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A ball of mass \( 100 \, \text{g} \) is dropped from a height \( h = 10 \, \text{cm} \) 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 \( \frac{h}{2} \). The spring constant is ______________ \( \text{N m}^{-1} \). (Use \( g = 10 \, \text{m s}^{-2} \))

A block of mass 'm' (as shown in figure) moving with kinetic energy \( E \) compresses a spring through a distance \( 25 \, \text{cm} \) when, its speed is halved. The value of spring constant of used spring will be \( nE \, \text{N m}^{-1} \) for \( n = \) __________.

In the given figure, the block of mass \( m \) is dropped from the point ‘A’. The expression for kinetic energy of block when it reaches point ‘B’ is:

A mass \( 0.9 \, \text{kg} \), attached to a horizontal spring, executes SHM with an amplitude \( A_{1} \). When this mass passes through its mean position, then a smaller mass of \( 124 \, \text{g} \) is placed over it and both masses move together with amplitude \( A_{2} \). If the ratio \( {A_{1}}/{A_{2}} \) is \( \dfrac{\alpha}{\alpha - 1} \), then the value of \( \alpha \) will be ______.

A car covers AB distance with first one-third at velocity \( v_1 \, \text{ms}^{-1} \), second one-third at \( v_2 \, \text{ms}^{-1} \) and last one-third at \( v_3 \, \text{ms}^{-1} \). If \( v_3 = 3v_1 \), \( v_2 = 2v_1 \) and \( v_1 = 11 \, \text{ms}^{-1} \) then the average velocity of the car is ________ \( \text{ms}^{-1} \).

The identical spheres each of mass \( M \) are placed at the corners of a right angled triangle with mutually perpendicular sides equal to \( 3 \, \text{m} \) each. Taking point of intersection of mutually perpendicular sides as origin, the magnitude of position vector of centre of mass of the system will be \( \sqrt{x} \, \text{m} \). The value of the \( x \) is:

A bullet of mass \( 200\,\text{g} \) having initial kinetic energy \( 90\,\text{J} \) is shot inside a long swimming pool as shown in the figure. If it’s kinetic energy reduces to \( 40\,\text{J} \) within \( 1\,\text{s} \), the minimum length of the pool, the bullet has to travel so that it completely comes to rest is:

A particle of mass \( 500 \, \text{gm} \) is moving in a straight line with velocity \( v = b x^{5/2} \). The work done by the net force during its displacement from \( x = 0 \) to \( x = 4 \, \text{m} \) is: (Take \( b = 0.25 \, \text{m}^{-3/2}\, \text{s}^{-1} \))

A ball is released from rest from point \( P \) of a smooth semi-spherical vessel as shown in figure. The ratio of the centripetal force and normal reaction on the ball at point \( Q \) is \( A \) while angular position of point \( Q \) is \( \alpha \) with respect to point \( P \). Which of the following graphs represent the correct relation between \( A \) and \( \alpha \) when ball goes from \( Q \) to \( R \)?

A stone tide to a string of length \( L \) is whirled in a vertical circle with the other end of the string at the centre. At a certain instant of time, the stone is at its lowest position and has a speed \( u \). The magnitude of change in its velocity, as it reaches a position where the string is horizontal, is \( \sqrt{ \, x \left( u^{2} - g L \right) } \). The value of \( x \) is-

A pendulum is suspended by a string of length \( 250 \, \text{cm} \). The mass of the bob of the pendulum is \( 200 \, \text{g} \). The bob is pulled aside until the string is at \( 60^\circ \) with Vertical as shown in the figure. After releasing the bob, the maximum velocity attained by the bob will be ________ \( \text{m s}^{-1} \). (if \( g = 10 \, \text{m /s}^{2} \))

A block of metal weighing \( 2 \, \text{kg} \) is resting on a frictionless plane (as shown in figure). It is struck by a jet releasing water at a rate of \( 1 \, \text{kg s}^{-1} \) and at a speed of \( 10 \, \text{m s}^{-1} \). Then, the initial acceleration of the block, in \( \text{m s}^{-2} \), will be:

The distance of centre of mass from end A of a one dimensional rod (AB) having mass density \( \rho = \rho_0 \left(1 - \dfrac{x^2}{L^2}\right) \, \text{kg/m} \) and length \( L \) (in meter) is \( \dfrac{3L}{\alpha} \, \text{m} \). The value of \( \alpha \) is ________. (where \( x \) is the distance from end A)

Arrange the four graphs in descending order of total work done; where \( W_1 \), \( W_2 \), \( W_3 \) and \( W_4 \) are the work done corresponding to figure a, b, c and d respectively.

