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A humbling fraction of physics boils down to direct application of simple harmonic motion, the description of oscillating objects. Learn the basis for springs, strings, and quantum fields.

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 \( g = 9.8 \text{ m/s}^2. \)

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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 \( g = 9.80 \text{ m/s}^2. \)

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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 pendulum’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.

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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.)

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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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