Power

A body of mass 2 kg begins to move under the influence of time dependent force $\vec F=\left(2t\hat i+6t^2\hat j\right)$ N, where $\hat i$ and $\hat j$ are unit vectors along $x$ and $y$-axis respectively. The power produced by the force at $t=2$ s is ________ W.

A particle of mass $m$ is moving in a circular path of constant radius $r$ such that its centripetal acceleration ($a$) is varying with time $t$ as $a=k^2rt^2$, where $k$ is a constant. The power delivered to the particle by the force acting on it is given as

Water falls from a 40 m high dam at the rate of $9\times10^4$ kg per hour. Fifty percentage of gravitational potential energy can be converted into electrical energy. Using this hydro electric energy number of 100 W lamps, that can be lit, is :

(Take $g=10$ ms$^{-2}$)

Sand is being dropped from a stationary dropper at a rate of 0.5 kgs$^{-1}$ on a conveyor belt moving with a velocity of 5 ms$^{-1}$. The power needed to keep the belt moving with the same velocity will be :

A particle of mass $m$ is initially at rest at the origin. It is subjected to a force and starts moving along the $x$-axis. Its kinetic energy $K$ changes with time as $dK/dt = \gamma t$, where $\gamma$ is a positive constant of appropriate dimensions. Which of the following statements is (are) true?

A person trying to lose weight by burning fat lifts a mass of 10 kg upto a height of 1 m 1000 times. Assume that the potential energy lost each time he lowers the mass is dissipated. How much fat will he use up considering the work done only when the weight is lifted up ? Fat supplies $3.8\times10^7$ J of energy per kg which is converted to mechanical energy with a 20% efficiency rate. Take g = 9.8 ms$^{-2}$ :

A car of weight W is on an inclined road that rises by 100 m over a distance of 1 km and applies a constant frictional force $\dfrac W{20}$ on the car. While moving uphill on the road at a speed of 10 ms$^{-1}$, the car needs power P. If it needs power $\dfrac P2$ while moving downhill at speed $v$ then value of $v$ is :

A particle of mass M is moving in a circle of fixed radius R in such a way that its centripetal acceleration at time t is given by $n^2Rt^2$ where n is a constant. The power delivered to the particle by the force acting on it, is :

A conductor lies along the z-axis at $-1.5\le z<1.5$ m and carries a fixed current of 10.0 A in $-\hat a_z$ direction (see figure). For a field $\vec B=3.0\times10^{-4}\,e^{-0.2x}\,\hat a_y$ T, find the power required to move the conductor at constant speed to $x=2.0$ m, $y=0$ m in $5\times10^{-3}$ s. Assume parallel motion along the $x$-axis.

A particle of mass 0.2 kg is moving in one dimension under a force that delivers a constant power 0.5 W to the particle. If the initial speed (in m s$^{-1}$) of the particle is zero, the speed (in m s$^{-1}$) after 5 s is