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Jerk is defined as the rate of change of acceleration of a particle. The velocity of a particle is v=v0sinωt , where v0and ωare constant, Find jerk as a function of time.
a. v0ω2sinωt
b. v0ω2cosωt
c. v0ω2sinωt
d. v0ω2cosωt

Answer
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Hint: The rate of change of acceleration is called jerk. This rate of change of acceleration can be calculated by differentiating the acceleration with respect to time. Since acceleration is the rate of change of velocity, it can be calculated by differentiating the velocity with respect to time. Therefore, to get the function for jerk the expression for velocity must be double differentiated.

Complete step by step answer:
It is given in the question that, the velocity of the particle is-
v=v0sinωt
Here, the velocity vis a function of time t. It is given that the terms v0and ωare constant. The term v0 generally refers to the initial velocity of the object, whereas ω refers to the frequency or the angular speed of the particle. At an instant, ω multiplied with the instantaneous time t gives the amount of angle covered by the object.

Therefore the acceleration of this particle can be determined by differentiating this function with respect to time,
a=dvdt
a=ddt(v0sinωt)
The equation for derivative of sinx is,
ddx(sinx)=cosx
Using this value in the above equation we get,
a=v0(cosωt)×ω
a=v0ωcosωt

Now, to calculate the jerk of the object, the expression of acceleration needs to be differentiated with respect to time.
Therefore,
j=dadtor j=d2vdt2
j=ddt(v0ωcosωt)
The term v0ω is constant. Therefore,
j=v0ωddt(cosωt)
The equation for derivative of cosx is,
ddx(cosx)=sinx

Using this value in the above equation we get,
j=v0ω(sinωt)×ω
j=v0ω2sinωt
The function defining the jerk of the particle is given by,
j=v0ω2sinωt

Hence, the correct answer is option (C).

Note: To calculate the jerk produced in a particle, the velocity must be double differentiated. Since velocity refers to the displacement of an object per unit time, the displacement of the object can also be used to calculate jerk. For this, the displacement of the particle must be differentiated three times. Thus, Jerk is the third derivative of displacement with respect to time.