Question

In the arrangement, spring constant $$k$$ has value $$2N{m}^{-1}$$, mass $$M=3kg$$ and mass $$M=1kg$$. Mass $$M$$ is in contact with a smooth surface. The coefficient of friction between two blocks is 0.1. The time period of SHM executed by the system is …

A) $$\pi \sqrt{6}$$
B) $$\pi \sqrt{2}$$
C) $$2\sqrt{2}\pi$$
D) $$2\pi$$

Answer 1

Hint: For small amplitude the two blocks oscillate together , with angular frequency $$\omega =\sqrt{{\displaystyle \frac{k}{M+m}}}$$hz. So the time period becomes $$T=2\pi \sqrt{{\displaystyle \frac{M}{k}}}$$.

Formula used:- $$T=2\pi \sqrt{{\displaystyle \frac{M}{k}}}$$ where, T = Time- period of the spring block system
                                                   $$M$$ = Total mass attached to spring
Value of the spring constant is

                                                    $$k$$ = spring constant of the spring
Since total mass is the sum of both the blocks.
$$M$$=M+ m = $$4kg$$
$$k$$= $$2N{m}^{-1}$$
We have to find the time period of the system$$T=2\pi \sqrt{{\displaystyle \frac{M}{k}}}$$
Putting the value$$M$$= $$4kg$$,& $$k$$= $$2N{m}^{-1}$$in order to find the time period
$$T=2\pi \sqrt{{\displaystyle \frac{4}{2}}}$$=$$2\sqrt{2}\pi$$sec.

Hence option (C ) is the correct option.

Note :- Periodic motion : Any motion which repeats itself after a regular interval of time is called the periodic motion. Few examples are motion of planets around the sun , motion of the pendulum of wall clock.
-Oscillatory motion : The motion of a body is oscillatory if it moves back and forth ( to and fro) about a fixed point after a regular interval of time. This fixed position is called the mean position or the equilibrium position of the body.
-The periodic time of a hard spring is less as compared to that of a sift spring because the spring constant is large for a hard spring.
-For a system executing SHM, the mechanical energy which is the sum of potential energy and kinetic energy together remains constant.
-The frequency of oscillation of potential energy and kinetic energy is twice as of the displacement or velocity or acceleration of a particle executing simple harmonic motion.