Engineering

Physics

Basics of Simple Harmonic Motion

Question

A long plank P is placed on a horizontal frictionless surface and connected to two springs of spring constant k and 2k respectively. A cylinder of mass m is placed on the plank. The plank is displaced towards right (positive direction) and released at t = 0 and it oscillates with amplitude A. The cylinder does not slip on the plank during the oscillation

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Linked Question 1

The friction force acting on the cylinder as a function of time is shown as

A

B

C

D

Friction force acting on the cylinder

$f = - \frac{3}{7} kx = \frac{- 3}{7} kAcos ωt$

Linked Question 2

The maximum friction force acting on the cylinder during the oscillation is

A

$\frac{3 kA}{7}$

B

$\frac{2 kA}{3}$

C

3kA

D

$\frac{2 kA}{7}$

Let at certain instant of time displacement of plank be x towards right from the mean position.
For motion of plank

f – 3kx = 2ma₁

For linear motion of cylinder

– f = ma₂

For rotational

motion of cylinder

$fR = \frac{{mR}^{2}}{2} α$

For no slipping between plank and cylinder

$a_{1} = a_{2} - αR$

$\Rightarrow \frac{f}{2 m} - \frac{3 kx}{2 m} = - \frac{f}{m} - \frac{2 f}{m}$

$\Rightarrow f = \frac{3}{7} kx$

$f_{max} = \frac{3}{7} {kx}_{max} = \frac{3}{7} kA$