Question:
The rotational kinetic energy of a solid sphere of mass \(3\,\text{kg}\) and radius \(0.2\,\text{m}\) rolling down an inclined plane of height \(7\,\text{m}\) is (nearest to)
The rotational kinetic energy of a solid sphere of mass \(3\,\text{kg}\) and radius \(0.2\,\text{m}\) rolling down an inclined plane of height \(7\,\text{m}\) is (nearest to)
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For a solid sphere rolling without slipping,
\[
K_r=\frac27 mgh,
\qquad
K_t=\frac57 mgh.
\]
These results are frequently used in rolling motion problems.
Updated On: Jul 9, 2026
- \(80\,\text{J}\)
- \(36\,\text{J}\)
- \(40\,\text{J}\)
- \(60\,\text{J}\) \bigskip
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The Correct Option is D
Solution and Explanation
Concept:
For a body rolling without slipping,
\[
mgh
=
\frac12 mv^2+\frac12 I\omega^2.
\]
For a solid sphere,
\[
I=\frac{2}{5}mR^2.
\]
The rotational kinetic energy is
\[
K_r=\frac12 I\omega^2.
\]
Step 1: Express rotational kinetic energy in terms of translational kinetic energy. Since \[ v=R\omega, \] \[ K_r = \frac12\left(\frac25mR^2\right)\omega^2 = \frac15 mv^2. \] Also, \[ K_t = \frac12 mv^2. \] Hence, \[ K_r=\frac25K_t. \]
Step 2: Use conservation of energy. \[ mgh = K_t+K_r = K_t+\frac25K_t = \frac75K_t. \] Therefore, \[ K_t = \frac57 mgh. \] Thus, \[ K_r = \frac25\left(\frac57 mgh\right) = \frac27 mgh. \]
Step 3: Substitute the given values. \[ K_r = \frac27(3)(10)(7). \] \[ K_r = 60\,\text{J}. \]
Step 4: Write the final answer. \[ \boxed{K_r=60\,\text{J}} \] \[ \boxed{\text{Answer = (D)}} \]
Step 1: Express rotational kinetic energy in terms of translational kinetic energy. Since \[ v=R\omega, \] \[ K_r = \frac12\left(\frac25mR^2\right)\omega^2 = \frac15 mv^2. \] Also, \[ K_t = \frac12 mv^2. \] Hence, \[ K_r=\frac25K_t. \]
Step 2: Use conservation of energy. \[ mgh = K_t+K_r = K_t+\frac25K_t = \frac75K_t. \] Therefore, \[ K_t = \frac57 mgh. \] Thus, \[ K_r = \frac25\left(\frac57 mgh\right) = \frac27 mgh. \]
Step 3: Substitute the given values. \[ K_r = \frac27(3)(10)(7). \] \[ K_r = 60\,\text{J}. \]
Step 4: Write the final answer. \[ \boxed{K_r=60\,\text{J}} \] \[ \boxed{\text{Answer = (D)}} \]
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