Work, Energy and Power — JEE Main PYQs

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 $g = 10 \text{ m/s}^2$)

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 $1 \text{ kg}$ 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 $4 \text{ m/s}$. The work done by the frictional force in time $2 \text{ s}$ is __________ J. (Take $g = 10 \text{ m/s}^2$)

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 $g = 10\ \text{m/s}^2$)

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 ________.

[Figure: An inclined plane of height $h$ leading down to a vertical circular loop of radius $R$ at its base]

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.

A particle is released from height S above the surface of the earth. At certain height its kinetic energy is three times its potential energy. The height from the surface of the earth and the speed of the particle at that instant are respectively.

[Figure: block A (m₁ = 10 kg) moving with velocity v towards block B (m₂ = 5 kg) which has a spring of k = 3000 N/m attached, both on a horizontal surface]

Consider two blocks A and B of masses $m_1 = 10$ kg and $m_2 = 5$ kg that are placed on a frictionless table. The block A moves with a constant speed $v = 3$ m/s towards the block B kept at rest. A spring with spring constant $k = 3000$ N/m is attached with the block B as shown in the figure. After the collision, suppose that the blocks A and B, along with the spring in constant compression state, move together, then the compression in the spring is, (Neglect the mass of the spring)

A block of mass 1 kg, moving along $x$ with speed $v_i = 10$ m/s enters a rough region ranging from $x = 0.1$ m to $x = 1.9$ m. The retarding force acting on the block in this range is $F_r = -kx$ N, with $k = 10$ N/m. Then the final speed of the block as it crosses rough region is.

A block of mass 25 kg is pulled along a horizontal surface by a force at an angle $45°$ with the horizontal. The friction coefficient between the block and the surface is 0.25. The block travels at a uniform velocity. The workdone by the applied force during a displacement of 5 m of the block is :

An object of mass 1000 g experiences a time dependent force $\vec{F} = \left(2t\,\hat{i} + 3t^2\,\hat{j}\right)N$. The power generated by the force at time t is:

Which one of the following forces cannot be expressed in terms of potential energy ?

A block of mass 2 kg is attached to one end of a massless spring whose other end is fixed at a wall. The spring-mass system moves on a frictionless horizontal table. The spring's natural length is 2 m and spring constant is 200 N/m. The block is pushed such that the length of the spring becomes 1 m and then released. At distance $x$ m ($x < 2$) from the wall, the speed of the block will be

A bob of mass m is suspended at a point O by a light string of length $l$ and left to perform vertical motion (circular) as shown in figure. Initially, by applying horizontal velocity $v_0$ at the point 'A', the string becomes slack when, the bob reaches at the point 'D'. The ratio of the kinetic energy of the bob at the points B and C is ________.

[Figure: vertical circle with centre O; A at the bottom, D at the top; points B (lower right) and C (upper right) on the circle with OB and OC each making 60° with the horizontal through O; horizontal velocity $v_0$ applied at A]

A ball of mass 100 g is projected with velocity 20 m/s at $60^{\circ}$ with horizontal. The decrease in kinetic energy of the ball during the motion from point of projection to highest point is

A body of mass 100 g is moving in circular path of radius 2 m on vertical plane as shown in figure. The velocity of the body at point A is 10 m/s. The ratio of its kinetic energies at point B and C is :

[Figure: a vertical circle with centre O; A is the lowest point, D the highest point; C lies above the horizontal diameter and B below it, with OB making $30^{\circ}$ with the horizontal and angle between OC and OB equal to $90^{\circ}$]

(Take acceleration due to gravity as $10\ \mathrm{m/s^2}$)

A force $\vec{F} = 2\hat{i} + b\hat{j} + \hat{k}$ is applied on a particle and it undergoes a displacement $\hat{i} - 2\hat{j} - \hat{k}$. What will be the value of b, if work done on the particle is zero.

A force $f = x^2y\hat{i} + y^2\hat{j}$ acts on a particle in a plane $x + y = 10$. The work done by this force during a displacement from $(0, 0)$ to $(4\,\mathrm{m}, 2\,\mathrm{m})$ is _____ Joule (round off to the nearest integer)

A ball having kinetic energy KE, is projected at an angle of $60°$ from the horizontal. What will be the kinetic energy of ball at the highest point of its flight ?

A force $F = \alpha + \beta x^2$ acts on an object in the x-direction. The work done by the force is 5 J when the object is displaced by 1 m. If the constant $\alpha = 1\,\mathrm{N}$ then $\beta$ will be

A bead of mass 'm' slides without friction on the wall of a vertical circular hoop of radius 'R' as shown in figure. The bead moves under the combined action of gravity and a massless spring (k) attached to the bottom of the hoop. The equilibrium length of the spring is 'R'. If the bead is released from top of the hoop with (negligible) zero initial speed, velocity of bead, when the length of spring becomes 'R', would be (spring constant is 'k', g is accleration due to gravity)

[Figure: a vertical circular hoop of diameter 2R resting on the ground, with a spring attached from the bottom of the hoop to the bead at the top]

Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R

Assertion A: In a central force field, the work done is independent of the path chosen.

Reason R: Every force encountered in mechanics does not have an associated potential energy.