A bag of sand of mass \( 9.8 \, \text{kg} \) is suspended by a rope. A bullet of \( 200 \, \text{g} \) travelling with speed \( 10 \, \text{ms}^{-1} \) gets embedded in it, then loss of kinetic energy will be:

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

A body of mass \( 8 \, \text{kg} \) and another of mass \( 2 \, \text{kg} \) are moving with equal kinetic energy. The ratio of their respectively momenta will be:

A block of mass \( 2 \, \text{kg} \) moving on a horizontal surface with speed of \( 4 \, \text{m s}^{-1} \) enters a rough surface ranging from \( x = 0.5 \, \text{m} \) to \( x = 1.5 \, \text{m} \). The retarding force in this range of rough surface is related to distance by \( F = -kx \) where \( k = 12 \, \text{N m}^{-1} \). The speed of the block as it just crosses the rough surface will be:

A batsmen hits back a ball of mass \( 0.4 \, \text{kg} \) straight in the direction of the bowler without changing its initial speed of \( 15 \, \text{m s}^{-1} \). The impulse imparted to the ball is \( \_\_\_\_\_ \, \text{N s} \).

As per the given figure, two blocks each of mass \( 250 \, \text{g} \) are connected to a spring of spring constant \( 2 \, \text{N} \, \text{m}^{-1} \). If both are given velocity \( v \) in opposite directions, then maximum elongation of the spring is:

A \( 0.5 \, \text{kg} \) block moving at a speed of \( 12 \, \text{m s}^{-1} \) compresses a spring through a distance \( 30 \, \text{cm} \) when its speed is halved. The spring constant of the spring will be ________ \( \text{N m}^{-1} \).

A body of mass \( M \) at rest explodes into three pieces, in the ratio of masses \( 1:1:2 \). Two smaller pieces fly off perpendicular to each other with velocities of \( 30 \, \text{m s}^{-1} \) and \( 40 \, \text{m s}^{-1} \) respectively. The velocity of the third piece will be:

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

Two bodies of mass \( 1\,\text{kg} \) and \( 3\,\text{kg} \) have position vectors \( \hat{i}+2\hat{j}+\hat{k} \) and \( -3\hat{i}-2\hat{j}+\hat{k} \) respectively. The magnitude of position vector of centre of mass of this system will be similar to the magnitude of vector:

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 \( a = k^{2} r t^{2} \), where \( k \) is a constant. The power delivered to the particle by the force acting on it is given as:

A pendulum of length \( 2 \, \text{m} \) consists of a wooden bob of mass \( 50 \, \text{g} \). A bullet of mass \( 75 \, \text{g} \) is fired towards the stationary bob with a speed \( v \). The bullet emerges out of the bob with a speed \( \dfrac{v}{3} \) and the bob just completes the vertical circle. The value of \( v \) is ______ \( \text{m s}^{-1} \). (if \( g = 10 \, \text{m} / \text{s}^{2} \) ).

If momentum of a body is increased by \( 20\% \), then its kinetic energy increases by

In two different experiments, an object of mass \( 5 \, \text{kg} \) moving with a speed of \( 25 \, \text{m s}^{-1} \) hits two different walls and comes to rest within (i) \( 3 \) second, (ii) \( 5 \) seconds, respectively.
Choose the correct option out of the following:

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:

Two blocks of masses \( 10 \, \text{kg} \) and \( 30 \, \text{kg} \) are placed on the same straight line with coordinates \( (0, 0) \, \text{cm} \) and \( (x, 0) \, \text{cm} \) respectively. The block of \( 10 \, \text{kg} \) is moved on the same line through a distance of \( 6 \, \text{cm} \) towards the other block. The distance through which the block of \( 30 \, \text{kg} \) must be moved to keep the position of centre of mass of the system unchanged is:

A balloon has mass of \( 10 \, \text{g} \) in air. The air escapes from the balloon at a uniform rate with velocity \( 4.5 \, \text{cm}/\text{s} \). if the balloon shrinks in \( 5 \, \text{s} \) completely. Then, the average force acting on that balloon will be (in dyne).

An object is thrown vertically upwards. At its maximum height, which of the following quantity becomes zero?

A body of mass \( 0.5 \, \text{kg} \) travels on straight line path with velocity \( v=(3x^{2}+4)\, \text{m/s} \). The net workdone by the force during its displacement from \( x=0 \) to \( x=2\, \text{m} \) is:

What percentage of kinetic energy of a moving particle is transferred to a stationary particle when it strikes the stationary particle of \( 5 \) times its mass?
(Assumes the collision to be head-on elastic collision)

A man of \( 60 \, \text{kg} \) is running on the road and suddenly jumps into a stationary trolly car of mass \( 120 \, \text{kg} \). Then, the trolly car starts moving with velocity \( 2 \, \text{m}\,\text{s}^{-1} \). The velocity of the running man was ________ \( \text{m}\,\text{s}^{-1} \), when he jumps into the car.

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JEE Main 2022 29th June Evening Shift

In the given figure, the block of mass \( m \) is dropped from the point ‘A’. The expression for kinetic energy of block when it reaches point ‘B’ is: