Classical Mechanics

Simple Harmonic Motion

Simple Harmonic Motion: Level 4-5 Challenges

         

Consider the earth as a uniform sphere of mass MM And Radius RR. 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 TsT s, then find the value of T100\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 = 10ms210ms^{-2}

  • Radius of Earth = 6400km6400 km.

This is an entry for the problem writing party.

A block of mass mm is gently attached to the spring and released at time t=0t=0, when the spring has its free length. During subsequent motion of the block, the displacement of the block xx with respect to time tt is considered. Find 03x dt\displaystyle\int_0^3 x ~\mathrm{d}t

Details and Assumptions

  • Neglect air resistance.
  • kk is the spring constant.
  • m=10 kg,k=34 Nm1,g=9.8 ms2m = 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=(atbx) along the x-axisF = (at - bx) \text{ along the } x\text{-axis} Initially the mass lies at the origin at rest. In the definition of force (given) xx refers to the xx-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

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

Find the value of a+b+ca+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 kk and 2k2k 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 mm and angle of inclination θ=60 °\theta = \SI{60}{\degree} is attached to two springs of spring constant kLk_\textrm{L} on the left, and kR=3kLk_\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=3 kgm=\SI{3}{\kilo\gram}
  • kL=18N/mk_\textrm{L}= \frac18 \si[per-mode=symbol]{\newton\per\meter}.
Try my set.
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