Classical Mechanics

Simple Harmonic Motion

Linear restoring force - perturbation analysis

         

As shown in the figure above, a physical pendulum consists of a disc of radius R=5.0 cm R = 5.0 \text{ cm} and mass m=1.0 kg m = 1.0 \text{ kg} fixed at the end of a massless rod. The other end of the rod is pivoted about point P P on the ceiling. The distance from the pivot point to the center of mass of the bob is L=4 m. L =4 \text{ m}. Initially the bob is released from rest from a small angle θ0 \theta_0 with respect to the vertical. Find the period of the bob.

Assumptions and Details

  • The gravitational acceleration is g=9.8 m/s2. g = 9.8 \text{ m/s}^2.

What is the frequency of a simple pendulum with arm length l=2.00 m l = 2.00 \text{ m} that is in an elevator accelerating upward at a rate of a=1.00 m/s2 a = 1.00 \text{ m/s}^2?

Assumptions and Details

  • Assume that the amplitude of the simple pendulum is very small.
  • The gravitational acceleration is g=9.80 m/s2. g = 9.80 \text{ m/s}^2.

A physical pendulum consists of a 6.0 m 6.0 \text{ m} long stick with mass m=100 g m = 100 \text{ g} joined to the ceiling, as shown in the above figure. What is the pendulum’s period of oscillation about point A A at the tip of the stick?

Assumptions and Details

  • The gravitational acceleration is g=9.8 m/s2. g= 9.8 \text{ m/s}^2.
  • Assume that the amplitude of the pendulum is very small.

There is a half-ring of density ρ=4 kg/m \rho = 4 \text{ kg/m} and radius R=2 m R = 2 \text{ m}. When it is perturbed by small angle θ\theta , it oscillates. If the ring's period can be expressed as T=2πab s, T = 2\pi\sqrt{\frac{a}{b}} \text{ s}, where aa and bb are coprime positive integers, what is the value of a+b?a+b?

The gravitational acceleration is g=10 m/s2.g=10\text{ m/s}^2.
Assume that 2=1.5\sqrt{2} = 1.5 and π=3. \pi = 3 . (This is a model of a roly-poly.)

A disk of mass M=1 kg M = 1 \text{ kg} and radius R=4 m R = 4 \text{ m} oscillates on a rod. If the disk's period can be expressed as T=2πab s, T = 2\pi\sqrt{\frac{a}{b}} \text{ s}, where aa and bb are coprime positive integers, what is the value of a+b?a+b?

Assume that θ \theta is very small.
The moment of inertia on the axis is given by I=32MR2. I = \frac{3}{2}MR^2.
The gravitational acceleration is g=10 m/s2.g=10\text{ m/s}^2.

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