Classical Mechanics

# Linear restoring force - perturbation analysis

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

Assumptions and Details

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

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

Assumptions and Details

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

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

Assumptions and Details

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

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

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

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

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

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