Question

The springs shown in the figure are all unstretched in the beginning when a man starts pulling the block. The man exerts a constant force F on the block. Find the amplitude and the frequency of the motion of the block.

Answer 1

As the block of mass M is pulled, a net resultant force is exerted by the three springs opposing the motion of the block.

Now, springs k₂ and k₃ are in connected as a series combination.
Let k₄ be the equivalent spring constant.

\[\therefore \frac{1}{k_4} = \frac{1}{k_2} + \frac{1}{k_3} = \frac{k_2 + k_3}{k_2 k_3}\] 

\[ k_4  = \frac{k_2 k_3}{k_2 + k_3}\]

k₄ and k₁ form a parallel combination of springs. Hence, equivalent spring constant k = k₁ + k₄.

\[= \frac{k_2 k_3}{k_2 + k_3} +  k_1 \] 

\[ = \frac{k_2 k_3 + k_1 k_2 + k_1 k_3}{k_2 + k_3}\] 

\[ \therefore \text { Time  peiod },   T = 2\pi\sqrt{\frac{M}{k}}\] 

\[         = 2\pi\sqrt{\frac{M  \left( k_2 + k_3 \right)}{k_2 k_3 + k_1 k_2 + k_1 k_3}}\]

(b) Frequency \[\left( v \right)\] is given by,

\[v = \frac{1}{T}\]

\[= \frac{1}{2\pi}\sqrt{\frac{k_2 k_3 + k_1 k_2 + k_1 k_3}{M\left( k_2 + k_3 \right)}}\]

(c) Amplitude ( x ) is given by,

\[x = \frac{F}{k} = \frac{F\left( k_2 + k_3 \right)}{k_1 k_2 + k_2 k_3 + k_1 k_3}\]