
A rigid body rotates about a fixed axis with variable angular velocity equal to at where and are constants. The angle through which it rotates before it comes to rest is:
(A)
(B)
(C)
(D)
Answer
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Hint: Here, using the angular velocity formula, which is the angle per unit time, the angle through which the rigid body rotates before it comes to rest is computed. The angular velocity is zero at rest, so we will calculate the time and put it in the angle equation.
Complete Step-by-Step Solution:
Let us suppose the angle through which it rotates be
According to the question,
The angular velocity is given as
Now, if the body is at rest, the angular velocity of such a body is zero, i.e.,
We can now equate the value of angular velocity such that
Upon solving, we get
Also, we know that angular velocity is also measured as angle per unit time, therefore,
Rearranging the terms, we get
Now, we will integrate both the sides
Upon further solving, we get
Now, let us integrate the above expression
Let us simplify and get the final answer
Hence, the correct option is (B.)
Additional Information: Angular velocity is the quantity of a vector and is described as the angular displacement change rate that specifies the object's angular velocity or rotational velocity and the axis around which the object is rotating. At a given period of time, the amount of change in the angular displacement of the particle is called angular velocity. The angular velocity vector's track is vertical to the rotation plane, in a direction usually indicated by the right-hand rule.
Note: In the above question, we have found out the angle by integrating the equation of between the limits to . We could have also landed to the same result by putting the value of in the equation after integrating it.
Complete Step-by-Step Solution:
Let us suppose the angle through which it rotates be
According to the question,
The angular velocity is given as
Now, if the body is at rest, the angular velocity of such a body is zero, i.e.,
We can now equate the value of angular velocity such that
Upon solving, we get
Also, we know that angular velocity is also measured as angle per unit time, therefore,
Rearranging the terms, we get
Now, we will integrate both the sides
Upon further solving, we get
Now, let us integrate the above expression
Let us simplify and get the final answer
Hence, the correct option is (B.)
Additional Information: Angular velocity is the quantity of a vector and is described as the angular displacement change rate that specifies the object's angular velocity or rotational velocity and the axis around which the object is rotating. At a given period of time, the amount of change in the angular displacement of the particle is called angular velocity. The angular velocity vector's track is vertical to the rotation plane, in a direction usually indicated by the right-hand rule.
Note: In the above question, we have found out the angle by integrating the equation of
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