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JEE Mains | PYQ

A block of mass \( 10 \, \text{kg} \) starts sliding on a surface with an initial velocity of \( 9.8 \, \text{m s}^{-1} \). The coefficient of friction between the surface and block is \( 0.5 \). The distance covered by the block before coming to rest is:[use \( g = 9.8 \, \text{m s}^{-2} \)]

In the arrangement shown in figure \( a_1 \), \( a_2 \), \( a_3 \) and \( a_4 \) are the accelerations of masses \( m_1 \), \( m_2 \), \( m_3 \) and \( m_4 \) respectively. Which of the following relation is true for this arrangement?

A system to \( 10 \) balls each of mass \( 2 \, \text{kg} \) are connected via massless and unstretchable string. The system is allowed to slip over the edge of a smooth table as shown in figure. Tension on the string between \( 7^{\text{th}} \) and \( 8^{\text{th}} \) ball is \( \_\_\_\_ \, \text{N} \) when \( 6^{\text{th}} \) ball just leaves the table.

A fully loaded boeing aircraft has a mass of \( 5.4 \times 10^{5}\ \text{kg} \). Its total wing area is \( 500\ \text{m}^{2} \). It is in level flight with a speed of \( 1080\ \text{km/h} \). If the density of air \( \rho \) is \( 1.2\ \text{kg m}^{-3} \), the fractional increase in the speed of the air on the upper surface of the wing relative to the lower surface in percentage will be: \( (g = 10\ \text{m/s}^{2}) \)

The time taken by an object to down \( 45^\circ \) rough inclined plane is \( n \) times as it takes to slide down a perfectly smooth \( 45^\circ \) incline plane. The coefficient of kinetic friction between the object and the in line plane is:

A child of mass \( 5 \, \text{kg} \) is going round a merry-go-round that makes \( 1 \) rotation in \( 3.14 \, \text{s} \). The radius of the merry-go-round is \( 2 \, \text{m} \). The centrifugal force on the child will be:

A body of mass \( 1 \, \mathrm{kg} \) rests on a horizontal floor with which it has a coefficient of static friction \( \dfrac{1}{\sqrt{3}} \). It is desired to make the body move by applying the minimum possible force \( F \, \mathrm{N} \). The value of \( F \) will be _______. (Round off to the Nearest Integer) [Take \( g = 10 \, \mathrm{m\,s^{-2}} \)]

A wooden block, initially at rest on the ground, is pushed by a force which increases linearly with time \( t \). Which of the following curve best describes acceleration of the block with time:

A tennis ball is dropped on to the floor from a height of \( 9.8 \, \text{m} \). It rebounds to a height \( 5.0 \, \text{m} \). Ball comes in contact with the floor for \( 0.2 \, \text{s} \). The average acceleration during contact is ______ \( \text{ms}^{-2} \). (Given \( g = 10 \, \text{ms}^{-2} \))

A person is standing in an elevator. In which situation, he experiences weight loss?

A boy of mass \( 4 \, \mathrm{kg} \) is standing on a piece of wood having mass \( 5 \, \mathrm{kg} \). If the coefficient of friction between the wood and the floor is \( 0.5 \), the maximum force that the boy can exert on the rope so that the piece of wood does not move from its place is _______ \( \mathrm{N} \). (Round off to the Nearest Integer) [Take \( g = 10 \, \mathrm{m\,s^{-2}} \)]

Consider a block kept on an inclined plane (inclined at \( 45^\circ \)) as shown in the figure. If the force required to just push it up the incline is \( 2 \) times the force required to just prevent it from sliding down, the coefficient of friction between the block and inclined plane (\( \mu \)) is equal to:

A \( 2 \, \text{kg} \) brick begins to slide over a surface which is inclined at an angle of \( 45^\circ \) with respect to horizontal axis. The co-efficient of static friction between their surfaces is:

A large drum having radius \( R \) is spinning around its axis with angular velocity \( \omega \), as shown in figure. The minimum value of \( \omega \) so that a body of mass \( M \) remains stuck to the inner wall of the drum, taking the coefficient of friction between the drum surface and mass \( M \) as \( \mu \), is:

All surfaces shown in figure are assumed to be frictionless and the pulleys and the string are light. The acceleration of the block of mass \( 2 \, \text{kg} \) is:

