Work, Energy and Power

A solid cylinder of radius $R$ rolls without slipping with a center of mass speed $v_0=\sqrt{\dfrac{gR}3}$ on a horizontal surface with a vertical edge, as shown in the figure. Here, $g$ is the acceleration due to the gravity. At the moment when the cylinder loses contact with the surface due to rotation around the corner, the speed of its center of mass is:

After the collision the disk starts to rotate around point $C$ in the $XY$ plane. The maximum change in the height (in m) of its center $O$ is:

A spherical ball of mass 2 kg falls from a height of 10 m and is brought to rest after penetrating 10 cm into sand. The average force exerted by sand on the ball is ________ N.
(Take $g=10$ m/s²)

A 1 kg block subjected to two simultaneous forces $\left(2\hat i+3\hat j+4\hat k\right)$ N and $\left(3\hat i-\hat j-2\hat k\right)$ N is moved a distance of 25 m along $\left(3\hat i-4\hat j\right)$ direction. The work done in this process is ________ J.

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 mass of 1 kg is kept on a inclined plane with 30° inclination with respect to horizontal plane and it is at rest initially. Then the whole assembly is moved up with constant velocity of 4 m/s. The work done by the frictional force in time 2 s is ________ J. (Take $g=10$ m/s²)

A particle of charge $q$ and mass $m$ is projected from origin with an initial velocity $\vec v=\left(\dfrac{v_0}{\sqrt2}\hat x+\dfrac{v_0}{\sqrt2}\hat y\right)$. There exists a uniform magnetic field $\vec B=B_0\hat z$ and a space varying electric field $\vec E=E_0e^{-\lambda x}\hat x$ within the region $0\le x\le L$. After travelling a distance such that $x$-coordinate has changed from $x=0$ to $x=L$, the change in the kinetic energy is ________.

The rain drop of mass 1 g, starts with zero velocity from a height of 1 km. It hits the ground with a speed of 5 m/s. The work done by the unknown resistive force is ________ J.

(take $g=10$ m/s²)

A smooth inclined plane ends in a vertical circular loop, as shown in the figure. A small body is released from height $h$ as shown. If the body exerts a force of three times its weight on the plane at the highest point of circle then the height $h=\alpha R$. The value of $\alpha$ is ________.

A body of mass 1 kg moves along a straight line with a velocity $v=2x^2$. The work done by the body during displacement from $x=0$ to 5 m is ________ J.

An audio transmitter (T) and a receiver (R) are hung vertically from two identical massless strings of length 8 m with their pivots well separated along the $X$ axis. They are pulled from the equilibrium position in opposite directions along the $X$ axis by a small angular amplitude $\theta_0 = \cos^{-1}(0.9)$ and released simultaneously. If the natural frequency of the transmitter is 660 Hz and the speed of sound in air is 330 m/s, the maximum variation in the frequency (in Hz) as measured by the receiver (Take the acceleration due to gravity $g = 10\ \mathrm{m/s^2}$) is _____.

A particle of mass $m$ is under the influence of the gravitational field of a body of mass $M\ (\gg m)$. The particle is moving in a circular orbit of radius $r_0$ with time period $T_0$ around the mass $M$. Then, the particle is subjected to an additional central force, corresponding to the potential energy $V_c(r) = m\alpha/r^3$, where $\alpha$ is a positive constant of suitable dimensions and $r$ is the distance from the center of the orbit. If the particle moves in the same circular orbit of radius $r_0$ in the combined gravitational potential due to $M$ and $V_c(r)$, but with a new time period $T_1$, then $(T_1^2 - T_0^2)/T_1^2$ is given by

[$G$ is the gravitational constant.]

A table tennis ball has radius $(3/2)\times10^{-2}\,\mathrm{m}$ and mass $(22/7)\times10^{-3}\,\mathrm{kg}$. It is slowly pushed down into a swimming pool to a depth of $d = 0.7\,\mathrm{m}$ below the water surface and then released from rest. It emerges from the water surface at speed $v$, without getting wet, and rises up to a height $H$. Which of the following option(s) is(are) correct?

[Given: $\pi = 22/7$, $g = 10\,\mathrm{m\,s^{-2}}$, density of water $= 1\times10^3\,\mathrm{kg\,m^{-3}}$, viscosity of water $= 1\times10^{-3}$ Pa-s.]

A slide with a frictionless curved surface, which becomes horizontal at its lower end, is fixed on the terrace of a building of height $3h$ from the ground, as shown in the figure. A spherical ball of mass $m$ is released on the slide from rest at a height $h$ from the top of the terrace. The ball leaves the slide with a velocity $\vec u_0 = u_0\hat x$ and falls on the ground at a distance $d$ from the building making an angle $\theta$ with the horizontal. It bounces off with a velocity $\vec v$ and reaches a maximum height $h_1$. The acceleration due to gravity is $g$ and the coefficient of restitution of the ground is $1/\sqrt3$. Which of the following statement(s) is(are) correct?

A disk of radius $R$ with uniform positive charge density $\sigma$ is placed on the $xy$ plane with its center at the origin. The Coulomb potential along the $z$-axis is
$$V(z) = \frac{\sigma}{2\epsilon_0}\left(\sqrt{R^2 + z^2} - z\right).$$
A particle of positive charge $q$ is placed initially at rest at a point on the $z$ axis with $z = z_0$ and $z_0 > 0$. In addition to the Coulomb force, the particle experiences a vertical force $\vec F = -c\,\hat k$ with $c > 0$. Let $\beta = \dfrac{2c\epsilon_0}{q\sigma}$. Which of the following statement(s) is(are) correct?

Given below are two statements : One is labelled as Assertion (A) and the other is labelled as Reason (R).

Assertion (A) : In an uniform magnetic field, speed and energy remains the same for a moving charged particle.

Reason (R) : Moving charged particle experiences magnetic force perpendicular to its direction of motion.

A particle experiences a variable force $\vec F=\left(4x\hat i+3y^2\hat j\right)$ in a horizontal $x$-$y$ plane. Assume distance in meters and force is newton. If the particle moves from point $(1,2)$ to point $(2,3)$ in the $x$-$y$ plane, then Kinetic Energy changes by :

A ball of mass 100 g is dropped from a height $h=10$ cm on a platform fixed at the top of a vertical spring (as shown in figure). The ball stays on the platform and the platform is depressed by a distance $\dfrac h2$. The spring constant is ______ $\mathrm{Nm^{-1}}$.

(Use $g=10\ \mathrm{ms^{-2}}$)

A stone of mass m, tied to a string is being whirled in a vertical circle with a uniform speed. The tension in the string is

Potential energy as a function of $r$ is given by $U=\dfrac A{r^{10}}-\dfrac B{r^5}$, where r is the interatomic distance. A and B are positive constants. The equilibrium distance between the two atoms will be :