Answer
418.2k+ views
Hint
For solving this question, we have to use the formula for the kinetic energy in the rotational mechanics, which relates the kinetic energy to the moment of inertia and the angular velocity of the body.
Formula Used: The formula used in this solution is
$ K = \dfrac{1}{2}I{\omega ^2} $, where $ K $ is the kinetic energy, $ I $ is the moment of inertia, and $ \omega $ is the angular velocity of a body rotating about an axis.
Complete step by step answer
We know that the kinetic energy in the rotational mechanics is given by the relation
$ K = \dfrac{1}{2}I{\omega ^2} $
According to the question, the kinetic energy is
$ K = 9J $
Also, the moment of inertia of the body is $ 2kg - {m^2} $
$ \therefore $ $ I = 2kg - {m^2} $
Substituting these in the above equation, we get
$ 9 = \dfrac{1}{2} \times 2 \times {\omega ^2} $
Or $ 9 = {\omega ^2} $
Finally, taking the square root we get
$ \omega = 3rad/\operatorname{s} $
So, the angular velocity of the body is $ 3rad/\sec $
Hence, the correct answer is option (B), $ 3 $.
Additional Information
There exists an analogy between rotational and linear motion. All the quantities in rotational mechanics are analogous to the corresponding linear quantities.
This analogy is listed below:
-Angular displacement, $ \theta $ $ \approx $ Linear displacement, $ x $
-Angular velocity, $ \omega $ $ \approx $ Linear velocity, $ v $
-Angular acceleration, $ \alpha $ $ \approx $ Linear acceleration, $ a $
-Torque, $ \tau $ $ \approx $ Force, $ F $
-Angular momentum, $ L $ $ \approx $ Linear momentum, $ p $
So, it is not required to remember the equations related to the kinematics and dynamics of the rotational motion. We just need to replace the quantities in the equations of linear motion with the analogous rotational quantities to get the corresponding equations of the rotational motion.
Note
If we do not remember the formula of the kinetic energy in the rotational mechanics, then also very easily we can derive the formula. We know that the kinetic energy in the translational motion is given by $ K = \dfrac{1}{2}m{v^2} $. Using the analogy given above, we see that the moment of inertia $ I $ is the rotational analogue of the mass $ m $, and the angular velocity $ \omega $ is analogous to the linear velocity $ v $. Replacing with these in the above formula, we get $ K = \dfrac{1}{2}I{\omega ^2} $.
For solving this question, we have to use the formula for the kinetic energy in the rotational mechanics, which relates the kinetic energy to the moment of inertia and the angular velocity of the body.
Formula Used: The formula used in this solution is
$ K = \dfrac{1}{2}I{\omega ^2} $, where $ K $ is the kinetic energy, $ I $ is the moment of inertia, and $ \omega $ is the angular velocity of a body rotating about an axis.
Complete step by step answer
We know that the kinetic energy in the rotational mechanics is given by the relation
$ K = \dfrac{1}{2}I{\omega ^2} $
According to the question, the kinetic energy is
$ K = 9J $
Also, the moment of inertia of the body is $ 2kg - {m^2} $
$ \therefore $ $ I = 2kg - {m^2} $
Substituting these in the above equation, we get
$ 9 = \dfrac{1}{2} \times 2 \times {\omega ^2} $
Or $ 9 = {\omega ^2} $
Finally, taking the square root we get
$ \omega = 3rad/\operatorname{s} $
So, the angular velocity of the body is $ 3rad/\sec $
Hence, the correct answer is option (B), $ 3 $.
Additional Information
There exists an analogy between rotational and linear motion. All the quantities in rotational mechanics are analogous to the corresponding linear quantities.
This analogy is listed below:
-Angular displacement, $ \theta $ $ \approx $ Linear displacement, $ x $
-Angular velocity, $ \omega $ $ \approx $ Linear velocity, $ v $
-Angular acceleration, $ \alpha $ $ \approx $ Linear acceleration, $ a $
-Torque, $ \tau $ $ \approx $ Force, $ F $
-Angular momentum, $ L $ $ \approx $ Linear momentum, $ p $
So, it is not required to remember the equations related to the kinematics and dynamics of the rotational motion. We just need to replace the quantities in the equations of linear motion with the analogous rotational quantities to get the corresponding equations of the rotational motion.
Note
If we do not remember the formula of the kinetic energy in the rotational mechanics, then also very easily we can derive the formula. We know that the kinetic energy in the translational motion is given by $ K = \dfrac{1}{2}m{v^2} $. Using the analogy given above, we see that the moment of inertia $ I $ is the rotational analogue of the mass $ m $, and the angular velocity $ \omega $ is analogous to the linear velocity $ v $. Replacing with these in the above formula, we get $ K = \dfrac{1}{2}I{\omega ^2} $.
Recently Updated Pages
Mark and label the given geoinformation on the outline class 11 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
When people say No pun intended what does that mea class 8 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Name the states which share their boundary with Indias class 9 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Give an account of the Northern Plains of India class 9 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Change the following sentences into negative and interrogative class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Advantages and disadvantages of science
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Trending doubts
Bimbisara was the founder of dynasty A Nanda B Haryanka class 6 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Which are the Top 10 Largest Countries of the World?
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
10 examples of evaporation in daily life with explanations
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Give 10 examples for herbs , shrubs , climbers , creepers
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
How do you graph the function fx 4x class 9 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference Between Plant Cell and Animal Cell
![arrow-right](/cdn/images/seo-templates/arrow-right.png)