Question:
A solid cylinder rolls down an incline without slipping. Its acceleration depends on
A solid cylinder rolls down an incline without slipping. Its acceleration depends on
Show Hint
For rolling without slipping,
\[
a=\frac{g\sin\theta}{1+\dfrac{I}{mR^2}}.
\]
For a solid cylinder,
\[
I=\frac12 mR^2
\]
which gives
\[
a=\frac23 g\sin\theta.
\]
Notice that the mass cancels out.
Updated On: Jul 29, 2026
- Only the mass of the cylinder
- Only gravitational acceleration
- Both the mass of the cylinder and the angle of inclination
- Both the angle of inclination and gravitational acceleration
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The Correct Option is D
Solution and Explanation
Concept:
For a rigid body rolling down an incline without slipping,
\[
a=\frac{g\sin\theta}{1+\dfrac{I}{mR^2}}.
\]
Thus the acceleration depends on \(g\), the angle of inclination \(\theta\), and the shape of the body.
Step 1: Write the moment of inertia of a solid cylinder. \[ I=\frac12 mR^2. \] Substituting into the rolling acceleration formula, \[ a = \frac{g\sin\theta} {1+\dfrac12}. \] \[ a = \frac{2}{3}g\sin\theta. \]
Step 2: Identify the quantities on which acceleration depends. From \[ a=\frac{2}{3}g\sin\theta, \] the acceleration depends on \[ g \] and \[ \theta. \] It does not depend on \[ m \] or \[ R. \]
Step 3: Choose the correct option. Since \[ a=\frac{2}{3}g\sin\theta, \] the acceleration depends on both gravitational acceleration and the angle of inclination. Therefore, \[ \boxed{\text{Acceleration depends on }g\text{ and }\theta} \] \[ \boxed{\text{Answer = (D)}} \]
Step 1: Write the moment of inertia of a solid cylinder. \[ I=\frac12 mR^2. \] Substituting into the rolling acceleration formula, \[ a = \frac{g\sin\theta} {1+\dfrac12}. \] \[ a = \frac{2}{3}g\sin\theta. \]
Step 2: Identify the quantities on which acceleration depends. From \[ a=\frac{2}{3}g\sin\theta, \] the acceleration depends on \[ g \] and \[ \theta. \] It does not depend on \[ m \] or \[ R. \]
Step 3: Choose the correct option. Since \[ a=\frac{2}{3}g\sin\theta, \] the acceleration depends on both gravitational acceleration and the angle of inclination. Therefore, \[ \boxed{\text{Acceleration depends on }g\text{ and }\theta} \] \[ \boxed{\text{Answer = (D)}} \]
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