A body of mass \(10 \, \text{kg}\) is moving with an initial speed of \(20 \, \text{m/s}\). The body stops after \(5 \, \text{s}\) due to friction between body and the floor. The value of the coefficient of friction is: (Take acceleration due to gravity \(g = 10 \, \text{m s}^{-2}\))

A bullet of \( 10 \, \text{g} \) leaves the barrel of gun with a velocity of \( 600 \, \text{m/s} \). If the barrel of gun is \( 50 \, \text{cm} \) long and mass of gun is \( 3 \, \text{kg} \), then value of impulse supplied to the gun will be:

A block of mass \( m \) is placed on a surface having vertical cross section given by \( y=\dfrac{x^{2}}{4} \). If coefficient of friction is \( 0.5 \), the maximum height above the ground at which block can be placed without slipping is:

A uniform metal chain of mass \( m \) and length ‘\( L \)’ passes over a massless and frictionless pulley. It is released from rest with a part of its length ‘\( l \)’ is hanging on one side and rest of its length ‘\( L-l \)’ the other side of the pully. At a certain point of time, when \( l=\dfrac{L}{x} \), the acceleration of the chain is \( \dfrac{g}{2} \). The value of \( x \) is ______.

\( 100 \) balls each of mass \( m \) moving with speed \( v \) simultaneously strike a wall normally and reflected back with same speed, in time \( t \, \text{s} \). the total force exerted by the balls on the wall is:

A block is sliding down on an inclined plane of slope \( \theta \) and at an instant \( t = 0 \) this block is given an upward momentum so that it starts moving up on the inclined surface with velocity \( u \). The distance \( (S) \) travelled by the block before its velocity become zero, is ________.
\( (g = \text{gravitational acceleration}) \)

Two blocks (\( m = 0.5 \, \text{kg} \) and \( M = 4.5 \, \text{kg} \)) are arranged on a horizontal frictionless table as shown in figure. The coefficient of static friction between the two blocks is \( \frac{3}{7} \). Then the maximum horizontal force that can be applied on the larger block so that the blocks move together is ______ \( \text{N} \). (Round off to the Nearest Integer) [Take \( g \) as \( 9.8 \, \text{m s}^{-2} \)]

Two blocks \( (P \text{ and } Q) \) with respectively masses \( 2 \,\text{kg} \) and \( 1.5 \,\text{kg} \) are joined by a massless thread. These blocks are mounted on a frictionless pully which is fixed on the edge of a cube \( (S) \), as shown in the figure below. Below \( P \) is positioned on the top surface which has no friction and block \( Q \) is in contact with side-surface, having coefficient friction \( \mu \). The cube \( (S) \) moves towards the right with acceleration of \( \frac{g}{2} \), where \( g \) is gravitational acceleration. During this movement the block \( P \) and \( Q \) remain stationary. The value of \( \mu \) is ______. (take \( g=10\,\text{m/s}^2 \))

A particle moves in \( x{-}y \) plane under the influence of a force \( \vec{F} \) such that its linear momentum is \( \vec{p}(t)=\vec{i}\cos(kt)-\vec{j}\sin(kt) \). If \( k \) is constant, the angle between \( \vec{F} \) and \( \vec{P} \) will be:

A block of mass \( M \) placed inside a box descends vertically with acceleration ‘\( a \)’. The block exerts a force equal to one-fourth of its weight on the floor of the box. The value of ‘\( a \)’ will be:

When a body slides down from rest along a smooth inclined plane making an angle of \( 30^\circ \) with the horizontal, it take time \( T \). When the same body slides down from the rest along a rough inclined plane making the same angle and through the same distance, it takes time \( \alpha T \), where \( \alpha \) is a constant greater than \( 1 \). The co-efficient of friction between the body and the rough plane is \(\dfrac{1}{\sqrt{x}}(\dfrac{a^2-1}{a^2}) \) where \(x=\)___________.

