Question
Question: Starting from the origin, a body oscillates simple harmonically with a period of 2 s. After what tim...
Starting from the origin, a body oscillates simple harmonically with a period of 2 s. After what time will its kinetic energy be 75% of the total energy?
Solution
In this question, we will use the formula of kinetic energy for a body oscillating simple harmonically; this will help us to get the expression for total energy of the body. Also, we will use these values in the direct expression of displacement of waves. This will give us the required answer. We will also see the basics of simple harmonic motion, wave and energy, for our better understanding.
Formula used:
K=21mω2(A2−x2)
T=21mω2A2
x=Asinωt
Complete step by step solution:
We know that a simple harmonic progressive wave is a wave that continuously advances in a given direction without the change of form and also, the particles of the medium perform simple harmonic motion about their mean position with the same amplitude and period, when the waves pass over them.
Kinetic energy of S.H.M is given by:
K=21mω2(A2−x2)
Now, the total energy is given by:
T=21mω2A2
According to the question, we have:
21mω2(A2−x2)=10075×21mω2A2
Now, by solving the above equation to find the value of x, we get:
\eqalign{& {A^2} - \dfrac{3}{4}{A^2} = {x^2} \cr
& \Rightarrow x = \dfrac{A}{2} \cr}
As we know, for a simple harmonic motion, where x is the displacement is given by:
x=Asinωt
By substituting the value of x is the above equation we get:
2A=Asinωt
\eqalign{& \Rightarrow \sin \dfrac{\pi }{6} = \sin \omega t \cr
& \Rightarrow \omega t = \dfrac{\pi }{6} \cr}
⇒(T2π)t=6π
Now, we have T=2s, by putting this value in above equation, we get:
∴t=61s
Therefore, we get the required answer, which gives us the time at which the kinetic energy of the body will be 75% of its total energy.
Additional information:
As we know, a wave can be described as a disturbance that travels through a medium, transporting energy from one location to another location without transporting matter.
Simple harmonic motion is defined as a special type of periodic motion where the restoring force (force applied in the opposite direction) on the moving object is directly proportional to the object's displacement magnitude and acts towards the object's equilibrium position. Here, the motion is sinusoidal in time and demonstrates a single resonant frequency.
As we know that energy is defined as the ability to do work. Energy can be found in many things and can take different forms, like-kinetic energy is the energy of motion, and potential energy is energy due to an object's position or structure. Total energy of a system is given by the sum of kinetic energy and potential energy.
We also know that, according to the law of conservation of energy, energy can neither be created nor be destroyed; rather it can only be transferred from one form to another. Here, in this question electrical energy is transferred to heat energy.
Note:
It should be remembered that in the simple harmonic progressive wave the particles in the medium show simple harmonic motion. There are other systems as well which show simple harmonic motion like- simple pendulum.