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Simple Harmonic Motion (V) Circular Motion The Simple Pendulum Physics 1D03 - Lecture 34 1 SHM and Circular Motion Uniform circular motion about in the xy plane, radius A, angular velocity : (t) = 0 + t (similar to, x=xo+vt) A and so x A cos A cos( 0 t ) y A sin A sin( 0 t ) Hence, a particle moving in one dimension can be expressed as an ‘imaginary’ particle moving in 2D (circle), or vice versa the ‘projection’ of circular motion can be viewed as 1D motion. Physics 1D03 - Lecture 34 2 x A cos A cos(o t ) y A sin A sin( o t ) Compare with our expression for 1-D SHM. x A cos(t ) Result: SHM is the 1-D projection of uniform circular motion. Physics 1D03 - Lecture 34 3 Phase Constant, θo For circular motion, the phase constant is just the angle at which the motion started. A o Physics 1D03 - Lecture 34 4 Example An object is moving in circular motion with an angular frequency of 3π rad/s, and starts with an initial angle of π/6. If the amplitude is 2.0m, what is the objects angular position at t=3sec ? What are the x and y values of the position at this time ? Physics 1D03 - Lecture 34 5 Simple Pendulum Gravity is the “restoring force” taking the place of the “spring” in our block/spring system. L Instead of x, measure the displacement as the arc length s along the circular path. θ T Write down the tangential component of F=ma: Restoring force mg sin d 2s m 2 mat mg sin( ) dt But s L mg s mg sin θ d 2 g 2 sin dt L Physics 1D03 - Lecture 34 6 Simple Pendulum Using sin(θ)~θ for small angles, we have the following equation of motion: L d 2 g 2 dt L Which give us: θ T g L ------------------------------------------------------------------------Hence: or: gT 2 L 2 4 2 g mg 2 4 L 2 g L T2 Application - measuring height - finding variations in g → underground resources Physics 1D03 - Lecture 34 7 Actually: SHM: Simple pendulum: d 2x 2 x 2 dt d 2 g sin 2 dt L The pendulum is not a simple harmonic oscillator! However, take small oscillations: sin (radians) if is small. Then d 2 g g sin 2 dt L L d 2 2 2 dt Physics 1D03 - Lecture 34 8 For small : This looks like d 2 g 2 dt L d 2x 2 x , with angle instead of x. 2 dt The pendulum oscillates in SHM with an angular frequency g L and the position is given by (t ) o cos(t ) phase constant amplitude (2 / period) Physics 1D03 - Lecture 34 9 Question: A geologist is camped on top of a large deposit of nickel ore, in a location where the gravitational field is 0.01% stronger than normal. the period of his pendulum will be a) longer b) shorter (and by how much, in percent?) Physics 1D03 - Lecture 34 10 Application Pendulum clocks (“grandfather clocks”) often have a swinging arm with an adjustable weight. Suppose the arm oscillates with T=1.05sec and you want to adjust it to 1.00sec. Which way do you move the weight? ? Physics 1D03 - Lecture 34 11 Question: A simple pendulum hangs from the ceiling of an elevator. If the elevator accelerates upwards, the period of the pendulum: a) Gets shorter b) Gets larger c) Stays the same Question: What happens to the period of a simple pendulum if the mass m is doubled? Physics 1D03 - Lecture 34 12 SHM and Damping – EXTRA !!! SHM: x(t) = A cos ωt Motion continues indefinitely. Only conservative forces act, so the mechanical energy is constant. Damped oscillator: dissipative forces (friction, air resistance, etc.) remove energy from the oscillator, and the amplitude decreases with time. x t x t Physics 1D03 - Lecture 34 13 A damped oscillator has external nonconservative force(s) acting on the system. A common example is a force that is proportional to the velocity. f = bv where b is a constant damping coefficient F=ma give: dx d 2x kx b m 2 dt dt For weak damping (small b), the solution is: x x(t ) Ae b t 2m eg: green water cos(t ) A e-(b/2m)t t Physics 1D03 - Lecture 34 14