A body of mass \( 1 \, \text{kg} \) is suspended with the help of two strings making angles as shown in figure. Magnitudes of tensions \( T_1 \) and \( T_2 \), respectively, are (in \( \text{N} \)) : (Take acceleration due to gravity \( 10 \, \text{m}/\text{s}^2 \))

A car moving with a speed of \( 54 \,\text{km/h} \) takes a turn of radius \( 20 \,\text{m} \). A simple pendulum is suspended from the ceiling of the car. Determine the angle made by the string of the pendulum with the vertical during the turning.
\([ \text{Take } g=10 \,\text{m/s}^{2} ]\)

In the given figure the blocks A, B and C weigh \( 4 \, \text{kg} \), \( 6 \, \text{kg} \) and \( 8 \, \text{kg} \) respectively. The co-efficient of sliding friction between any two surfaces is \( 0.5 \). The force \( F \) required to slide the block C with constant speed is ______ \( \text{N} \).
(Use \( g = 10 \, \text{m/s}^2 \))

The position vector of a particle related to time \( t \) is given by \( \vec{r} = (10 t \hat{i} + 15 t^{2} \hat{j} + 7 \hat{k}) \, \text{m} \). The direction of net force experienced by the particle is:

As shown in figure, a \( 70 \, \text{kg} \) garden roller is pushed with a force of \( \vec{F} = 200 \, \text{N} \) at an angle of \( 30^\circ \) with horizontal. The normal on the roller is:
(Given \( g = 10 \, \text{m s}^{-2} \))

As per given figure, a weightless pulley \( P \) is attached on a double inclined frictionless surfaces. The tension in the string (massless) will be (if \( g = 10 \, \text{m} \, /\text{s}^{2} \))

In the given arrangement of a doubly inclined plane two blocks of masses \( M \) and \( m \) are placed. The blocks are connected by a light string passing over an ideal pulley as shown. The coefficient of friction between the surface of the plane and the blocks is \( 0.25 \). The value of \( m \), for which \( M = 10 \, \text{kg} \) will move down with an acceleration of \( 2 \, \text{m} \,/ \text{s}^{2} \), is: (take \( g = 10 \, \text{m} \, /\text{s}^{2} \) and \( \tan 37^\circ = \frac{3}{4} \) )

A body of weight \( 200 \, \text{N} \) is suspended from a tree branch through a chain of mass \( 10 \, \text{kg} \). The branch pulls the chain by a force equal to (if \( g = 10 \, \text{m}/\text{s}^{2} \) )

A car of mass '\( m \)' moves on a banked road having radius '\( r \)' and banking angle \( \theta \). To avoid slipping from banked road, the maximum permissible speed of the car is \( v_{0} \). The coefficient of friction \( \mu \) between the wheels of the car and the banked road is:

The time taken by a block of mass \( m \) to slide down from the highest point to the lowest point on a rough inclined plane is \( 50\% \) more compared to the time taken by the same block on identical inclined smooth plane. Both inclined planes are at \( 45^\circ \) with the horizontal. The coefficient of kinetic friction between the rough inclined surface and block is

A system of two blocks of masses \( m = 2 \, \text{kg} \) and \( M = 8 \, \text{kg} \) is placed on a smooth table as shown in figure. The coefficient of static friction between two blocks is \( 0.5 \). The maximum horizontal force \( F \) that can be applied to the block of mass \( M \) so that the blocks move together will be:

A force acts for \( 20 \, \text{s} \) on a body of mass \( 20 \, \text{kg} \), starting from rest, after which the fore ceases and then body describes \( 50 \, \text{m} \) in the next \( 10 \, \text{s} .\) The value of force will be:

A light string passing over a smooth light pulley connects two blocks of masses \( m_{1} \) and \( m_{2} \) (where \( m_{2} > m_{1} \)). If the acceleration of the system is \( \frac{g}{\sqrt{2}} \), then the ratio of the masses \( \frac{m_{1}}{m_{2}} \) is:

A modern grand – prix racing car of mass \( m \) is travelling on a flat track in a circular act of radius \( R \) with a speed \( v \). If the coefficient of static friction between the tyres and the track is \( \mu_s \), then the magnitude of negative lift \( F_L \) acting downwards on the car is: (Assume forces on the four tyres are identical and \( g \)=acceleration due to gravity)

A mass of \( 10 \, \text{kg} \) is suspended vertically by a rope of length \( 5 \, \text{m} \) from the roof. A force of \( 30 \, \text{N} \) is applied at the middle point of rope in horizontal direction. The angle made by upper half of the rope with vertical is \( \theta = \tan^{-1}(x \times 10^{-1}) \). The value of \( x \) is ___________.
(Given, \( g = 10 \, \text{m} / \text{s}^2 \) )

