Question

For a particle executing SHM. The KE ‘$K$’ is given by $ K = {K_0}{\cos ^2}wt $ . The maximum value of PE is
(A) $ {K_0} $
(B) Zero
(C) $ \dfrac{{{K_0}}}{2} $
(D) $ \dfrac{{{K_0}}}{4} $

Answer 1

Hint: We first try to understand the term SHM which is a short form of simple harmonic motion. The simple harmonic motion is defined as a periodic motion in which the restoring force is directly proportional to the magnitude of the displacement of the object. Using the property of simple harmonic motion we will evaluate the maximum potential energy.

Complete step by step solution:
Here in this question, we have given that the kinetic energy of the particle is performing simple harmonic motion having a kinetic energy of $ K = {K_0}{\cos ^2}wt $ .
We know that in the case of simple harmonic motion the total energy always converses. The total energy of the particle or object can be given as the sum of the kinetic energy and potential energy together.
As discussed the total energy is conserved for a body performing the simple harmonic motion. Hence if the kinetic energy $ \;K.E $ increases then the potential energy $ \;P.E $ tends to decrease. Similarly, if the kinetic energy decreases then the potential energy increases,
Hence if kinetic energy becomes zero then the potential energy becomes maximum. Therefore for maximum potential energy
 $ \Rightarrow K = {K_0}{\cos ^2}wt = 0 $
It occurs when the $ \cos wt = 1 $ , hence the potential energy becomes maximum and equals to
 $ \therefore {\left( {P.E} \right)_{\max }} = {K_0} $
Hence the option (A) is the correct answer.

Note:
The law of conservation of energy states that the total energy is neither created nor destroyed. It can only be converted from one form to another. Hence we know that if a body is at rest it consists of some energy exits in it which are known as potential energy and when the starts moving then this potential energy starts converting into kinetic energy.