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

$1\textrm{ kg}$ is dropped from a height $H=2\textrm{ m}$. Estimate how long the ball is in contact with the ground (in seconds)?

A ball of mass**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 $9.8\textrm{ m/s}^2$.