Physics 1D03 - Lecture 341 Simple Harmonic Motion ( V ) Circular Motion The Simple Pendulum

Physics 1D03 - Lecture 342 SHM and Circular Motion  A Uniform circular motion about in the xy plane, radius A, angular velocity  :  (t) =  0 +  t (similar to, x=x o +vt) and so 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 343 Compare with our expression for 1-D SHM. Result: SHM is the 1-D projection of uniform circular motion.

Physics 1D03 - Lecture 344 Phase Constant, θ o  A  For circular motion, the phase constant is just the angle at which the motion started.

Physics 1D03 - Lecture 345 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 346 Simple Pendulum Gravity is the “restoring force” taking the place of the “spring” in our block/spring system. Instead of x, measure the displacement as the arc length s along the circular path. Write down the tangential component of F=ma: L θ T mg sin θ Restoring force s mg

Physics 1D03 - Lecture 347 Simple Pendulum L θ T mg Using sin(θ)~θ for small angles, we have the following equation of motion: Which give us: Hence: Application - measuring height - finding variations in g → underground resources or:

Physics 1D03 - Lecture 348 Simple pendulum: SHM: The pendulum is not a simple harmonic oscillator! However, take small oscillations: (radians) if  is small. Then Actually:

Physics 1D03 - Lecture 349 This looks like For small  :, with angle  instead of x. The pendulum oscillates in SHM with an angular frequency and the position is given by amplitude phase constant (2  / period)

Physics 1D03 - Lecture 3410 a)longer b)shorter 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 (and by how much, in percent?)

Physics 1D03 - Lecture 3411 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 3412 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 3413 t x x t 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. SHM and Damping – EXTRA !!!

Physics 1D03 - Lecture 3414 For weak damping (small b), the solution is: f =  bv where b is a constant damping coefficient x t A damped oscillator has external nonconservative force(s) acting on the system. A common example is a force that is proportional to the velocity. eg: green water A e -(b/2m)t F=ma give: