Classical Mechanics
# Simple Harmonic Motion

An inventor designs a pendulum clock using a bob with mass 200 g at the end of a thin wire of length 23 cm. Instead of swinging back and forth, the bob is to move in a horizontal circle, making a fixed angle 27° with the vertical. This is called a **conical pendulum** because the suspending wire traces out a cone. Find the period \(T\) of this pendulum.

**Note:** Use \(\pi = 3.1416\) and \(g = 9.81\) m/s\(^2\).

A clock is made out of a disk of radius \(R = 10~\mbox{cm}\) which is hung by a point on its edge and oscillates. All of a sudden, a circular part right next to the hanging point of radius \(\frac{R}{2}\) falls off, but the clock continues oscillating. What is the absolute value of the difference **in s** between the periods of oscillation before and after the part fell off?

**Details and assumptions**

- Gravitational acceleration is \(g = 9.81~\mbox{m/s}^2\)
- Amplitude of the vertical oscillations is small
- The axis of rotation of the disc is horizontal all the time
- The disc is homogeneous

**Details and assumptions**

- Simplify the question by modeling the ball as an ideal spring of spring constant \(100 \textrm{ Nm}|\).
- If necessary, take the gravitational acceleration to be \(9.8\textrm{ m/s}^2\).

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