In the light of the above statements, choose the most appropriate answer from the options given below

A body of mass 4 kg is placed on a plane at a point P having coordinate $(3, 4)$ m. Under the action of force $\vec{F} = \left(2\hat{i} + 3\hat{j}\right)\,\mathrm{N}$, it moves to a new point Q having coordinates $(6, 10)$ m in 4 sec. The average power and instantaneous power at the end of 4 sec are in the ratio of :

If a rubber ball falls from a height h and rebounds upto the height of h/2. The percentage loss of total energy of the initial system as well as velocity ball before it strikes the ground, respectively, are :

A body of $m$ kg slides from rest along the curve of vertical circle from point $A$ to $B$ in friction less path. The velocity of the body at $B$ is:

[Figure: a vertical circle with centre marked; point A on the upper right of the circle at $45°$ above the horizontal radius, point B at the bottom of the circle; the vertical diameter is shown dashed]

(given, $R = 14~m$, $g = 10~m/s^2$ and $\sqrt{2} = 1.4$ )

A body of mass 50 kg is lifted to a height of 20 m from the ground in the two different ways as shown in the figures. The ratio of work done against the gravity in both the respective cases, will be :

[Figure: Case-1 — block of M = 50 kg pulled straight up a height h = 20 m; Case-2 — the same mass moved along a ramp inclined at 30° to the same height h = 20 m]

A body is moving unidirectionally under the influence of a constant power source. Its displacement in time t is proportional to :

Four particles $A, B, C, D$ of mass $\frac{m}{2}, m, 2m, 4m$, have same momentum, respectively. The particle with maximum kinetic energy is :

A bullet of mass $50g$ is fired with a speed 100 m/s on a plywood and emerges with 40 m/s. The percentage loss of kinetic energy is :

When kinetic energy of a body becomes 36 times of its original value, the percentage increase in the momentum of the body will be :

A stationary particle breaks into two parts of masses $m_A$ and $m_B$ which move with velocities $v_A$ and $v_B$ respectively. The ratio of their kinetic energies $(K_B : K_A)$ is :

Three bodies A, B and C have equal kinetic energies and their masses are 400 g, 1.2 kg and 1.6 kg respectively. The ratio of their linear momenta is :

[Figure: a 5 kg block at the top of a frictionless ($\mu = 0$) inclined plane of length 10 m at $30°$; at the bottom, a horizontal surface of length 2 m with $\mu = 0.5$ leads to a spring of $k = 100\ \mathrm{N/m}$ fixed to a wall]

A block is simply released from the top of an inclined plane as shown in the figure above. The maximum compression in the spring when the block hits the spring is :

A particle of mass m moves on a straight line with its velocity increasing with distance according to the equation $v = \alpha\sqrt{x}$, where $\alpha$ is a constant. The total work done by all the forces applied on the particle during its displacement from $x = 0$ to $x = d$, will be :

A force $(3x^2 + 2x - 5)$ N displaces a body from $x = 2$ m to $x = 4$ m. Work done by this force is ________ $J$.

A simple pendulum of length 1 m has a wooden bob of mass 1 kg. It is struck by a bullet of mass $10^{-2}$ kg moving with a speed of $2 \times 10^{2}\ \mathrm{ms^{-1}}$. The bullet gets embedded into the bob. The height to which the bob rises before swinging back is. (use $g = 10\ \mathrm{m/s^2}$)

Two bodies of mass 4 g and 25 g are moving with equal kinetic energies. The ratio of magnitude of their linear momentum is :

A block of mass 100 kg slides over a distance of 10 m on a horizontal surface. If the co-efficient of friction between the surfaces is 0.4, then the work done against friction (in $J$) is :

The potential energy function (in $J$) of a particle in a region of space is given as $U = (2x^2 + 3y^3 + 2z)$. Here $x$, $y$ and $z$ are in meter. The magnitude of $x$-component of force (in $N$) acting on the particle at point $P$ (1, 2, 3) m is :

A particle is placed at the point A of a frictionless track ABC as shown in figure. It is gently pushed towards right. The speed of the particle when it reaches the point B is : (Take $\mathrm{g} = 10~\mathrm{m/s^2}$).

[Figure: a curved track with point A at height 1 m, point B at height 0.5 m inside a valley, and point C at the top on the right]

A block of mass $1$ kg is pushed up a surface inclined to horizontal at an angle of $60°$ by a force of $10\ N$ parallel to the inclined surface as shown in figure. When the block is pushed up by $10\ m$ along inclined surface, the work done against frictional force is :

[$g = 10\ m/s^2$]

[Figure: incline at $60°$ to the horizontal with block $M$ on it, coefficient of static friction $\mu_s = 0.1$, force of $10\ N$ applied up along the incline]

An artillery piece of mass $M_1$ fires a shell of mass $M_2$ horizontally. Instantaneously after the firing, the ratio of kinetic energy of the artillery and that of the shell is:

A body of mass 2 kg begins to move under the action of a time dependent force given by $\vec{F} = \left(6t\,\hat{i} + 6t^2\,\hat{j}\right) N$. The power developed by the force at the time $t$ is given by: