Rotational Power

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.

To maintain a rotor at a uniform angular speed of 200 rad/s, an engine needs to transmit a torque of 180 N m. What is the power required by the engine? (Note: Uniform angular velocity in the absence of friction implies zero torque. In practice, applied torque is needed to counter frictional torque). Assume that the engine is 100% efficient.

Q. Engine of a car supplies constant power to the wheels of car, then

1. displacement of the car is proportional to $t^2$.
2. acceleration of the car is proportional to $t^{-1}$.
3. velocity of the car is proportional to $t^{1/2}$.
4. displacement of the car is proportional to $t^3$.

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