Period of Small Oscillations

Classical Mechanics Level 3

Mass of Planck is m m and that of the solid cylinder is 8 m 8m . Springs are light. The planck are slightly displaced from equilibrium and then released.
Find the period of small oscillations ( in seconds ) of the planck.
There is no slipping at any contact point .
The ratio of the the mass of the planck and stiffness of the spring is m k = 2 π 2 \frac{m}{k}=\frac{2}{π^{2}}


The answer is 4.

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1 solution

The linear velocity of the plank is related to the angular velocity of the cylinder as

v = 2 R ω v=2R\omega , where R R is the radius of the cylinder.

Total mechanical energy of the system is

2 × 1 2 k x 2 + 1 2 m v 2 + 2\times \dfrac 12 kx^2+\dfrac 12 mv^2+

1 2 ( 1 2 × 8 m R 2 + 8 m R 2 ) ω 2 \dfrac 12 (\frac 12 \times 8mR^2+8mR^2)\omega^2

= k x 2 + 2 m v 2 =kx^2+2mv^2

Where x x is the displacement of the plank. Since this energy remains conserved, we have

k x v + 2 m v d v d t = 0 kxv+2mv\dfrac{dv}{dt}=0

⟹ d v d t = − k 2 m x = − π 2 4 x \implies \dfrac {dv}{dt}=-\dfrac {k}{2m}x=-\dfrac {π^2}{4}x

Hence the period of motion of the system is

T = 2 π 4 π 2 = 4 T=2π\sqrt {\dfrac {4}{π^2}}=\boxed 4 units.

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