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A block of mass \( 25 \, \text{kg} \) is pulled along a horizontal surface by a force at an angle \( 45^\circ \) 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 \( 5 \, \text{m} \) of the block is:

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

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

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 __________.

A block of mass \( 2 \, \text{kg} \) 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 \( 2 \, \text{m} \) and spring constant is \( 200 \, \text{N/m} \). The block is pushed such that the length of the spring becomes \( 1 \, \text{m} \) and then released. At distance \( x \, \text{m} \) (\( x < 2 \)) from the wall, the speed of the block will be:

The center of mass of a thin rectangular plate (fig - $x$) with sides of length $a$ and $b$, whose mass per unit area ($\sigma$) varies as $\sigma = \frac{\sigma_0 x}{ab}$ (where $\sigma_0$ is a constant), would be __________.

Consider a circular disc of radius \( 20 \, \text{cm} \) with centre located at the origin. A circular hole of radius \( 5 \, \text{cm} \) is cut from this disc in such a way that the edge of the hole touches the edge of the disc. The distance of centre of mass of residual or remaining disc from the origin will be:

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

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

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 acceleration due to gravity).

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 force \( \vec{F} = 2\hat{i} + b\hat{j} + \hat{k} \) 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 \( 0 \).

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

As shown below, bob \( A \) of a pendulum having massless string of length \( 'R' \) is released from \( 60^\circ \) to the vertical. It hits another bob \( B \) of half the mass that is at rest on a frictionless table in the center. Assuming elastic collision, the magnitude of the velocity of bob \( A \) after the collision will be (take \( g \) as acceleration due to gravity.)

A rod of length \( 5L \) is bent right angle keeping one side length as \( 2L \).

The position of the centre of mass of the system:
(Consider \( L = 10\,\text{cm} \)).

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

A body of mass \( 'm' \) connected to a massless and unstretchable string goes in vertical circle of radius \( 'R' \) under gravity \( g \). The other end of the string is fixed at the center of circle. If velocity at top of circular path is \( n\sqrt{gR} \), where, \( n \ge 1 \), then ratio of kinetic energy of the body at bottom to that at top of the circle is:

Consider two blocks A and B of masses \( m_1 = 10 \, \text{kg} \) and \( m_2 = 5 \, \text{kg} \) that are placed on a frictionless table. The block A moves with a constant speed \( v = 3 \, \text{m/s} \) towards the block B kept at rest. A spring with spring constant \( k = 3000 \, \text{N/m} \) 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 force \( F = \alpha + \beta x^2 \) acts on an object in the x-direction. The work done by the force is \( 5 \, \text{J} \) when the object is displaced by \( 1 \, \text{m} \). If the constant \( \alpha = 1 \, \text{N} \) then \( \beta \) will be:

A sand dropper drops sand of mass \( m(t) \) on a conveyer belt at a rate proportional to the square root of speed \( (v) \) of the belt, i.e. \( \dfrac{dm}{dt} \propto \sqrt{v} \). If \( P \) is the power delivered to run the belt at constant speed then which of the following relationship is true?

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JEE Main 2025 28th Jan Morning Shift

The center of mass of a thin rectangular plate (fig - $x$) with sides of length $a$ and $b$, whose mass per unit area ($\sigma$) varies as $\sigma = \frac{\sigma_0 x}{ab}$ (where $\sigma_0$ is a constant), would be __________.