The time period of vertical simple harmonic motion of the block shown in the diagram is T=(x π/y) s. Find (x +y). Given that the spring is massless and ideal with k=20 N / m, the pulleys are massless and smooth and the block's mass is m=5 kg, the inclined plane is at θ=30° to the horizontal and the thread is inextensible and massless.

Question

The time period of vertical simple harmonic motion of the block shown in the diagram is T=(x π/y) s. Find (x +y). Given that the spring is massless and ideal with k=20 N / m, the pulleys are massless and smooth and the block's mass is m=5 kg, the inclined plane is at θ=30° to the horizontal and the thread is inextensible and massless. (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

At equilibrium position, if the extension of the spring is

x_{0}

\Rightarrow 2 k x_{0} = m g

, on downward (or upward) displacement

s

of the block, the net extension of the spring becomes

x = x_{0} + 2 s

(or

x = x_{0} - 2 s

),
hence, from the dynamical equation

2 k x - m g = ma

\Rightarrow 2 k (x_{0} + 2 s) - m g = ma

\Rightarrow 4 k s = - m \frac{d ^{2} s}{d t ^{2}}

\Rightarrow T = 2 π \frac{m}{4 k}