) 27h (d) 6h (e) h/3 Linear motion topics 4 The position of an object with zero initial velocity and subject to constant gravitational acceleration -g is z = -gt2/2. Thus, if the elapsed time is 3t, the height fallen must be 9h. An arrow shot vertically into the air rises to a height h before it falls back to Earth. If the speed of the arrow as it leaves the bow is doubled, how high could the arrow rise? (a) 8h (b) h (c) 16h (d) 2h (e) 4h The position of an object with zero final velocity (when the arrow reaches its maximum height) and subject to constant gravitational acceleration -g is h = -(vf2 - vi2)/2g = vi2/2g. Thus, if the initial velocity is doubled, then the maximum height increases by a factor of 22 = 4. A girl is standing in an elevator traveling upwards at a constant speed v. She observes that it takes a time t for a penny to drop from her hand to the floor of the elevator. If the elevator were traveling downwards at a constant speed of 2v, how long would it take for the penny to drop? (a) t (b) 2t (c) 4t (d) t/2 (e) none of [a]-[d] Because the velocity of the elevator is constant in both situations, then the time for relative motion is unchanged. Hence, it takes t whether the elevator is at rest or moving with constant velocity. A projectile is fired at an angle of 35o above the horizontal. At the highest point in its trajectory, its speed is 200 m/s. If air resistance is neglected, what is the initial horizontal component of the projectile's velocity (in m/s)? (a) 0 (b) 200 cos35o (c) 200 sin35o (d) 200/cos35o (e) 200 At the highest point in the trajectory, the velocity is entirely horizontal. Hence, the horizontal component must be 200 m/s throughout the flight of the projectile. A golfball is shot horizontally from the top of a vertical cliff of height h. If the initial speed of the ball is v, how far away from the bottom of the cliff does the ball land? (a) v(2h/g)2 (b) gt2/2 (c) v(2h/g)1/2 (d) vg / 2h (e) none of (a)-(d) Resolving the motion into components: horizontal distance = R = vt vertical distance = h = gt2 /2. ---> R = v(2h / g)1/2. A car moving with constant speed in a circular path experiences a centripetal acceleration ac. If the speed of the car increases by a factor of three, but the circular path remains the same, what is the new centripetal acceleration in terms of the original ac? (a) 3ac (b) ac/9 (c) ac/3 (d) 9ac (e) ac 2 Since the centripetal acceleration is proportional to v /R, then the acceleration increases by 32=9 if the speed increases by 3. Two cars are travelling in concentric circles of radius R and 2R, as in the diagram. Linear motion topics 5 2R 4R Each car completes its own circle in the same time T. What is the centripetal acceleration of the outer car divided by that of the inner car? (a) 1 (b) 2 (c) 1/2 (d) 4π2 (e) 8π2 In general, ac = v2/r with v = 2πr/T, so that ac = (2πr/T)/r = 4π2r/T2. Hence, ac(outer)/ac(inner) = 2R / R = 2. A Ferris wheel ride at a circus has a diameter of 20 m. What must be the speed of the perimeter of the wheel to produce a centripetal acceleration (at the perimeter) the same as that of gravity (use g = 10 m/s2; answers quoted in m/s)? (a) 10 (b) 14 (c) 200 (d) 20 (e) 100 Centripetal acceleration ac = v2/R. Demanding ac = 10 m/s2, v = (acR)1/2 = (10 x 20 / 2)1/2 = 10 m/s. An object at the end of a string is swung in a circular path at constant speed with a period T. If the period is shortened to T/2 without changing the radius of the circle, what is the new centripetal acceleration in terms of the original acceleration a? (a) 4a (b) a/2 (c) a/4 (d) a (e) 2a The centripetal acceleration a is equal to 2πv/T. In turn, the velocity is 2πR/T, so the acceleration is 4π2R/T2. Thus, if the period is decreased by a factor of 2, the acceleration is increased by a factor of 4. A car travels at constant speed on a circular test track of radius R, completing each lap around the track in time T. The centripetal acceleration of the car, ac, is at the limit where the tires start to skid. If the test track were three times as large (i.e., had a radius of 3R), what would be the shortest period in which the car could complete a lap without its acceleration exceeding the same ac as for the smaller track? (a) 3T (b) 9T (c) T / 3 (d) √3 T (e) none of (a - d) The speed of the car is 2πR / T, so the centripetal acceleration is proportional to R / T2, or T2 ~ R /ac. If the centripetal acceleration is to remain unchanged in the new track, then the period must be at least √3 T. The total linear momentum of a system of particles will be conserved if: (a) the positions of the particles do not change with respect to each other (b) one particle is at rest (c) no external force acts on the system (d) the internal forces equal the external forces (e) the particles do not rotate about their axes. Force is the rate of change of linear momentum. Hence, there must be no external force for the rate of change of momentum to be zero. Linear motion topics 6 An 80 kg man on ice skates pushes a 40 kg boy, also on skates, away from him with a force of 100 N. What is the force exerted by the boy on the man? (a) 200 N (b) 100 N (c) 50 N (d) 40 N (e) 0 From Newton's third law, action = reaction, so the force of the boy on the man must be 100 N. A mass travelling with an initial velocity vi is brought to rest by a force F. If the initial velocity is doubled to 2vi, what force is necessary to bring the object to rest in the same distance as the first situation? (a) F (b) 2F (c) 3F (d) 4F (e) 8F 2 2 Using a = (vf - vi ) / 2d, then vi -> 2vi implies a -> 4a and hence F -> 4F. A constant force F is applied for a time T to a body initially at rest. The object covers a distance D during the time T. If the force is doubled and applies for 2T, what distance is covered by the object, starting from rest? (a) D (b) 2D (c) 4D (d) 6D (e) 8D If the force is doubled, so is the acceleration. Using d = at2/2, the distance becomes (2a) (2T)2 / 2 = 8 (aT2/2) = 8D. Having decided to elope, you construct a rope from nylon stockings, down which you will slide from your second-floor bedroom. The rope can withstand a maximum tension of 300 N without breaking. Your mass is 61.2 kg. What is the smallest acceleration down which you can slide without breaking the rope? (a) 9.8 m/s2 (b) 4.9 m/s2 (c) 0 m/s2 (d) 2.4 m/s2 (e) 19.6 m/s2 The net force on the rope is mg - ma = 300N. Thus a = g - 300/m = 9.8 - (300/61.2) = 4.9 m/s2. A vertical rope is attached to an object of mass M. What is the tension in the rope in order to give the mass an upward acceleration of 3g? (a) 3g (b) 4g (c) Mg (d) 3Mg (e) 4Mg A free-body diagram shows that the net force on the mass is T - Mg, which must give the object an upward acceleration of 3g. From F = Ma, T - Mg = 3Mg or T = 4Mg. The weight of an object on the Moon is 1/6 the weight of the same object on the Earth. If an object moves with a speed v and kinetic energy K on the Earth, what is the kinetic energy of the same object moving with speed v on the Moon? (a) 36K (b) 6K (c) K (d) K /6 (e) K /36 Kinetic energy is the same, K, as it depends on mass and velocity, not weight. The acceleration due to gravity on the Moon is only 1/6 that on the Earth. An object with a weight of 60 N on Earth is transported to the Moon. What is its mass as measured on Linear motion topics 7 the Moon? Use g = 10 m/s2 on Earth. (a) 6 kg (b) 1 kg (c) 0.6 kg (d) 60 kg (e) 10 kg Mass on Earth = w / g = 60 / 10 = 6 kg. The same mass must be measured on the Moon. Two masses (m and 2m) are attached to one another by a string as illustrated. A force F acts on mass m to accelerate the whole system. What is the magnitude of the net force on m? 2m m F (a) F /3 (b) F (c) 2F /3 (d) F /2 (e) 3F /2 The overall acceleration of the blocks is a = F /3m. But this acceleration is the result of the net force on a given block. Hence, the net force required to produce an acceleration of a on m is Fnet = ma = mF /(3m) = F /3. Two masses (m and 2m) are attached to one another by a string as illustrated. A force F acts on mass m to accelerate the whole system. What is the magnitude of the force on mass 2m? 2m m F (a) F /3 (b) F (c) 2F /3 (d) F /2 (e) 3F /2 The overall acceleration of the blocks is a = F /3m. The net force required to produce an acceleration of a on 2m is F = 2ma = 2mF /(3m) = 2F /3. Two masses (m1 and m2) are connected by a massless string and accelerated uniformly on a frictionless surface, as shown. The ratio of the tensions T1/T2 is given by: m2 m1 T1 T2 (a) m1/m2 (b) m2/m1 m1 (c) (m1+m2)/m2 T1 (d) m1/(m1+m2) m2 T1 (e) m2/(m1+m2) T2 At m1, an unbalanced force results in an acceleration T1 = m1a. At m2, an unbalanced force results in an acceleration T2 = T1 + m2a = (m1+m2)a. Hence, T1/T2 = m1/(m1+m2). A picture frame of mass 0.5 kg is held in place by a massless wire making an angle of 20o with respect to the horizontal. What is the tension in the wire (in Newtons)? Linear motion topics 8 T 20 o (a) 7.2 N (b) 4.9 N (c) 2.5 N (d) 14.3 N (e) 0.84 N Consider the forces at the nail. Taking components in the vertical direction T T 20 o mg 2T sin 20o= mg hence T = = 7.2 N. A vertical rope is attached to an object that has a mass of 40 kg and is at rest. The tension in the rope needed to give the object an upward speed of 3.5 m/s in 0.70 seconds is (a) 392 N (b) 192 N (c) 532 N (d) 592 N (e) 140 N The free-body diagram for the mass is T mg The tension in the string is unbalanced, such that T - mg = ma or T = m(g - a) The acceleration of the object is a = (vf - vi)/t = 3.50 / 0.7 = 5 m/s2. Hence, T = 40•(9.8 - 5) = 192 N. The radius of the planet Xion is three times that of the Earth, although the densities of the planets are the same. What is the acceleration due to gravity on the surface of Xion? (a) 27g (b) g (c) 3g (d) g / 3 (e) g / 9 2 From the gravitational force expression F = G M1M2/R , the acceleration due to gravity on the surface of a planet is ag = G M / R2. Since the mass of a planet with constant density grows like R3, then the gravitational acceleration must grow like R3/R2 = R. Hence, the acceleration due to gravity on Xion must be 3g. Two identical springs, each with force constant k are attached in series k k What is the effective force constant of the springs, taken together? (a) k (b) 2k (c) k / 2 (d) 4k (e) k / 4 If a single spring is stretched by a distance x, then the spring system must stretch by 2x. But the force exerted by the individual spring is kx , and must be constant along the Linear motion topics 9 system. If the effective force constant of the system is equal to the applied force divided by the displacement from equilibrium, then the effective force constant is kx / 2x = k / 2. Two identical springs, each with force constant k, are attached in parallel. What is the effective force constant of the springs, taken together? k k (a) 2k (b) 4k (c) k (d) k/4 (e) k/2 Because the springs are in parallel, it takes twice as much force to displace them a distance x as it does to displace a single spring by an amount x. Hence, the effective spring constant is double that of a single spring: 2k. The coefficients of friction between a 10 kg block and a table are µs = 0.8 and µk = 0.6. If a horizontal force F = 100 N is applied to the block, what is the net force that the block experiences? (Use g = 10 m/s2 ) F (a) 20 N (b) 40 N (c) 60 N (d) 80 N (e) 100 N The maximum force from static friction is 0.8 mg = 0.8 • 10 • 10 = 80 N, which is less than the applied force. Hence, the block will slide, but is opposed by the force from kinetic friction of 0.6 mg = 0.6 • 10 • 10 = 60 N. The net force on the block is then 40 N. A 10 kg block is at rest on a level surface, where the coefficient of static friction is 0.5 and that for kinetic friction is 0.3. A horizontal force of 40 N is applied to the block. What is the frictional force on the object? (a) 98 N (b) 40 N (c) 49 N (d) 29 N (e) 0 N The minimum force required to move the object is µsmg = 10 x 9.8 x 0.5 = 49 N. This threshold is larger than the applied force, so the frictional force must just be the applied force itself, 40 N. A block of mass m is pulled across a rough surface with a coefficient of kinetic friction T ! m µk, as shown in the drawing. The tension in the rope pulling the block is T. What is the frictional force on the block if it is pulled at a constant non-zero acceleration a? (a) µkmg (b) T cosθ - ma (c) µk(T - mg) (d) µkTsinθ Linear motion topics 10 (e) none of (a)-(d) The frictional force reduces the applied force to produce a constant acceleration: ma = Tcosθ - f ----> f = T cosθ - ma. A spring under a compression x has a potential energy Vo. When the compression is doubled 2x, the potential energy stored in the spring is (a) Vo (b) 2Vo (c) 3Vo (d) 8Vo (e) None of [a-d] is correct. The potential energy of a spring is quadratic in the displacement from equilibrium. Thus, if the compression is doubled, the potential energy must increase by a factor of four. A car of mass 1000 kg rolls without friction down Burnaby Mountain for a total vertical distance of 300 m. If the car started from rest at the top of the hill, what is its speed at the bottom of the hill in km/hr? (a) 195 (b) 54.2 (c) 76.7 (d) 2940 (e) 276 Using conservation of energy, mgh = mv2/2 -----> v = √(2gh). = √(2 x 9.8 x 300) = √5880 = 76.7 m/s. Converting to km/hr, v = 76.7 x 3600 / 1000 = 276 km/hr. A 50 gram piece of cake is dragged across a 2 meter long strip of sandpaper, losing about half of its mass by the time it reaches the end of the strip. Only just enough force is applied to the cake to keep it moving. How much work was done in moving the cake? (Take g = 10 m/s2 and the coefficient of friction µ for cake on sandpaper to be 0.8.) (a) 60 J (b) 0.3 J (c) 0.6 J (d) 0.4 J (e) 0.8 J The average weight of the cake is (mg + mg /2) / 2 = 3mg /4. Hence, the average frictional force is just f = µ(3mg /4). Multiplying the average force to obtain the work gives W = fD = µ(3mg /4)D = 0.8 • 0.75 • 0.050 • 10 • 2 = 0.60 J. The potential energy experienced by a ball at the bottom of a particular wine glass is ax4, where a = 1 J/m4. What force is experienced by the ball at x = -2 m? (a) 16 N (b) 32 N (c) -32 N (d) -16 N (e) none of (a) - (d) The relation between force and potential is F = - dU / dx. In this problem, F = - d(ax4) / dx, or F = -4ax3. At x = -2, then F = - 4 • 1 • (-2)3 = + 32 kg-m/s2. A particle is subject to a conservative force that does not depend on distance x. How does its potential energy change with x? (a) 1/x2 (b) x (c) 1/x (d) constant (e) x2 Starting with the definition of work W = F • Δx, the work must be proportional to x if F is constant. The work done on a system raises its potential energy, so we expect ΔU ~ x. Using a cable, an engine applies a constant force to move a mass m across a frictionless table in time t, starting from rest. The average power generated by the Linear motion topics 11 engine during this process is P. If the same engine delivered the same force to a mass 2m in moving it across the table, what would be the average power expended by the engine (in terms of the original P)? (a) P (b) √2 P (c) P / 2 (d)P / √2 (e) 2P The force and the distance, and hence the work done by the engine, is the same in both situations. The average power then depends upon the time t required for the motion, via d = at2/2 --> t = (2d /a)1/2. Since the force is constant, the acceleration experienced by 2m is half of that experienced by m, and hence t2m = √2 tm. Therefore, P2m =Pm / √2. The power required to keep a car moving at a constant speed v against turbulent drag is P. What is the power required to move the car at 2v? (a) P (b) 2P(c) 4P (d) 8P (e) none of (a) - (d) The drag force in the presence of turbulence grows quadratically with the speed Fdrag ~ v2. But power is given by P = F•v, so the power grows like v3. Thus, if the velocity is doubled, the power increases by 23 = 8. What is the exponent n in the expression [power] ∝ [speed]n for a drag force that varies with speed as Fdrag ∝ v 2? (a) 0 (b) 1 (c) 2 (d) 3 (e) 4 The power P associated with a force F acting on an object moving with velocity v is P = F•v. In this problem, F ∝ v 2, so P ∝ v 2•v ∝ v 3. Four squares of equal mass and dimension L x L are arranged as shown. What is the y-component of their centre-of-mass? y L L x (a) L/2 (b) 3L/2 (c) L (d) 3L/4 (e) L/4 The centre-of-mass of each block lies at its centre, so the cm of the whole system is y = (1/4M) • {3 • (1/2) ML + (3/2)ML} = (L/4) • (3/2 + 3/2) = 3/4 L. Linear motion topics 12 2D A uniform rectangular sheet of height 4 D and width 2 D has a square hole cut in it, as in the diagram. Find the height of the centre-of-mass position of this object, as measured from the bottom of the sheet. 4D (a) (5/3)D (b) 3D (c) (7/3)D (d) 2D (e) none of (a) - (d) Consider the problem as a sheet of positive mass plus a hole of negative mass. The sheet has a cm position of 2D, and the hole has 3D. Thus, the position of the cm in the z-direction is r = [8D2 • 2D - (√2 D)2•3D ] / [ 8D2 - (√2 D)2 ] = [16D3 - 6D3] / [ 8D2 - 2D2] = (10/6) D = (5/3) D. Two objects have the same mass m and velocities (v,0) and (0,v). What is the magnitude of their centre-of-mass velocity? (a) v/2 (b) 0 (c) √2v (d) 2v (e) v/√2 Since the total momentum Ptot is equal to Σimivi, then Vcm = Ptot / Mtot = Mtot-1 Σimivi. With Mtot = 2m, the centre-of-mass velocity is 1/2 (v, v). The magnitude of this vector must be 1/2 • (v2 + v2)1/2 = v / √2. A car of mass m travels at a speed v towards a stationary truck of mass 3m. What is the speed of the centre-of-mass of the system? (a) v (b) v/2 (c) 3v/4 (d) v/4 (e) v/3 The centre-of-mass velocity vcm is given by vcm = mtot-1 Σimivi = (4m)-1 (mv + 3m•0) = v/4. Two cars leave an intersection at the same time, with the same speed v. Car A heads due north, and car B heads due east. What is the relative velocity of car B as seen from car A? (a) √2 v, south-east (b) √2 v, north-east (c)v, east (d) 2v, south-east (e) 2v, north-east The velocity vectors for cars A and B are (0, v) and (v, 0) respectively, for x=east and y=north. Correcting for the motion of car A, the velocity of B is (v, 0) - (0,v) = (v, -v), which has a magnitude of √2 v and points to the south-east. An object of mass m, moving on a frictionless surface, collides with, and sticks to, a second object of mass m, initially at rest. Together, they recoil with a total kinetic Linear motion topics 13 energy K. If the stationary object had a mass of 2m, instead of m, what would be the kinetic energy of the pair? (a) K/3 (b) 2K (c) 3K/2 (d) 2K/3 (e) K/2 By conservation of momentum, the momentum of the pair of objects is p, and this does not depend on the mass of the struck object. After the collision, the kinetic energy K of the first case is K = p2/2(2m) = p2/4m. In the second case, the kinetic energy K2 is K2 = p2/2(3m) = p2/6m. Thus, K2/K = 4/6 = 2/3. An object of mass m, moving on a frictionless surface, collides with, and sticks to, a second object of mass m, initially at rest. Together, they recoil with a total kinetic energy K. If the stationary object had a mass of 2m, instead of m, what would be the kinetic energy of the pair? (a) K/3 (b) 2K (c) 3K/2 (d) 2K/3 (e) K/2 By conservation of momentum, the momentum of the pair of objects is p, and this does not depend on the mass of the struck object. After the collision, the kinetic energy K of the first case is K = p 2 /2(2m) = p 2 /4m. In the second case, the kinetic energy K2 is K2 = p 2 /2(3m) = p 2 /6m. Thus, K2/K = 4/6 = 2/3. Object A with mass m is travelling in the positive x direction when it collides inelastically with a body of mass 2m and comes to a complete stop. If object A has an initial velocity v, what is the kinetic energy of object B? (a) mv2 /2 (b) mv2 /4 (c) mv2 (d) 0 (e) 2mv2 Initial momentum = mv = final momentum = pB Kinetic energy of B is therefore K = pB2 / 2mB = (mv)2 /4m = mv2 /4. Two masses M and 5M are at rest on a frictionless horizontal table with a compressed (massless) spring between them. When the spring and masses are released, what share of the spring's potential energy is carried off as kinetic energy by mass M? (a) 4/5 (b) 1/5 (c) 1/6 (d) 1/2 (e) 5/6 Each mass recoils with the same momentum p. Hence, the kinetic energies of the two masses are p2/ 2M and p2/ 10M, respectively, for a total of 6p2/ 10M = 3p2/ 5M. The fraction of this energy carried by mass M is (p2/ 2M) / (3p2/ 5M) = 5/6. Two identical masses are hung from massless strings of the same length. One mass is released from a height h above its free-hanging position, and strikes the second mass which is initially at rest. The two masses stick and move off together. To what height H Linear motion topics 14 do they rise? h H (a) 3h/4 (b) h/4 (c) h (d) h/2 (e) none of (a)-(d) Before collision, equate energies to find momentum p p2 / 2m = mgh --> p2 = 2m2gh. After the collision, the momentum of the pair is still p, but the height is found from p2 / 2(2m) = 2mgH H = p2 / (8m2g) = (2m2gh) / (8m2g) = h / 4. A steady stream of 0.02 kg balls roll off a table and bounce elastically on a weigh scale as shown in the diagram. If the balls drop through a height of 0.1 m in reaching the scale, and they arrive at 0.5 s intervals, what is the average force measured by the scales (in kg-m/s2)? 0.1 m (a) 5.6 (b) 1.4 (c) 0.056 (d) 0.028 (e) 0.11 The kinetic energy of each ball as it hits the scales is mgh, from which the velocity can be found from (1/2) mv2 = mgh or v2 = 2gh or v = (2 • 9.8 • 0.1)1/2 = 1.4 m/s. Thus, the momentum of the ball is mv = 0.02 • 1.4 = 0.028 kg-m/s. The change in momentum at the scale is double the incident momentum, or 0.056 kg-m/s. Since this change occurs every 0.5 seconds, then the force (rate of change of momentum) is 0.056 / 0.5 = 0.11 N. Linear motion topics 15 Linear motion - problems A student driving up Gaglardi way at 72 km/hr is late for class. When he is 100 m from the Curtis/Gagliardi stoplight, the light turns yellow. He hits the gas and accelerates at 3 m/s2. But after 2 seconds, he decides that he can't make the light in time and jams on the brakes. He decelerates uniformly, coming to a stop at the light. What is his deceleration? During acceleration vi = 72 km/hr = 72 x 103 / 3.6 x 103 = 20 m/s a = 3 m/s t=2s The distance he covers during acceleration is d = vit + at2/2 = 20 • 2 + 0.5 • 3 • 22 = 46 m and his final velocity is vf = vi + at = 20 + 3 • 2 = 26 m/s. During deceleration the distance he covers is 100 - 46 = 54 m His deceleration is therefore a = (vf2 - vi2)/2a = (0 - 262) / (2 • 54) = -6.3 m/s2. Linear motion topics 16 The geometry of the "shoot-the-can" demonstration done in class is approximately tin can line of sight blow tube 30 o 10 m The projectile is aimed at an angle of 30o with respect to the horizontal, and the horizontal distance between the end of the blow tube and the can is 10 m. How far will the can fall before the projectile strikes it, assuming that the can starts to fall as soon as the projectile leaves the tube? The speed of the projectile when it leaves the tube is 24 m/s (neglect air resistance). The horizontal velocity, vx, of the projectile as it leaves the tube is v = 24 cos 30o = 24 • 0.866 = 20.8 m/s. x Thus, the projectile covers the horizontal distance to the falling can in a time t, given by t = d / vx = 10 / 20.8 = 0.48 s. Since the can starts off with zero velocity, then during time t, the can has fallen a height given by h = -gt2 / 2, where g is the acceleration due to gravity. Thus, h = - 9.8 • (0.48)2 / 2 = - 1.13 m. Linear motion topics 17 The space shuttle orbiter is moving in a circular orbit with a speed of 7 km/s and a period of 80 minutes. In order to return to Earth, the orbiter fires its engines opposite to its direction of motion. The engines provide a deceleration in this direction of 20 m/s2. What is the magnitude and direction of the total acceleration of the orbiter when the engines are first ignited? The centripetal acceleration is given by ac = 2πv / T = 2π • 7x103 / (80 • 60) = 9.16 m/s2 Because ac is perpendicular to the direction of motion, the total acceleration can be found from pythagoras theorem to be atot = (ac2 + atan2)1/2 = (9.162 + 202)1/2 = 22.0 m/s2. The angle of the acceleration vector is away from the direction of motion, in the general direction of the Earth, making an angle with respect to the vertical obtained from tanθ = 20.0 / 9.16 = 2.18 or θ = 65.4o. Linear motion topics 18 An object moves horizontally with a position given as a function of time t by x(t) = at 3 - bt, where a and b are positive constants. (a) What is the velocity and acceleration of the object? (b) What is the initial direction of the motion? (c) At what position (in terms of a and b) will the object reverse direction? For those students who have not yet encountered this result in their calculus class, the slope of the function at n is given by na t n-1. The slope of a polynomial is equal to the sum of the slopes of individual terms. (a) To find velocity, one takes the slope of the position vs time graph. Here, the slope of each contribution gives v(t) = 3at 2 - b. (b) The velocity is initially negative, with v(0) = -b. (c) Solving for the condition v(t) = 0, one has 3at 2 - b = 0 3at 2 = b or t = (b / 3a)1/2 Substituting back into the position equation gives: x(t) = at 3 - bt = (at 2 - b) t = [a•(b /3a) - b] (b / 3a)1/2 = -(2b /3) (b / 3a)1/2. Linear motion topics 19 In a binary star system, two stars with masses m1 and m2 rotate about their common R2 R1 centre of mass. Assume that the orbits are circular, with radii R1 and R2, such that the distance D = R1 + R2 between the stars is constant. (i) Establish that R1 = Gm2T2 / (4π2D2), where T is the period of the orbit. (ii) Find the sum of the masses m1 + m2 in terms of D and T. (i) Each star experiences an acceleration due to gravity of F = GMm / D2. Equating this force to mac, with ac = v2/R, gives mv2/R = GMm / D2 or v2/R = GM / D2 But the velocity of each star is given by v = 2πR / T, so that (2πR1 / T)2/R1 = GM2 / D2so that or 4π2R1 / T2 = GM2 / D2 or R1 = Gm2T2 / (4π2D2). (ii) Using the result from part (i), we can write each mass as m2 = 4π2D2R1 / GT2 which gives a total mass of m1 + m2 = 4π2D2R1 / GT2 + 4π2D2R2 / GT2 = 4π2D3 / GT2. Linear motion topics 20 The viscous drag force exerted by a stationary fluid on a spherical object of radius R is F = 6πηRv at low speed (Stokes' law) F = (ρ/2)ACDv2 at higher speeds, where the symbols are defined in Lec. 9. Let’s apply this to a spherical cell one micron in radius, moving in water with η = 10-3 kg/m•s and ρ = 103 kg/m3. Take the cell to have the same density as water, and let its drag coefficient be 0.5. (a) Plot the two forms of the drag force as a function of cell speed up to 100 µm/s. Quote the forces in pN. (b) Find the speed at which the linear and quadratic drag terms are the same. For the linear drag term, the constants are numerically equal to 6πηR = 6π • 10-3 • 10-6 = 6π • 10-9 kg/s. For the quadratic drag term, we have (ρ/2)ACD = 0.5 • 103 • π 10-12 • 0.5 = (π/4) • 10-9 kg/m. (a) Both expressions vanish when v = 0. By definition, the linear term increases fastest at small v. However, even when v = 100 µm/s, the linear term still dominates: linear drag force = 6π • 10-9 • 10-6 kg-m/s = 6π • 10-15 N = 6π • 10-3 pN = 0.0188 pN. quadratic drag term = (π/4) • 10-9 (10-6)2 = (π/4) • 10-21 N = (π/4) • 10-9 pN = 7.8 • 10-10 pN. (b) Equating the two drag forces gives 6π • 10-9 v = (π/4) • 10-9 v 2 or v = 6π / (π/4) = 24 m/s. Linear motion topics 21 A stone of weight w is thrown vertically upward into the air with an initial speed vo. Suppose that the (constant) air drag force f dissipates an amount fy as the stone travels a distance y. (a) Show that the maximum height reached by the stone is h = vo2 / {2g (1 + f/w)}. (b) Show that the speed of the stone upon impact with the ground is v = vo {(w-f) / (w+f)}1/2. (a) The initial kinetic energy mvo2/2 is converted to gravitational potential energy mgh = wh and lost to friction fh by the time the object reaches a height h: mvo2/2 = wh + fh = (w+f)h. Thus (w/g)vo2/2 = (w+f)h and h = (w / 2g) vo2 / (w+f) = vo2 / {2g (1 + f/w)}. (b) On the trip back down, the potential energy wh is converted to kinetic energy mv2/2 and lost to friction fh: wh = fh + mv2/2 = fh + wv2 / 2g Thus wv2 / 2g = (w-f)h = (w-f) vo2 / {2g (1 + f/w)}. Re-arranging v2 = (w-f) vo2 / {w (1 + f/w)} = { (w-f) / (w+f) } vo2 and finally v = vo {(w-f) / (w+f)}1/2. Linear motion topics 22 As shown in the diagram, a box of mass 200 kg is hung from a crane hook using a rope that can bear a tension of 2,000 N before it breaks. What is the minimum value of θ required to avoid breaking the rope when the box is lifted? Draw a free-body diagram of the box, and take the acceleration due to gravity to be 10 m/s2. ! ! Let the tension in the rope be T. The free-body diagram for the box is T ! T ! mg In the vertical direction, the forces are balanced by mg = 2T sinθ, where mg = 10 x 200 = 2000 N. If the tension is less than 2,000 N, then sinθ > mg / 2T = 2000 / 4,000 = 0.5. The angle whose sine is 0.5 is 30 degrees. Linear motion topics 23 A spherical ball of mass m sits in a tilted box as shown, where θ is the angle of tilt. A B ! There are two contact points with the box, labelled A and B. In terms of mg, what is the magnitude of the force on the ball at point A? First, we observe that contact forces are perpendicular to the walls of the box. Resolving the components of the forces at each contact point: Vertical: Horizontal fA sinθ + fB cosθ = mg fA cosθ = fB sinθ Solve (2) for fB fB = fA cosθ / sinθ and substitute into (1) fA sinθ + fA cosθ • cosθ / sinθ = mg fA (sin2θ + cos2θ) / sinθ = mg By the trigonometric identity sin2θ + cos2θ = 1 we are left with fA = mg sinθ. (1) (2) (3) Linear motion topics 24 A worker applies a horizontal force of 550 N to a 110 kg box which is at rest on a carpet. The coefficients of static and kinetic friction between the box and the carpet are 0.6 and 0,4, respectively. What is the frictional force exerted by the carpet on the box (use g = 10 m/s2)? The maximum force that can be generated by static friction is fmax = µsN where N is the normal force mg. In this problem, m = 110 kg, so fmax = 0.6 • 110 • 10 = 660 N. Since fmax is less than the applied force, the box does not move and the frictional force is equal to 550 N. A 10 kg block slides down an incline plane having an angle of 50o with respect to the horizontal. If the coefficient of friction between the block and the plane is 0.3, what is the acceleration of the block (in m/s2)? (9 marks, 00-1M) 50 o N fmax = µN mg 50 o Balancing forces perpendicular to the plane gives N = mg cos50o Thus, the maximum frictional force is fmax = µN = µmg cos50o. The in-plane component of mg, which gives rise to the acceleration, is mg sin50o. The acceleration a arises from the net in-plane force: ma = mg sin50o - µmg cos50o or a = g (sin50o - µ cos50o). Substituting the numerical values for this situation gives a = 9.81 (sin50o - 0.3 • cos50o) = 5.6 m/s2. Linear motion topics 25 A horizontal force is applied to a block of mass m sitting on a plane which makes an angle of 60o with respect to the horizontal, as illustrated. The coefficient of static m F 60 o friction between the block and the plane is 1.0. What is the minimum value of F required to stop the block from moving? Draw a free-body diagram and quote your answer in terms of mg. The free-body diagram for the block looks like y x N f = µN F mg Choose a coordinate system aligned with the plane. In the x-direction: f + F cos60o = mg sin60o. (1) In the y-direction N = F sin60o + mg cos60o. (2) Friction equation reads f = N since µ = 1. (3) Substituting (1) and (2) into (3) yields mg sin60o - F cos60o = F sin60o + mg cos60o. Solving for F gives F sin60o + F cos60o = mg sin60o - mg cos60o. F (sin60o + cos60o) = mg (sin60o - cos60o) F = mg (sin60o - cos60o) / (sin60o + cos60o) or finally F = 0.27 mg. Linear motion topics 26 A force F of 100 N is applied to a box of mass 5 kg resting on the floor as shown in the diagram. Both coefficients of friction (static and dynamic) between the box and the floor are 0.95. What is the magnitude and direction of the frictional force between the box and the floor. Applied force 30 o First, resolve the components of the applied force in the vertical and horizontal directions: x-direction: F cos 30o = 100 • 0.866 = 86.6 N y-direction: F sin 30o = 100 • 0.5 = 50 N The normal force in the vertical direction opposes both the weight and the vertical component of F N mg F sin 30 o and thus has the value N = mg + F sin 30o = 5 • 9.8 + 50 = 99 N. The maximum value of the frictional force is µN, which is 0.95 • 99 = 94.1 N. Since this value is larger than the applied force in the x-direction, then the frictional force must be the same as the applied force, 86.6 N. It acts to the left, in the diagram. Linear motion topics 27 A block of mass m is at rest on a horizontal plane, as shown in the diagram, and the coefficient of static friction between the block and the plane is µ. A massless string is attached to the block, making an angle of 45o with respect to the horizontal plane. What minimum force F must be applied through the string in order to move the block horizontally? Draw a free-body diagram of the block, and express F in the simplest expression involving m, µ and g, the acceleration due to gravity. F 45 o µ The free-body diagram of the block is: N F 45 o f mg Resolving the components of the force: y-direction: N + F sin 45 = mg (1) x-direction: f = F cos 45 (2) In addition, the maximum frictional force f is related to the normal force N through f = µN (3) Solving Eq. (1) for N, we find N = mg - F sin 45 (4) Equating (2) and (3), F cos 45 = µN (5) and substituting (4) gives F cos 45 = µ(mg - F sin 45) (6) Solving for F: F cos 45 + µF sin 45 = µmg F = µmg/(µsin 45 + cos 45) F = µmg/(µ/√2 + 1/√2) F = √2µmg/(1+µ). Linear motion topics 28 A thin rope of mass M hangs over the edge of a table. The fraction of its total length L lying on the table experiences a frictional force with the table governed by a coefficient of friction µ. For a given value of µ, what is the minimum fraction f of the rope that can lie on the table without slipping over the edge? Assume that the edge is smooth and does not contribute to the frictional force. The fraction of the rope on the table top has a mass of mtop = fM and exerts a normal force N = fMg on the table. This results in a frictional force Ffriction = µfMg on the upper part of the rope. The friction acts against the force of gravity pulling the segment of the rope over the edge: mass of rope over the edge = (1-f)M Fgravity = (1-f)Mg. Thus, the forces are balanced at (1-f)Mg = µfMg, or 1-f = µf or f(1+µ) = 1 which gives f = 1 / (1+µ). Linear motion topics 29 An object moves along the x-axis subject to a potential energy V(x) of the form V(x) = Vo(-x2/2 + x4/4). (i) (9 marks) At what values of x does the object feel no net force? (ii) (6 marks) Suppose that the object is displaced very slightly from the position(s) that you calculated in part (i). Describe the motion of the object. (i) Force is related to potential energy by F = -dV(x) / dx. If the force vanishes, then dV(x) / dx = 0. Here, dV(x) / dx = Vo d( -x2/2 + x4/4 ) / dx = Vo (-x + x3). This expression vanishes when (-x + x3) = x(x2 - 1) = 0 or x=0 x = +/- 1. (ii) The potential energy at the zero force positions is V(0) = 0 V(+/- 1) = Vo(-1/2 + 1/4) = -Vo/4. V is a minimum at x = +/- 1, so that displacements from equilibrium will just oscillate. V is not a minimum at x = 0, so the solution is unstable. Linear motion topics 30 A small block of mass m slides without friction along a loop-the-loop track of radius R, after being released from an initial height of 2R. To what height h inside the loop will 2R R the block slide before it loses contact with the track? The initial potential energy is 2mgR. Hence, at any height h within the loop, the kinetic energy is determined by 2mgR = mv2/2 + mgh or v2 = 2[2gR - gh] or v2 = 2g[2R - h]. The centripetal acceleration ac experienced by the block is then ac = v2 /R = 2g[2 - h/R]. When this acceleration is greater than the component of g in the radial direction, the block remains in contact with the track: ac > gcos ! ! g But cosθ is given by R ! h-R cosθ = (h - R) / R = h/R - 1. Hence, the condition for contact is ac > g cosθ --> 2g[2 - h/R] > g (h/R - 1) 4 - 2h/R > h/R - 1 3h/R > 5 h > (5/3)R. Linear motion topics 31 A planet of mass m executes a circular orbit of radius R about a very heavy star of mass M. (a) In terms of M, R and the gravitational constant G, what is the speed vo of the planet in its orbit? (b) What is the total energy of the planet? (c) In terms of vo, what must its speed vesc be to escape from the orbit to infinity? (a) The speed can be obtained from the force balance between the gravitational attraction GmM/R2 and the centripetal force mv 2/R: mv o2 /R = GmM/R2 or v o 2 = GM/R v o = (GM/R)1/2. (b) The total energy is the sum of its potential and kinetic parts, which have opposite signs here: E = -GmM/R + mv o 2/2 = -GmM/R + GmM/ 2R = -GmM/ 2R. (c) For the planet to escape, it must overcome the potential energy binding it at radius R; that is mvesc2/2 = GmM/R or v esc 2 = 2GM/R = 2vo2. Thus, vesc = √2 vo. Linear motion topics 32 An object which is restricted to moving horizontally, is subject to a quadratic drag force in the form F = c2v2. (a) Find the power needed to maintain an object at constant speed in the presence of this force. (b) Find the power (in horsepower) needed to overcome this drag force for a car travelling at 100 km/hr if its cross-sectional area is 2 m2 and drag coefficient is 0.5. Your answer in (b) does not include internal friction, tire friction etc. (a) At a constant speed, the drag force is constant and the resulting power is just P = Fv = c2v3. (b) To do this problem, we need the density of air at ρ = 1.29 kg/m3 and v = 100 • 103 / 3600 = 27.8 m/s. Thus, P = (ρ /2) A CD v 3 = 0.5 • 1.29 • 2 • 0.5 • (27.8)3 = 13.9 x 103 W = 13.9 x 103 / 746 = 19 hp. Show that the terminal speed of a falling spherical object is given by vterm = [(mg/c2) + (c1/2c2)2 ]1/2 - (c1/2c2) when both the linear and quadratic terms in the drag force are taken into account. This question is very easy. Balancing the gravitational force mg against the drag force c1v + c2v2 gives the terminal velocity. Thus c1v + c2v2 = mg or c2v2 + c1v - mg = 0, which has the usual quadratic solution v = {-c1 ±[c12 + 4c2mg]1/2} / 2c2. This can be rearranged to give vterm = [(mg/c2) + (c1/2c2)2 ]1/2 - (c1/2c2). Linear motion topics 33 Consider three different power-law forms of the drag force with magnitudes: F1/2 = a v 1/2 (square root) F1 = b v 1 (linear) 3/2 F3/2 = c v (3/2 power). Travelling horizontally from an initial speed vo, an object subjected to one of these drag forces would come to rest at xmax = (2m /3a)vo3/2 (square root) xmax = mvo / b (linear) xmax = 2m vo1/2/c (3/2 power). (a) Determine the coefficents a, b and c for a cell of mass 10-14 kg whose drag force is measured to be 5 pN when travelling at 10 microns/second. (b) Find the maximum displacement that the cell could reach for each force if vo = 1 µm/s. (c) Why should xmax in part (b) have the order (shortest to longest) that it does? (a) Calculate the unknown constants for each of the forces, including the units appropriate for each: square root 5 • 10-12 = a (10-5)1/2 a = 1.58 • 10-9 N(s/m)1/2 linear 5 • 10-12 = b 10-5 b = 5 • 10-7 N•s/m 3/2 power 5 • 10-12 = c (10-5)3/2 c = 1.58 • 10-4 N(s/m)3/2. (b) Calculating the maximum distances travelling is now simple: square root xmax = (2m /3a)vo3/2 xmax = (2x10-14 / [3 • 1.5x10-9]) (10-6)3/2 = 4.2x10-15m = 4.2x10-5 Å linear 3/2 power xmax = mvo / b = 10-14 • 10-6 / 5x10-7 = 2x10-14 m = 2x10-4 Å xmax = 2m vo1/2/c xmax = 2 10-14 (10-6)1/2 / 1.58x10-4= 1.26x10-13 m = 1.26x10-3 Å. [kg / N(s/m)]•[m/s]3/2 = m kg (m/s) / (N•s/m) = m kg (m/s)1/2 / [N (s/m)3/2] = m Of the three forces, the square root force rises fastest with v. Conversely, it dies more slowly that the other forces as the cell slows down. Thus, it has a higher average value Linear motion topics 34 than the other forces, and brings the cell to rest faster, explaining why xmax is smallest for the square root force, and largest for the 3/2-power force. An object of mass m moves in a circular path of radius R on a horizontal plane. It is subject to turbulent drag. At what range or values of R and instantaneous speed v does it experience an acceleration which makes an angle of 45o with respect to a tangent to the circle? The object experiences a centripetal force mac perpendicular to its direction of motion, and a drag force opposite to its direction of motion: motion drag ac For the acceleration to make an angle of 45o with respect to the direction of motion, the drag force and the centripetal force must have the same magnitude. Thus, in magnitude, FDRAG = FCEN. But FDRAG = c2v2 and FCEN = mac = mv 2/R so c2v2 = mv 2/R or R = m / c2. There is no constraint on the velocity for this condition to occur, only on the radius. Linear motion topics 35 An object of mass mass m is attached to one end of a massless string of length L. The top end of the string is attached to the ceiling. Keeping the string straight, the object is drawn to the side and released from rest at an angle θo with respect to the vertical. As a function of the angle θ (with respect to the vertical), what is the tension in the string once the mass begins to move? First, find the kinetic energy of the object by calculating the decrease in the potential energy. The height difference is Δh = L (cosθ – cosθo) show a diagram so that ΔU = -mgL (cosθ – cosθo) By conservation of energy ΔK = -ΔU = mgL (cosθ – cosθo). But the change in kinetic energy is just mv 2/2 because the object is released from rest. Thus mv 2/2 = mgL (cosθ – cosθo) or v 2/L = 2g (cosθ – cosθo). Because the motion is in a circle, the centripetal acceleration is ac = v 2/L = 2g (cosθ – cosθo) and the centripetal force is mac = 2mg (cosθ – cosθo). The tension in the string provides the centripetal force as well as balancing the component of the gravitational force mg cosθ. Thus, the total force provided by the string is T = 2mg (cosθ – cosθo) + mg cosθ = mg (3cosθ – 2cosθo). Linear motion topics 36 An ideal massless spring hangs motionless, its top end attached to a rigid support as in one of the class demonstrations. The spring has a spring constant k and the bottom end of the unstretched spring, which is free, defines z = 0 of a coordinate system. You attach a mass m to the bottom end and carefully lower it with your hand until it reaches a new equilibrium position -h, where h is a positive number. In terms of m, g, and k, what non-conservative work, if any, was done by the system in this process? When the mass is in equilibrium, the net force vanishes. The forces are: F = kx from the spring F = mg from gravity and kh mg Thus kh = mg or h = mg/k. The non-conservative work is governed by Wnon-con = Δ(K + U) In this process, there is no change in kinetic energy, so ΔK = 0. However, the change in potential energy is ΔUspring = +kh 2/2 = +k (mg/k)2 / 2 = +(mg)2 / 2k. and ΔUgrav = -mgh = -mg (mg/k) = - (mg)2 / k. Thus, ΔUtotal = ΔUspring + ΔUgrav = +(mg)2 / 2k - (mg)2 / k = - (mg)2 / 2k. The non-conservative work is then Wnon-con = - (mg)2 / 2k. Linear motion topics 37 A bullet travelling at 400 m/s is shot horizontally at a sand-filled tin sitting on a fence 2 m high. The bullet stops inside the can, causing it to fall off the fence. (a) What is the can's velocity just as it topples off the fence? (b) How far from the bottom of the fence does it hit the ground? The masses of the bullet and the can are mB = 10 g and mC = 2 kg, respectively. You can ignore the spatial size of the can compared to the height of the fence. (a) Solve the bullet-can collision problem using conservation of momentum: pB + pC = pB+C mBvB + mCvC = mB+CvB+C 0.01 • 400 + 0 = 2.01vB+C or vB+C = 4.00 / 2.01 = 2.0 m/s. (b) It takes time T for the can to hit the ground. In the vertical (y) direction, the equation of motion reads: d = viT + aT2/2. Here, vi = 0, d = -2 m, a = -g = -9.81 m/s2. Hence, -2 = 0 - 9.8T2/2 ---> T = √(2 x 2 / 9.8) = 0.64 s. There is no acceleration in the horizontal (x) direction, so x = vT = 2 x 0.64 = 1.28 m. Linear motion topics 38 An object of mass m moves on a frictionless surface with a speed v in the +x direction. It explodes into two fragments of equal mass, one of which travels in the +x direction with speed 3v. What kinetic energy is liberated in the explosion? (8 marks) Strategy: apply conservation of momentum to determine the velocity of the second fragment. initial momentum = mv momentum of forward particle = (m/2)•3v = (3/2)mv By conservation of momentum, the velocity of the second fragment must be (m/2)v2 = mv - (3/2)mv = -mv/2 Thus v2 = -v. To find the kinetic energy released, we compare: initial kinetic energy = mv2/2 kinetic energy of fragment #1 = (1/2)•(m/2)•(3v)2 = (9/4)mv2 kinetic energy of fragment #2 = (1/2)•(m/2)•(v)2 = (1/4)mv2 final kinetic energy = {9/4 + 1/4}mv2 = 5/2 mv2 Thus, the kinetic energy released is ΔK = {5/2 - 1/2}mv2 = 2 mv2. Linear motion topics 39 (a) Find the centre-of-mass position of the Earth-Jupiter-Sun system in the two configurations shown in the diagram. Choose the centre of the Sun as the coordinate system origin. The centres of the objects lie on a straight line. (b) Comparing configurations I and II, how much does the position of the Sun change relative to the centre-of-mass of each configuration? Config I E ------- S ------- J Config 2 J ------- E -------- S 24 Masses: Earth = 5.98 x 10 kg Jupiter = 1.90 x 1027 kg Sun = 1.99 x 1030 kg Distances: Earth-Sun = 1.50 x 108 km Jupiter-Sun = 7.78 x 108 km The total mass of the system is essentially the mass of the Sun: mTOT = 5.98 x 1024 + 1.90 x 1027 + 1.99 x 1030 kg = 1.99 x 1030 kg. In configuration I, Σimiri = -(5.98 x 1024 x 1.50 x 108) + 0 + (1.90 x 1027 x 7.78 x 108) = -8.97 x 1032 + 1.48 x 1036 = 1.48 x 1036 kg•km. Hence, xcm = 1.48 x 1036 / 1.99 x 1030 = 7.42 x 105 km. In configuration II, Σimiri = -8.97 x 1032 - 1.48 x 1036 = -1.48 x 1036 kg•km. Hence, xcm = -1.48 x 1036 / 1.99 x 1030 = -7.42 x 105 km. (b) The total shift in the Sun's position is 1.48 x 106 km. Linear motion topics 40 A physics major skating on the surface of a frozen pond decides to use a flashlight to accelerate himself. The flashlight radiates a beam of light at 2 watts, with an average wavelength of 600 nm. (a) How many photons leave the flashlight per second? (b) How much momentum is carried by the photons per second? (c) If the mass of the skater is 70 kg, what is his acceleration? (a) The energy of a single photon is: E = hc / λ = 6.63 x 10-34 x 3 x 108 / 600 x 10-9 = 3.32 x 10-19 J. The number of photons leaving per second in 2 watts is 2 J / 3.32 x 10-19 J = 6.0 x 1018 sec-1. (b) The momentum of each photon is p = h / λ or E / c. Single photon momentum p = 6.63 x 10-34 / 600 x 10-9 = 1.05 x 10-27 kg-m/s. Total momentum radiated per second is 1.05 x 10-27 • 6.0 x 1018 = 6.66 x 10-9 kg-m/s. (Shorter way to do this is divide the power by the speed of light c). (c) Use conservation of momentum to calculate the change in momentum of the skater: a = F / m = (Δp/Δt)/m = 6.66 x 10-9 / 70 = 9.5 x 10-11 m/s2. Linear motion topics 41 Rockets burn up their fuel very rapidly as they accelerate, so that the total mass of a rocket decreases with height above the ground. In this problem, we study a particular rocket with an initial mass of 1.000 x 106 kg, which losses 1.0 x 105 kg in mass during the first 50 m of its flight. (a) Assuming that the loss in mass is uniform with height (i.e. so many kg per meter) find a numerical expression for the mass m of the rocket as a function of its height h for the first 50 m. (b) What is the work done by the rocket engines during the first 50 m of flight? Use g = 10 m/s2. (a) The mass as a function of height looks like 1.0 • 106 kg 0.9 • 106 kg m h 50 m The function is linear, of the form m = 1.00 x 106 - (h / 50) x 105 = (1.000 - 0.002h) x 106 kg. (b) The force due to gravity on the rocket is F = mg. The work done under the force vs distance curve is thus F W = (1/2) • 105 • 10 • 50 = 25 • 106 J. W = 0.9 • 106 • 10 • 50 = 450 • 106 J. 50 m Total work = 475 x 106 J. Linear motion topics 42 A uniform rectangular sheet of mass M has dimensions L x 2L. The coordinate origin is chosen to lie at the geometrical centre of the rectangle. A hole of radius L/2 is subsequently drilled in the block, the centre of the hole having coordinates (L/2, 0). What is the x-coordinate of the centre-of-mass position of the block with the hole in it? L 2L First, concentrate on the x-coordinate. We need to know the total mass after the hole is drilled: density = M / 2L2 Hence: mafter = density • area = (M / 2L2) • (2L2 - π[L/2]2) = M (1 - π/8). mhole = (M / 2L2) • π(L/2)2 = π/8 M. The position of the centre of mass is given by Rcm = (msheetRsheet - mholeRhole) / mafter = (0 - [π/8] M [L/2]) / [M (1 - π/8)] = - (πL/16) / (1 - π/8) = πL / (16 - 2π) = -0.323L. Linear motion topics 43 A boat moving straight towards shore plows into a soft sandy beach, coming to a stop in a distance L. As it moves through the sand, it experiences a coefficient of kinetic friction that grows with distance x as µk = µo x / L. Find the work done by friction as a function of distance x. Take the normal force on the boat to be mg, where m is the boat's mass. L The frictional force is distance-dependent, and given by F = µk mg = (µomg /L) x. The work done on the boat is distance-dependent as well W = Fx but is not linear because the force is not constant. To evaluate the work, we look at F µomgx /L x As in the spring problem, the work is equal to the area under the curve, which is [area] = 1/2 [base] [height] = 1/2 x (µomg /L) x or W = (µomg /L) x 2 / 2. Linear motion topics 44 A cannonball of mass 5 kg is fired into the air at an angle of 60o with an initial velocity of 500.0 m/s. Exactly ten seconds later, it reaches its maximum height of 400 m at point A with a velocity of 100.0 m/s to the right. (a) Find the average acceleration of the cannonball in the first ten seconds. (b) Find the average power exerted by air resistance in the first ten seconds. For numerical simplicity, use g = 10 m/s2. A y vA vo 60o x (a) Acceleration is a vector and its average value must be determined component by component. v ! v i ,x v ! v i ,y ax = f ,x ay = f ,y "t "t 0 100.0 ! 500.0 cos 60 0.0 ! 500.0 sin60 0 = = 10 10 100.0 ! 250.0 !433.0 = = !15.0 m / s 2 = = !43.3 m / s 2 10 10 The magnitude of the average acceleration is then (15.02 + 43.32)1/2 = 45.8 m/s2. The angle of the acceleration vector with respect to the horizontal is tanθ = ay/ax = 43.3 / 15.0 = 2.89. Thus θ = arctan(2.89) = 71o. (b) The first challenge is to determine the change in the kinetic and potential energies. ΔK = mvf2/2 - mvi2/2 = (5 / 2) • (1002 - 5002) = -600,000 J and ΔU = mg Δh = 5 • 10 • 400 = +20,000 J. Thus, the change in the total mechanical energy is ΔE = ΔK + ΔU = -600,000 + 20,000 = -580,000 J. As expected, ΔE is negative because of dissipation. The average power exerted by the dissipative force is P = ΔE / Δt, which becomes P = 580,000 / 10 = 58,000 watts. Linear motion topics 45 As shown in the figure below, the two blocks have identical masses m. Block #1 experiences a sliding friction force with coefficient of friction µk. Find the acceleration of the blocks. T m block #2 block #1 m 30o We start by drawing free-body diagrams for both masses. Introducing the acceleration a, the FBD for block #2 is T m mg The weight of the mass is only partly compensated by the vertical tension,leaving a net force that accelerates the mass: mg = T + ma. (1) The force experienced by block #2 is more complex, because of the presence of friction, and the component of the weight: N = mg cosθ T mg f = µN mg sinθ Here, the tension is opposed by the frictional force as well as mgsinθ, leaving an unbalanced force that provides the acceleration. That is T = ma + f + mgsinθ = ma + µmgcosθ + mgsinθ Linear motion topics = m (a + µgcosθ + gsinθ) 46 (2) Substituting (2) into (1) gives mg = m (a + µgcosθ + gsinθ) + ma or g = 2a + µgcosθ + gsinθ whence 2a = g - µgcosθ - gsinθ or a = (g /2) • ( 1 - µcosθ - sinθ ). (3) Linear motion topics 47 Three objects of mass m, 2m and 3m are glued together in the shape of the letter L: a y m 3m a 2m x Each block is a square of length a to the side. Find the position of the centre-of-mass of the objects with respect to the coordinate system shown in the diagram. (8 marks) Each object acts as if its mass were concentrated at its centre. (mandatory) Thus, we can regard the system as three point masses, where the coordinates of the masses are: 3m at (a/2, a/2) 2m at (3a/2, a/2) m at (3a/2, 3a/2). From the definition of the centre of mass Rcm = Σimiri / mtotal, (mandatory) we determine the coordinates to be xcm = [(3m • a/2) + (2m • 3a/2) + (m • 3a/2)] / 6m = [ 3ma/2 + 3ma + 3ma/2 ] / 6m = 6ma / 6m =a and ycm = [(3m • a/2) + (2m • a/2) + (m • 3a/2)] / 6m = [ 3ma/2 + ma + 3ma/2 ] / 6m = 4ma / 6m = (2/3)a. Linear motion topics 48 In the diagram below, a ball hangs from the ceiling, and is held in a steady position by a horizontal force F of 8 N. The length of the string is 0.5 meter, and the angle θ is 30o. (i) Find the magnitude of the tension T in the string. (ii) If the ball is released (F = 0), what will be its speed when it reaches the bottom? Set the acceleration due to gravity equal to 10 m/s2, for numerical simplicity. θ T F m (a) The free-body diagram of the ball is θ T F mg In the z-direction, the force vectors must obey F = T sinθ = T sin30o = T /2. Thus T = 2F = 16 N. (b) The change in height experienced by the ball as it falls is Δh = 0.5 ( 1 - cos30o) = 0.067 m. 0.5 cos30o 0.5 m -> 0.5 - 0.5 cos30o = 0.5 (1 - cos30o) By conservation of mechanical energy, the increase in the kinetic energy must equal the decrease in the potential energy, hence (mandatory) mv 2 /2 - [initial K] = mgΔh. But the initial kinetic energy is zero, so v 2 /2 = g Δh, or v = [2g Δh ]1/2 = [2 • 10 • 0.067]1/2 = 1.16 m/s.