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Question: A body rotates about a fixed axis with an angular acceleration of \[3{\text{ }}rad \cdot {s^{ - 2}}\...

A body rotates about a fixed axis with an angular acceleration of 3 rad⋅s−23{\text{ }}rad \cdot {s^{ - 2}} . The angle rotated by it during the time when its angular velocity increases from 10 rad⋅s−110{\text{ }}rad \cdot {s^{ - 1}} to 20 rad⋅s−120{\text{ }}rad \cdot {s^{ - 1}} (in radian) is:
A. 5050
B. 100100
C. 150150
D. 200200

Explanation

Solution

This question is based on rotational motion. The study of motion is known as kinematics. The relationships between rotation angle (θ)\left( \theta \right) , angular velocity (ω)\left( \omega \right) , angular acceleration (α)(\alpha ) , and time (t)\left( t \right) are described by rotational motion kinematics. It excludes any forces or masses that could influence rotation (these are part of dynamics).

Complete answer:
Before we move to the question, we will first write all of the given values in the correct order.
Angular acceleration of the body is given as (α)=3rad⋅s−2(\alpha ) = 3rad \cdot {s^{ - 2}}
Initial angular velocity was (ω∘)=10rad⋅s−1({\omega _ \circ }) = 10rad \cdot {s^{ - 1}}
Final angular velocity (ω)=20rad⋅s−1\left( \omega \right) = 20rad \cdot {s^{ - 1}}
Explanation:-
We already know that the third Kinematic Equation, i.e.
v2−u2=2as{v^2} - {u^2} = 2as
In order to convert it to an angular form,
ω2−ω∘2=2αθ{\omega ^2} - \omega _ \circ ^2 = 2\alpha \theta
Now, we'll plug in all of the specified values into the equation above.
(20)2−(10)2=2×3×θ{\left( {20} \right)^2} - {\left( {10} \right)^2} = 2 \times 3 \times \theta
Evaluating the equation further we will get
⇒400−100=6×θ\Rightarrow 400 - 100 = 6 \times \theta
⇒300=6×θ\Rightarrow 300 = 6 \times \theta
∴θ=3006=50rad\therefore \theta = \dfrac{{300}}{6} = 50rad
Therefore, the angle rotated by it during the time when its angular velocity increases from 10 rad⋅s−110{\text{ }}rad \cdot {s^{ - 1}} to 20 rad⋅s−120{\text{ }}rad \cdot {s^{ - 1}} (in radian) is: 50 rad50\,rad

So, the correct option is: A. 5050

Note:
One thing that students should be aware of is that the dynamics of rotating motion are quite similar to those of linear or translational motion. Many of the equations for rotating object mechanics are related to the linear motion equations. Only rigid bodies are considered in rotational motion. A rigid body is a massed entity that has a rigid shape.