Classical Mechanics
# Simple Harmonic Motion

Consider the earth as a uniform sphere of mass \(M\) And Radius \(R\). Imagine a straight smooth tunnel made through the earth which connects any two points on its surface (The two points are not diametrically opposite). Determine the time that a particle would take to go from one end to the other through the tunnel. If Your answer is \(T s\), then find the value of \(\frac{T}{100}\) to the nearest integer.

**Details And Assumptions:**

Consider only gravitation force due to earth be acting on the particle during its motion along the tunnel.

Take acceleration due to gravity on surface of Earth = \(10ms^{-2}\)

Radius of Earth = \(6400 km\).

This is an entry for the problem writing party.

A block of mass \(m\) is gently attached to the spring and released at time \(t=0\), when the spring has its free length. During subsequent motion of the block, the displacement of the block \(x\) with respect to time \(t\) is considered. Find \[\displaystyle\int_0^3 x ~\mathrm{d}t\]

**Details and Assumptions**

- Neglect air resistance.
- \(k\) is the spring constant.
- \(m = 10\text{ kg},k = 34 \text{ Nm}^{-1}, g = 9.8 \text{ ms}^{-2}\)

A mass is subjected to a force \[F = (at - bx) \text{ along the } x\text{-axis}\]
Initially the mass lies at the origin at rest. In the definition of force (given) \(x\) refers to the \(x\)-coordinate of the mass and **t** refers to the time elapsed.

Find the x - coordinate of the mass after a time of **4 seconds**.

**Assumptions**

1) **All the values are in SI units.**

2) **Take the mass = 1 kg, a = 1 N/s, b = 1 N/m**

Find the value of \(a+b+c\).

**Details and Assumptions**:

Treat the pulley as a disc.

The block and pulley are oscillating in the same phase with the same frequency.

Pulley has sufficient friction for

*Pure Rolling*and mass=m & strings are light and inextensible.Gravity is present.

Zig-zag lines in fig represents spring of spring constant \(k\) and \(2k\) as shown.

a is a square-free integer and the fraction is in simplest form.

A smooth wedge of mass \(m\) and angle of inclination \(\theta = \SI{60}{\degree}\) is attached to two springs of spring constant \(k_\textrm{L}\) on the left, and \(k_\textrm{R} = 3k_\textrm{L}\) on the right. The wedge rests on a smooth frictionless plane. Find the period of oscillation of the wedge in seconds.

Give your answer to 3 decimal places.

**Details and Assumptions**:

- The springs are perpendicular to the respective sides they are facing.
- The spring on the left is constrained to compress and extend along its length. It is attached to the wedge by a frictionless roller that can move along the hypotenuse.
- \(m=\SI{3}{\kilo\gram}\)
- \(k_\textrm{L}= \frac18 \si[per-mode=symbol]{\newton\per\meter}\).

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