Question

Two wheels $A$ and $B$ are released from rest from points $X$ and $Y$ respectively on an inclined plane as shown in figure. Which of the following statement(s) is/are incorrect?

• A. Wheel $B$ takes twice as much time to roll from $Y$ to $Z$ than that of wheel $A$ from $X$ to $Z$.
• B. At point $Z$ velocity of wheel $A$ is four times that of wheel $B$
• C. Acceleration of the wheel $A$ is four times that of wheel $B$
• D. Both wheel take same time to arrive at point $Z$.

Answer 1

Answer: (B), (C), (D)

The correct options are
B At point $Z$ velocity of wheel $A$ is four times that of wheel $B$
C Acceleration of the wheel $A$ is four times that of wheel $B$
D Both wheel take same time to arrive at point $Z$.

The wheels will start pure rolling after being released from rest.

Let $v$ and $w$ be the linear and angular velocities respectively.

$v=rw$ (pure rolling)

Work-energy theorem: $W=\Delta K.E$

$MgH=\frac{1}{2}M{v}^2+\frac{1}{2}I{w}^2$

$MgH=\frac{1}{2}M{v}^2+\frac{1}{2}\times M{R}^2{w}^2$

$MgH=\frac{1}{2}M{v}^2+\frac{1}{2}\times M{v}^2$

$⟹v=\sqrt{gH}$

Thus $\frac{v_B}{v_A}=\frac{\sqrt{g\left(4h\right)}}{\sqrt{g\left(h\right)}}⟹v_B=2v_A$

Also acceleration of the body rolling without slipping on the inclined plane, $a=\frac{gsin\theta }{1+\frac{I}{M{R}^2}}$ where $I=KM{R}^2$ and $K=1$

Thus $a_A=a_B$

Now $v=0+at⟹t=\frac{v}{a}$

$\frac{t_B}{t_A}=\frac{v_B}{a_B}\times \frac{a_A}{v_A}=\frac{2v}{v}$

$⟹t_B=2t_A$