Question

A body of mass 1 kg is executing SHM, its displacement y cm a t seconds is given by
\[y = 6\sin (100t + \dfrac{\pi }{4})\]
Its maximum kinetic energy is
A. 6 J
B. 18 J
C. 24 J
D. 36 J

Answer 1

Hint: The kinetic energy is $K.E. = \dfrac{1}{2}m{v^2}$. The kinetic energy will be maximum when velocity is maximum. For velocity we can differentiate the given equation with respect to time and then substitute its maximum value in the kinetic energy equation.

Complete step by step answer:
Given are the mass of the body m=1 kg and the equation of its displacement in terms of time t as
\[y = 6\sin (100t + \dfrac{\pi }{4})\]
The kinetic energy is given as
$K.E. = \dfrac{1}{2}m{v^2}$
So, the kinetic energy will be maximum when the velocity will be maximum since the mass is a constant. To find the velocity we can differentiate the given equation with respect to time.
\[v = \dfrac{{dy}}{{dt}} = 600\cos (100t + \dfrac{\pi }{4})\]
For the velocity to be maximum the cosine term needs to be maximum as it is the only variable quantity. The maximum value of the cosine is +1. Therefore,
\[{v_{\max }} = 600\dfrac{{cm}}{s} = 6\dfrac{m}{s}\]
 So the maximum kinetic energy will be given as,
\[ K.{E_{\max }} = \dfrac{1}{2}mv_{\max }^2 = \dfrac{1}{2}{.1.6^2} = 18J \\
 K.{E_{\max }} = 18J \\ \]

So, the correct answer is “Option B”.

Additional Information:
If it was given to calculate the time at which the kinetic energy or velocity will be maximum, then, the cosine term will be maximum i.e.
\[ \cos (100t + \dfrac{\pi }{4}) = 1 \Rightarrow 100t + \dfrac{\pi }{4} = 2n\pi \\
 t = \dfrac{{(2n - \dfrac{1}{4})\pi }}{{100}}\sec \\ \]

Note:
Do not get confused that the velocity will be dx/dt and not dy/dx. The body is performing SHM about the y axis so the displacement in x-direction will be symmetric. Hence the variation in velocity will be in the y-direction and therefore the velocity will be dy/dx.