Classical Mechanics

Simple Harmonic Motion

Simple Harmonic Motion: Level 4-5 Challenges


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

This Question is not Original

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.

1) All the values are in SI units.
2) Take the mass = 1 kg, a = 1 N/s, b = 1 N/m

If the time period of SHM of rectangular block can be represented as \[T=2\pi \sqrt{\frac{m(\sqrt a +b)}{ck}}\]

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.

Try its different variants - medium and easy

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.
  • \(m=\SI{3}{\kilo\gram}\)
  • \(k_\textrm{L}= \frac18 \si[per-mode=symbol]{\newton\per\meter}\).
Try my set.

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