Work and Power

Show that the vector field defined by the vector function $E=xyz\left(yz\hat{i}+xz\hat{j}+xy\hat{k}\right)$ is conservative.

A cylinder of mass $M$ and radius $R$ starts falling freely under gravity at $t=0$ as shown in figure. The tension in each string at any given time $t$ is $T.$ The instantaneous power developed by the gravitational force at time $t$ is $P.$ The linear acceleration of the cylinder is a. Which of the following are correct?

A $3 kg$ object has initial velocity $\left(6\widehat{\mathrm{i}}-2\widehat{\mathrm{j}}\right) \mathrm{m} {\mathrm{s}}^{-1}.$ The total work done $\left(\mathrm{in} \mathrm{J}\right)$ on the object if its velocity changes to $(8\widehat{\mathrm{i}}+4\widehat{\mathrm{j}}) \mathrm{m} {\mathrm{s}}^{-1}$ is

In the figure shown, initially the spring of negligible mass is in undeformed state and the block has zero velocity $E$ is a uniform electric field. Then :

The power output from a certain experimental car which is designed in shape like a cube is proportional to the mass $m$ of the car. The force of air friction on the car is proportional to $A{v}^2$, where $v$ is the speed of the car and $A$ the cross sectional area. On a level surface the car has a maximum speed $v_{\max }$. Assuming that all geometrically similar versions of this design have the same density, then which of the following is true?

A block of mass $2 \mathrm{kg}$ is attached to one end of a massless rod of length $\frac{1}{\mathrm{\pi }}\mathrm{m}$. The rod is fixed to a horizontal plane at the other end such that the block and rod are free to revolve on a horizontal plane. The coefficient of friction between the block and surface is $0.1$. Block is made to rotate with uniform angular speed by applying a constant external force in tangential direction on the block. The work done by external force till the rod rotates by $90°$ is

A force $\overrightarrow{F}=\left(y^2-x^2+z^2\right)\hat{i}+\left(3xy-5z\right)\hat{j}+4zk$ is applied on a particle. The work done by the force when the particle moves from the point $\left(0,0,0\right)$ to the point $\left(2,4,0\right)$ along the path shown in figure in S.I. units.

A block of mass$1 \mathrm{kg}$ is pulled along the curve path $ACB$ by a tangential force as shown in figure. The work done by the frictional force when the block moves from $A$ to $B$ is

A uniform, rigid, mass less (light) rod $AB$ of length ''$l$'" has two equal, masses "$m$" each as show. The rod is smoothly hinged at $A$.$C$ is the midpoint of the rod. The rod rotates in the vertical plane under critical condition.

Johnny and his sister Jane race up a hill. Johnny weighs twice as much as Jane and takes twice the time as long as Jane to reach the top. Compared to Jane:

A block of mass $50 \mathrm{kg}$ just slides over a distance of $1 \mathrm{m}$. If the coefficient of friction between their surfaces is $0.2$, the work done against friction is

A cubical vessel of height $1 \mathrm{m}$ is full of water. What is the amount of work done in pumping water out of the vessel?

[Take $g=10 {\mathrm{ms}}^{-2}$]

When a force is applied on a moving body, its motion is retarded. Then the work done is

Water falls down a $500.0 \mathrm{m}$ shaft to reach a turbine which generates electricity. How much water must fall per second in order to generate $1.00\times {10}^9$ watts of power ? (Assume $50\%$ efficiency of conversion and $g=10 \mathrm{m} {\mathrm{s}}^{-1}$)

A girl of mass $50 \mathrm{kg}$ swinging on a cradle. If she moves with a velocity of $2 \mathrm{m} {\mathrm{s}}^{-1}$ upwards in a direction making an angle $60°$ with the vertical, then the power generated is $\left(g=9.8 \mathrm{m} {\mathrm{s}}^{-2}\right)$

A body is initially at rest. It undergoes one-dimensional motion with constant acceleration. The power delivered to it at time $t$ is proportional to:

An engine develops $10 \mathrm{kW}$ of power. The time it will take in lifting a mass of  $200 \mathrm{kg}$ to a height of $40 \mathrm{m}. (\mathrm{g}=10 \mathrm{m}/{\mathrm{s}}^2)$ is: