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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.

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.

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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.
- \(m = 10\text{ kg},k = 34 \text{ Nm}^{-1}, g = 9.8 \text{ ms}^{-2}\)

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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**

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If the time period of SHM of rectangular block can be represented as
\[T=2\pi \sqrt{\frac{m(\sqrt a +b)}{ck}}\]
###### Try its different variants - medium and easy

Find the value of \(a+b\). Give your answer to 2 decimal places.

**Details and Assumptions**:

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.

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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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