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Question #42067

Difficulty: 8/10 | 🟢 Free

The Question

A long horizontal plank is moving on a smooth horizontal floor with an initial velocity $v_0 \hat{i}$ and a constant acceleration $-a_0 \hat{i}$ ( $a_0 > 0$ ). A solid cylinder of mass $m$ and radius $R$ is gently placed on the plank at $t=0$ with zero initial velocity and zero angular velocity. The coefficient of kinetic friction between the cylinder and the plank is $\mu$ . Let $P$ be the top-most point of the cylinder and $Q$ be the right-most point of the cylinder (at the same height as the center of mass). Determine the magnitude of the velocity of point $P$ relative to point $Q$ at the exact instant the cylinder begins pure rolling on the plank.

Options

A.
$\frac{2 \sqrt{2} \mu g v_0}{3 \mu g + a_0}$
B.
$\frac{\sqrt{2} \mu g v_0}{2 \mu g + a_0}$
C.
$\frac{2 \sqrt{2}}{3} v_0$
D.
$\frac{3 \mu g v_0}{3 \mu g + a_0}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Variable_Constraint

Develop pure symbolic logic skills by mastering algebraic constraints without relying on numerical data.

Concepts Behind This Question

Relative Velocity in 2D|Rigid Body Constraints|Vector Cross-Product

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to analyze rigid body kinematics in a mixed translational-rotational system while avoiding the trap of the accelerating reference frame.

A common mistake is failing to account for the plank's acceleration when determining the transition time to pure rolling, or incorrectly applying pseudo-forces if working in the non-inertial plank frame.

Furthermore, students often erroneously attempt to calculate the absolute velocities of points P and Q in the ground frame and subtract them, which is algebraically tedious and prone to sign errors, instead of recognizing the rigid body constraint $\vec{v}_{P/Q} = \vec{\omega} \times \vec{r}_{P/Q}$ which elegantly isolates the rotational component.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Motion in a Plane - Class 11 Part 1
Concept	Relative Velocity in 2D
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free