Question

Two solid cylinders P and Q of same mass and same radius start rolling down a fixed inclined plane from the same height at the same time. Cylinder P has most of its mass concentrated near its surface, while Q has most of its mass concentrated near the axis. Which statement(s) is (are) correct ?

• A. Both cylinders P and Q reach the ground at the same time.
• B. Cylinder P has larger linear acceleration than cylinder Q.
• C. Both cylinders reach the ground with same translational kinetic energy
• D. Cylinder Q reaches the ground with larger angular speed

Answer 1

The correct option is D Cylinder Q reaches the ground with larger angular speed


Translational motion :
$mgsin\theta -f=m{a}_{cm}...........\left(i\right)$
Rotational motion $fR=I_{cm}\alpha ..........\left(ii\right)$
Rolling without slipping $\alpha R=a_{cm}........\left(iii\right)$
From (ii) & (iii) $f=\frac{I_{cm}{a}_{cm}}{R^2}$
Put this in (i)
$mgsin\theta -\frac{I_{cm}{a}_{cm}}{R^2}=m{a}_{cm}$
$a_{cm}=\frac{mgsin\theta }{(\frac{I_{cm}}{R^2}+m)}$
As $I_P>I_Q$
So $a_P<a_Q$
So, $V_{cm\left(Q\right)}>v_{cm\left(P\right)}$
Hence ${\omega }_Q>{\omega }_P$