Question:
A solid sphere of mass $5\, kg$ rolls on a plane surfaces. Find its kinetic energy at an instant when its centre moves with speed $4 \,m / s$.
A solid sphere of mass $5\, kg$ rolls on a plane surfaces. Find its kinetic energy at an instant when its centre moves with speed $4 \,m / s$.
Updated On: Apr 10, 2023
- 56 J
- 45 J
- 75 J
- 105 J
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The Correct Option is A
Solution and Explanation
As we know, the rotational kinetic energy of a body is given by,
$KE =\frac{1}{2} m v^{2}\left[1+\frac{k^{2}}{R^{2}}\right]$
where, $k=$ radius of gyration
Given, $m=5 \,kg , v=4 \,m / s$ and $k=\sqrt{\frac{2}{5}} \,R$
Hence, $KE =\frac{5}{2} \times 4^{2}\left[1+\frac{2}{5}\right] $
$\Rightarrow \,KE =40 \times \frac{7}{5}=56 \,J$
$KE =\frac{1}{2} m v^{2}\left[1+\frac{k^{2}}{R^{2}}\right]$
where, $k=$ radius of gyration
Given, $m=5 \,kg , v=4 \,m / s$ and $k=\sqrt{\frac{2}{5}} \,R$
Hence, $KE =\frac{5}{2} \times 4^{2}\left[1+\frac{2}{5}\right] $
$\Rightarrow \,KE =40 \times \frac{7}{5}=56 \,J$
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Concepts Used:
System of Particles and Rotational Motion
- The system of particles refers to the extended body which is considered a rigid body most of the time for simple or easy understanding. A rigid body is a body with a perfectly definite and unchangeable shape.
- The distance between the pair of particles in such a body does not replace or alter. Rotational motion can be described as the motion of a rigid body originates in such a manner that all of its particles move in a circle about an axis with a common angular velocity.
- The few common examples of rotational motion are the motion of the blade of a windmill and periodic motion.