Classical Mechanics
# Simple Harmonic Motion

When a ball with mass $1\text{ kg}$ is hung on a $50\text{ cm}$ long spring, the length of the spring is increased by $10\text{ cm}$ in steady state, as shown in the far left diagram. Now, suppose that we pull this ball further down and let it go. Then it will show simple harmonic oscillations. What will be the periods of these oscillations when we pull down the ball by $10$ cm and $20$ cm, respectively, as shown in the middle and right diagrams above?

Gravitational acceleration is $10 \text{ m/s}^2$.

What is the ratio of the period of a pendulum on Mars to the period of the same pendulum on Earth?

Details and assumptions

Gravity on Earth is $9.8~m/s^{2}$

Gravity on Mars is $3.71~m/s^{2}$

$T=2\pi \sqrt{\frac{am}{bk}}$ for coprime positive integers $a,b$.

If the time period of simple harmonic motion of the rectangular block can be represented asFind $\sqrt{a^2+b^2}$ up to two decimal places.

**Details and Assumptions**

- Pulley is smooth and massless, and the strings are light and inextensible.
- Gravity is present.
- The crooked lines in the figure represents springs of spring constant $k$ and $2k$ as shown.

-Take the amplitude of oscillations to be very small.