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

A body of mass \( 14 \, \text{kg} \) initially at rest explodes and breaks into three fragments of masses in the ratio \( 2:2:3 \). The two pieces of equal masses fly off perpendicular to each other with a speed of \( 18 \, \text{m/s} \) each. The velocity of the heavier fragment is ______ \( \text{m/s} \).

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

The velocity at which \( 6 \, \text{kg} \) mass (shown in figure) strikes the ground when it is released from a height of \( 6 \, \text{m} \) above the ground is _____ \( \text{m/s} \). Assume pulley is massless and string is light and inextensible. (Take \( g = 10 \, \text{m/s}^2 \))

Potential energy \( (V) \) versus distance \( (x) \) is given by the graph. Rank various regions as per the magnitudes of the force \( (F) \) acting on a particle from high to low.

A small bob \( A \) of mass \( m \) is attached to a massless rigid rod of length \( 1 \, \text{m} \) pivoted at point \( P \) and kept at an angle of \( 60^\circ \) with vertical as shown in figure. At distance of \( 1 \, \text{m} \) below point \( P \), an identical bob \( B \) is kept at rest on a smooth horizontal surface that extends to a circular track of radius \( R \) as shown in figure. If bob \( B \) just manages to complete the circular path of radius \( R \) upto a point \( Q \) after being hit elastically by bob \( A \), then radius \( R \) is ______ \( \text{m} \).

In case of vertical circular motion of a particle by a thread of length \( r \) if the tension in the thread is zero at an angle \( 30^{\circ} \) shown in figure, the velocity at the bottom point \( (A) \) of the circular path is: \( (g = \text{gravitational acceleration}) \).

Given below are two statements:
Statement I: For a mechanical system of many particles total kinetic energy is the sum of kinetic energies of all the particles.
Statement II: The total kinetic energy can be the sum of kinetic energy of the center of mass w.r.t to the origin and the kinetic energy of all the particles w.r.t. the centre of mass as the reference.
In the light of the above statements, choose the correct answer from the options given below:

Two blocks of masses \( 2 \, \text{kg} \) and \( 1 \, \text{kg} \) respectively, are tied to the ends of a string which passes over a light frictionless pulley as shown in the figure below. The masses are held at rest at the same horizontal level and then released. The distance traversed by the centre of mass in \( 2 \, \text{s} \) is _______ \( \text{m} \). (Take \( g=10 \, \text{m/s}^{2} \))

The position of center of mass of three masses \( 2 \, \text{kg} \), \( 3 \, \text{kg} \) and \( 15 \, \text{kg} \) placed with respect to mid point \( (p) \) of normal bisector, as shown in the figure is ______.

Two blocks with masses \( 100 \, \text{g} \) and \( 200 \, \text{g} \) are attached to the ends of springs A and B as shown in figure. The energy stored in A is \( E \). The energy stored in B, when spring constants \( k_A \), \( k_B \) of A and B, respectively satisfy the relation \( 4k_A = 3k_B \), is:

A body of mass \( 2 \, \text{kg} \) is moving along \( x \)-direction such that its displacement as function of time is given by \( x(t)=(\alpha t^{2}+\beta t+\gamma )\), where \( \alpha = 1 \, \text{m/s}^{2} \), \( \beta = 1 \, \text{m/s} \) and \( \gamma = 1 \, \text{m} \). The work done on the body during the time interval \( t=2 \, \text{s} \) to \( t=3 \, \text{s} \), is ________ \( \text{J} \).

A mass of \( 1 \, \text{kg} \) is kept on an inclined plane with \( 30^\circ \) 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 __________ \( \text{J} \). (Take \( g=10 \, \text{m/s}^2 \))

In a perfectly inelastic collision, two spheres made of the same material with masses \( 15 \, \text{kg} \) and \( 25 \, \text{kg} \), moving in opposite directions with speeds of \( 10 \, \text{m/s} \) and \( 30 \, \text{m/s} \), respectively, strike each other and stick together. The rise in temperature (in \( ^\circ \text{C} \)), if all the heat produced during the collision is retained by these spheres, is: [Specific heat of sphere material \( 31 \, \text{cal/(kg} \cdot ^\circ\text{C)} \) and \( 1 \, \text{cal} = 4.2 \, \text{J} \)]

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

The rain drop of mass \( 1\,\text{g} \) starts with zero velocity from a height of \( 1 \,\text{km} \). It hits the ground with a speed of \( 5\,\text{m/s} \). The work done by the unknown resistive force is ______ \( \text{J} \). (Take \( g=10\,\text{m/s}^2 \))

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JEE Main 2026 6th April Morning Shift

The position of center of mass of three masses \( 2 \, \text{kg} \), \( 3 \, \text{kg} \) and \( 15 \, \text{kg} \) placed with respect to mid point \( (p) \) of normal bisector, as shown in the figure is ______.