Rotational Power

Q. A small block slides down from the top of hemisphere of radius $R=3\text{m}$ as shown in the figure.The height ${}^{\prime }{h}^{\prime }$ at which the block will lose contact with the surface of the sphere is $\text{(in m)}$.
(Assume there is no friction between the block and the hemisphere)

Q. Consider a situation in which a ring, a solid cylinder and a solid sphere roll down on the same inclined plane without slipping. Assume that they start rolling from rest and having identical diameter. The correct statement for this situation is:

1. All of them will have same velocity.
2. The cylinder has the greatest and the sphere has the least velocity of the centre of mass at the bottom of the inclined plane.
3. The ring has the greatest and the cylinder has the least velocity of the centre of mass at the bottom of the inclined plane.
4. The sphere has the greatest and the ring has the least velocity of the centre of mass at the bottom of the inclined plane.

Q. An elevator in a building can carry a maximum of $10$ persons, with the average mass of each person being $68\text{kg}$. The mass of the elevator itself is $920\text{kg}$ and it moves with a constant speed of $3{\text{ms}}^{-1}$. The frictional force opposing the motion is $6000\text{N}$. If the elevator is moving up with its full capacity, the power delivered by the motor to the elevator $(g=10{\text{ms}}^{-2})$ must be at least

1. $56300\text{W}$
2. $62360\text{W}$
3. $48000\text{W}$
4. $66000\text{W}$

Q. An automobile engine develops $100\text{hp}$ when rotating at a speed of $1800\text{rad/min}$. The torque delivered by the engine is:

1. $5487\text{W-s}$
2. $4487\text{W-s}$
3. $3487\text{W-s}$
4. $2487\text{W-s}$

Q.

Test if the following equations are dimensionally correct :

(a) $h=\frac{2Scos\theta }{\rho rg}$,

(b) $u=\sqrt{\frac{P}{rho}}$,

(c) $V=\frac{\pi P{r}^{4}t}{8\eta l}$,

(d) $v=\frac{1}{2\pi \sqrt{\frac{mgl}{I}}}$;

where h = height, S = surface tension, $\rho$ = density, P = pressure, V = volume, $\eta$ = coefficient of viscosity, v = frequency and I = moment of inertia.

Q.

The engine of a truck moving along a straight road delivers constant power. The distance travelled by the truck in time $t$ is proportional to

1. $t$

2. $t^2$

3. $\sqrt{t}$

4. $t^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$

Q. A uniform solid cylinder of radius $R=15\text{cm}$ rolls over a horizontal plane passing into an inclined plane forming an angle $\alpha ={30}^{\circ }$ with the horizontal. Find the maximum value of the velocity $v_{\circ }$ which still permits the cylinder to roll onto the inclined plane section without a jump. Assume there is no sliding between the cylinder and surfaces.

1. $1\text{m/s}$
2. 
   $0.5\text{m/s}$
3. $2\text{m/s}$
4. $4\text{m/s}$

Q.

A motorboat moves at a steady speed of$6m{s}^{-1}$ .Water resistance acting on it is $500N$ . Calculate the power of the engine.

Q. A puck of mass $m=1.50\text{kg}$ rotates in a circle of radius $r=20\text{cm}$ on a frictionless table while attached to a hanging cylinder of mass $M=2.50\text{kg}$ by means of a cord that extends through a hole in the table as shown in the figure. What speed of the puck is required to keep the cylinder at rest?

1. $0.85\text{m/s}$
2. $1.81\text{m/s}$
3. $2.52\text{m/s}$
4. $3.62\text{m/s}$

Q. A cone-shaped funnel is being rotated with constant angular velocity $\omega$. An object is placed on the inner wall of the funnel. The object can freely move along the surface of the cone, but during the rotation of the cone, it is in a state of equilibrium. Choose the correct statement(s).

1. Radius of circular path of the body $=\frac{g\cot \theta }{{\omega }^2}$
2. Radius of circular path of the body $=\frac{g\tan \theta }{{\omega }^2}$
3. The equilibrium is stable.
4. The equilibrium is unstable.

Q. Calculate the torque developed by an airplane engine whose output is $3200\text{HP}$ at an angular velocity of $2400\text{rev/min}$. $(1\text{HP}=746\text{J/s})$

1. $1273\text{N-m}$
2. $2984\text{N-m}$
3. $5000\text{N-m}$
4. $9498\text{N-m}$

Q. A force $F=10\text{N}$ is acting on pulley of mass $M=2\text{kg}$ and radius $R=1\text{m}$ as shown in the figure. If pulley is rotating with constant angular velocity $\omega =4\text{rad/s}$. Find the power delivered by the force $F$ to the pulley.

1. $20\text{J/s}$
2. $10\text{J/s}$
3. $30\text{J/s}$
4. $40\text{J/s}$

Q. Assertion: A particle of mass $m$ takes uniform horizontal circular motion inside a smooth funnel as shown. Normal reaction in this case is not $mg\cos \theta$.
Reason: Acceleration of particle is not along the surface of funnel.

1. Both Assertion and Reason are true and the Reason is correct explanation of the Assertion.
2. Both Assertion and Reason are true but Reason is not the correct explanation of Assertion.
3. Assertion is true, but the Reason is false.
4. Assertion is false, but the Reason is true.

