Work, Energy and Power

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 :

Velocity-time graph for a body of mass 10 kg is shown in figure. Work-done on the body in first two seconds of the motion is :

Two identical uniform discs roll without slipping on two different surfaces $AB$ and $CD$ (see figure) starting at $A$ and $C$ with linear speeds $v_1$ and $v_2$, respectively, and always remain in contact with the surfaces. If they reach $B$ and $D$ with the same linear speed and $v_1=3$ m/s, then $v_2$ in m/s is ($g=10\ \mathrm{m/s^2}$)

A particle of unit mass is moving along the $x$-axis under the influence of a force and its total energy is conserved. Four possible forms of the potential energy of the particle are given in column I ($a$ and $U_0$ are constants). Match the potential energies in column I to the corresponding statement(s) in column II.

An ideal monoatomic gas is confined in a horizontal cylinder by a spring loaded piston (as shown in the figure). Initially the gas is at temperature $T_1$, pressure $P_1$ and volume $V_1$ and the spring is in its relaxed state. The gas is then heated very slowly to temperature $T_2$, pressure $P_2$ and volume $V_2$. During this process the piston moves out by a distance $x$. Ignoring the friction between the piston and the cylinder, the correct statement(s) is(are)

A block of mass m = 0.1 kg is connected to a spring of unknown spring constant k. It is compressed to a distance $x$ from its equilibrium position and released from rest. After approaching half the distance $\left(\dfrac x2\right)$ from equilibrium position, it hits another block and comes to rest momentarily, while the other block moves with a velocity 3 ms$^{-1}$. The total initial energy of the spring is :

A particle is moving in a circle of radius r under the action of a force $F=\alpha r^2$ which is directed towards centre of the circle. Total mechanical energy (kinetic energy + potential energy) of the particle is (take potential energy = 0 for r = 0) :

Consider an elliptically shaped rail PQ in the vertical plane with OP = 3 m and OQ = 4 m. A block of mass 1 kg is pulled along the rail from P to Q with a force of 18 N, which is always parallel to line PQ (see the figure given). Assuming no frictional losses, the kinetic energy of the block when it reaches Q is $(n\times10)$ Joules. The value of $n$ is (take acceleration due to gravity $=10\ \mathrm{m\,s^{-2}}$)

A wire, which passes through the hole in a small bead, is bent in the form of quarter of a circle. The wire is fixed vertically on ground as shown in the figure. The bead is released from near the top of the wire and it slides along the wire without friction. As the bead moves from A to B, the force it applies on the wire is

When a rubber-band is stretched by a distance $x$, it exerts a restoring force of magnitude $F=ax+bx^2$ where $a$ and $b$ are constants. The work done in stretching the unstretched rubber-band by $L$ 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 small ball of mass m starts at a point A with speed $v_o$ and moves along a frictionless track AB as shown. The track BC has coefficient of friction $\mu$. The ball comes to stop at C after travelling a distance L which is :

The average mass of rain drops is $3.0\times10^{-5}$ kg and their average terminal velocity is 9 m/s. Calculate the energy transferred by rain to each square metre of the surface at a place which receives 100 cm of rain in a year.

A spring of unstretched length $l$ has a mass m with one end fixed to a rigid support. Assuming spring to be made of a uniform wire, the kinetic energy possessed by it if its free end is pulled with uniform velocity $v$ is :

A bullet looses $\left(\dfrac1n\right)^{\text{th}}$ of its velocity passing through one plank. The number of such planks that are required to stop the bullet can be :

A particle is released on a vertical smooth semicircular track from point X so that OX makes angle θ from the vertical (see figure). The normal reaction of the track on the particle vanishes at point Y where OY makes angle φ with the horizontal. Then :

The work done on a particle of mass $m$ by a force, $K\left[\dfrac{x}{\left(x^2+y^2\right)^{3/2}}\hat i+\dfrac{y}{\left(x^2+y^2\right)^{3/2}}\hat j\right]$ ($K$ being a constant of appropriate dimensions), when the particle is taken from the point $(a,0)$ to the point $(0,a)$ along a circular path of radius $a$ about the origin in the $x$-$y$ plane is

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

A bob of mass $m$, suspended by a string of length $l_1$, is given a minimum velocity required to complete a full circle in the vertical plane. At the highest point, it collides elastically with another bob of mass $m$ suspended by a string of length $l_2$, which is initially at rest. Both the strings are mass-less and inextensible. If the second bob, after collision acquires the minimum speed required to complete a full circle in the vertical plane, the ratio $\dfrac{l_1}{l_2}$ is