Question

The time period of a mass suspended from a spring is T. If the spring is cut into four equal parts and the same mass is suspended from one of the parts, then the new time period will be:
A. 0.25T
B. T
C. 0.5T
D. 2T

Answer 1

A motion in which a particle undergoes periodic motion is called Simple harmonic motion (S.H.M). Not every periodic motion is S.H.M but every S.H.M is periodic motion. The revolution of earth about the sun is an example of periodic motion but it is not simple harmonic. A motion is said to be simple harmonic only if the acceleration of the particle is the function of first power of displacement and having direction opposite of the displacement.

Formula used:
$T = 2\pi \sqrt{\dfrac{m}{k}}$

Complete answer:
When a mass is suspended with a spring, it executes simple harmonic motion if displaced slightly from the position. The time period of the oscillation is given by $T = 2\pi \sqrt{\dfrac{m}{k}}$where, ‘m’ is the mass of a particle suspended with a spring of stiffness ‘k’.
Now, when the mass of spring is reduces, its stiffness value will increase according to the relation $kl = constant.$
Thus, if length is reduced to one-fourth, the stiffness will become four times.
Now, when mass ‘m’ is suspended from one of the parts, we can treat the part as one of the springs with stiffness ‘4k’. Hence using $T = 2\pi \sqrt{\dfrac{m}{k}}$, we get;
$T_f = 2\pi \sqrt{\dfrac{m}{4k}} = \dfrac12 2\pi \sqrt{\dfrac{m}{k}} = \dfrac12 T = 0.5T$

So, the correct answer is “Option C”.

Note:
Mathematically we can say that if the motion is simple harmonic, it must follow the standard differential equation of simple harmonic motion which is given by $\dfrac{d^2x}{dt^2} = -\omega^2 x$. Students can make mistakes about the fact that every simple harmonic motion is periodic but not every periodic motion is simple harmonic. The revolution of a fan about its own axis is an example of simple harmonic motion.