Q. A force $F=10\text{N}$ is acting on pulley of mass $M=2\text{kg}$ and radius $R=1\text{m}$ as shown in the figure. If pulley is rotating with constant angular velocity $\omega =4\text{rad/s}$. Find the power delivered by the force $F$ to the pulley.

1. $20\text{J/s}$
2. $10\text{J/s}$
3. $30\text{J/s}$
4. $40\text{J/s}$

Q. A homogeneous solid cylindrical roller of radius $R$ and mass $M$ is pulled on a cricket pitch by a horizontal force. Assuming rolling without slipping, angular acceleration of the cylinder is:

1. $\frac{2F}{3mR}$
2. $\frac{3F}{2mR}$
3. $\frac{F}{3mR}$
4. $\frac{F}{2mR}$

Q. A mass ${}^{\prime }{m}^{\prime }$ supported by a massless string wound around a uniform hollow cylinder of mass $m$ and radius $R$. If the string does not slip on the cylinder, with what acceleration will the mass fall on release ?

1. g
2. $\frac{2g}{3}$
3. $\frac{5g}{6}$
4. $\frac{g}{2}$

Q. The angular velocity of a body is $\overrightarrow{\omega }=2\hat{i}+3\hat{j}+4\hat{k}$ and a torque $\overrightarrow{\tau }=\hat{i}+2\hat{j}+3\hat{k}$ acts on it. The rotational power will be

1. 20 W
2. 15 W
3. $\sqrt{17}W$
4. $\sqrt{14}W$

Q. Calculate the torque developed (in Nm) by an airplane engine whose output is $2000HP$ at an angular velocity of $2400rev/min$.

Q. Which of the following pairs of physical quantities have same dimensions?

1. Force and power
2. Torque and energy
3. Torque and power
4. Force and torque

Q. Due to application of a torque having a magnitude of 20 N-m, a ring is rotating at an angular velocity of 250 $\frac{rad}{s}$ at any given instant. What is the power delivered at this moment?

1. 2500 J/s
2. 1000 J/s
3. 1250 J/s
4. 5000 J/s

Q. A $70kg$ man stands in contact against the inner wall of a hollow cylindrical drum of radius $3m$ rotating about its vertical axis. The coefficient of friction between the wall and his clothing is $0.15$. What is the minimum rotational speed in $rad/s$ of the cylinder to enable the man to remain stuck to the wall (without falling) when the floor is suddenly removed? Take $g=10m{s}^{-2}$

Q. An electric motor rotates a wheel at a constant angular velocity $\omega$ while opposing torque is $\tau$. The power of that electric motor is :

1. $\frac{\tau \omega }{2}$
2. $\tau \omega$
3. $\frac{\tau }{\omega }$
4. $2\tau \omega$

Q. An automobile engine develops 100 kW when rotating at a speed of 1800 rev/min. What torque does it deliver

1. 350 N-m
2. 440 N-m
3. 531 N-m
4. 628 N-m

Q. A cubical block of side a is moving with velocity v on a horizontal smooth plane as shown in figure. It hits a ridge at point O. The angular speed of the block after it hits O is:

1. $\frac{3v}{4a}$
2. $\frac{3v}{2a}$
3. $\sqrt{\frac{3}{2}}a$
4. zero

Q. Calculate the torque developed by an airplane engine whose output is $3200\text{HP}$ at an angular velocity of $2400\text{rev/min}$. $(1\text{HP}=746\text{J/s})$

1. $9498\text{N-m}$
2. $1273\text{N-m}$
3. $2984\text{N-m}$
4. $5000\text{N-m}$

Q. Two metal spheres of the same material and radius $r$ are in contact with each other. The gravitational force of attraction between the spheres is given by:

1. $F=K{r}^4$
2. $F=K/r^3$
3. $F=K/4r^2$
4. $F=k{r}^2$

Q. A hollow sphere is projected horizontally along a rough surface with speed v and angular velocity ${\omega }_0$ find out the ratio $\frac{v}{{\omega }_0}$. So that the sphere stop moving after some time.

1. $\frac{2R}{3}$
2. $\frac{R}{3}$
3. $\frac{5R}{6}$
4. $\frac{2R}{7}$

Q. A ballet dancer is rotating about his own vertical axis. Without external torque if his angular velocity is doubled then his rotational kinetic energy is:

1. halved
2. quadrupled
3. unchanged
4. doubled

Q. A bullet dancer spins about a vertical axis at 60 rpm with his arm closed. Now he stretches his arms such that M.I increases by 50%. The new speed of revolution is

1. $80rpm$
2. $40rpm$
3. $90rpm$
4. $30rpm$

Q. A highly conducting solid sphere of radius R, density $\rho$ and specific heat is kept in an evacuated chamber. A parallel beam of electromagnetic radiation having uniform intensity I is incident on its surface. Assume that the surface of the sphere is perfectly black . Calculate the rate of increase of temperature

1. $\frac{3(I-2\sigma T^4)}{2R\rho s}$
2. $\frac{3(I-4\sigma T^4)}{4R\rho s}$
3. $\frac{4(I-4\sigma T^4)}{4R\rho s}$
4. $\frac{3(I+4\sigma T^4)}{4R\rho s}$