A solid cylinder P rolls without slipping from rest down an incli… - Vidyadip

MCQ

A solid cylinder $P$ rolls without slipping from rest down an inclined plane attaining a speed $v_p$ at the bottom. Another smooth solid cylinder $Q$ of same mass and dimensions slides without friction from rest down the inclined plane attaining a speed $v_q$ at the bottom. The ratio of the speeds $\frac{v_q}{v_p}$ is

A
$\sqrt{\frac{3}{4}}$
✓
$\sqrt{\frac{3}{2}}$
C
$\sqrt{\frac{2}{3}}$
D
$\sqrt{\frac{4}{3}}$

✓

Answer

Correct option: B.

$\sqrt{\frac{3}{2}}$

b
(b)

In presence of friction, cylinder rolls and by energy conservation, we have

Initial potential energy $=$ Kinetic energy at bottom

$m g h =( KE )_{\text {translation }}+( KE )_{\text {rotation }}$

$=\frac{1}{2} m v^2+\frac{1}{2} I \omega^2$

$=\frac{1}{2} m v^2+\frac{1}{2}\left(\frac{1}{2} m r^2\right) \omega^2$

$=\frac{3}{4} m v^2 \quad[\therefore v=r \omega]$

So, $\quad v_p=\sqrt{\frac{4}{3} g h}$

In absence of friction, cylinder slips without rolling.

Hence, $\frac{1}{2} m v^2=m g h$

$\Rightarrow \quad v_q=\sqrt{2 g h}$

So, ratio of $v_q / v_p=\frac{\sqrt{2 g h}}{\sqrt{\frac{4}{3} g h}}=\sqrt{\frac{3}{2}}$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

NEET STD 11 Science question papers→Gujarat Board STD 11 Science question papers→Gujarat Board question paper generator→

Similar questions

Magnification of a compound microscope is $30$. Focal length of eye-piece is $5\, cm$ and the image is formed at a distance of distinct vision of $25\, cm$. The magnification of the objective lens is

An empty thick conducting shell of inner radius $a$ and outer radius $b$ is shown in figure.If it is observed that the inner face of the shell carries a uniform charge density $-\sigma$ and the surface carries a uniform charge density $ '\sigma '$

If a point charge $q_A$ is placed at the center of the shell, then choose the correct statement $(s)$

A body falls freely under gravity. Its speed is $v$ when it has lost an amount $U$ of the gravitational energy. Then its mass is

A block is placed on a frictionless horizontal table. The mass of the block is m and springs are attached on either side with force constants ${K_1}$ and ${K_2}$. If the block is displaced a little and left to oscillate, then the angular frequency of oscillation will be

Express which of the following setups can be used to verify Ohm’s law

A metre stick is pivoted about its centre. A piece of wax of mass $20 \,g$ travelling horizontally and perpendicular to it at $5 \,m / s$ strikes and adheres to one end of the stick so that the stick starts to rotate in a horizontal circle. Given the moment of inertia of the stick and wax about the pivot is $0.02 \,kg m ^2$, the initial angular velocity of the stick is ........... $rad / s$

An electric bulb rated as $200\, {W}$ at $100 \,{V}$ is used in a circuit having $200\, {V}$ supply. The resistance $'R'$ that must be put in series with the bulb so that the bulb delivers the same power is $.....\,\Omega$

The position vectors of radius are $2\hat i + \hat j + \hat k$ and $2\hat i - 3\hat j + \hat k$ while those of linear momentum are $2\hat i + 3\hat j - \hat k.$ Then the angular momentum is

A telescope is used to observe two objects at a distance of $z = 10\ km$ which are $s = 0.12\ m$ apart and illuminated by light of wavelength $\lambda = 600\ nm$ . Estimate the diameter of the objective lens of the telescope if it can just resolve the two objects. Assume diameter $D > > \lambda $ and separation between objects $s < < z$ .....$cm$

Two ideal slits $S_1$ and $S_2$ are at a distance $d$ apart, and illuminated by light of wavelength $\lambda $ passing through an ideal source slit $S$ placed on the line through $S_2$ as shown. The distance between the planes of slits and the source slit is $D$. A screen is held at a distance $D$ from the plane of the slits. The minimum value of $d$ for which there is darkness at $O$ is