A body of mass \( 4 \, \text{kg} \) experiences two forces \( \vec{F}_1 = 5\hat{i} + 8\hat{j} + 7\hat{k} \) and \( \vec{F}_2 = 3\hat{i} - 4\hat{j} - 3\hat{k} \). The acceleration acting on the body is:

The time period of oscillation of a simple pendulum of length \( L \) suspended from the roof of a vehicle, which moves without friction down an inclined plane of inclination \( \alpha \), is given by:

Given below are two statements:
Statement (I): The limiting force of static friction depends on the area of contact and independent of materials.
Statement (II): The limiting forces of kinetic friction is independent of the area of contact and depends on materials.
In the light of the above statements, choose the most appropriate answer from the options given below:

Three block A, B and C are pulled on a horizontal smooth surface by a force of \( 80 \, \text{N} \) as shown in figure.
The tension \( T_1 \) and \( T_2 \) in the string are respectively:

A block of mass \( M \) slides down on a rough inclined plane with constant velocity. The angle made by the incline plane with horizontal is \( \theta \). The magnitude of the contact force will be:

A coin is placed on a disc. The coefficient of friction between the coin and the disc is \( \mu \). If the distance of the coin from the center of the disc is \( r \), the maximum angular velocity which can be given to the disc, so that the coin does not slip away, is:

A particle is projected with velocity \( V_{0} \) along \( x \)-axis. A damping force is acting on the particle which is proportional to the square of the distance from the origin i.e. \( m a = -\alpha x^{2} \). The distance at which the particle stops:

A smooth circular groove has a smooth vertical wall as shown in figure. A block of mass \( m \) moves against the wall with a speed \( v \). Which of the following curve represents the correct relation between the normal reaction on the block by the wall \( (N) \) and speed of the block \( (v) \)?

The normal reaction ‘\( N \)’ for a vehicle of \( 800 \, \text{kg} \) mass, negotiating a turn on a \( 30^\circ \) banked road at maximum possible speed without skidding is _________ \( \times 10^3 \, \text{kg m/s}^2 \). [ Given \( \cos 30^\circ = 0.87, \mu_s = 0.2 \) ]

The initial mass of a rocket is \( 1000 \, \text{kg} \). Calculate at what rate the fuel should be burnt so that the rocket is given an acceleration of \( 20 \, \text{m s}^{-2} \). The gases come out at a relative speed of \( 500 \, \text{m s}^{-1} \) with respect to the rocket:
[Use \( g = 10 \, \text{m /s}^{2} \)]

Given below are two statements:
Statement I : An elevator can go up or down with uniform speed when its weight is balanced with the tension of its cable.
Statement II : Force exerted by the floor of an elevator on the foot of a person standing on it is more than his/her weight when the elevator goes does with increasing speed.
In the light of the above statement, choose the correct answer form the options given below:

A block of mass \( m \) slides on the wooden wedge, which in turn slides backward on the horizontal surface. The acceleration of the block with respect to the wedge is:
Given \( m = 8 \, \text{kg} \), \( M = 16 \, \text{kg} \)
Assume all the surfaces shown in the figure to be frictionless

A stone is projected at angle \( 30^\circ \) to the horizontal. The ratio of kinetic energy of the stone at point of projection to its kinetic energy at the highest point of flight will be:

The boxes of masses \( 2 \, \text{kg} \) and \( 8 \, \text{kg} \) are connected by a massless string passing over smooth pulleys. Calculate the time taken by box of mass \( 8 \, \text{kg} \) to strike the ground starting from rest. (use \( g = 10 \, \text{m/s}^2 \))

A \( 0.5 \, \text{kg} \) mass is in contact against the inner wall of a cylindrical drum of radius \( 4 \, \text{m} \) rotating about its vertical axis. The minimum rotational speed of the drum to enable the mass to remain stuck to the wall (without falling) is \( 5 \, \text{rad/s} \). The coefficient of friction between the drum’s inner wall surface and mass is _________. (Take \( g=10 \, \text{m/s}^{2} \))

Two billiard balls of mass \( 0.05 \, \text{kg} \) each moving in opposite directions with \( 10 \, \text{ms}^{-1} \) collide and rebound with the same speed. If the time duration of contact is \( t = 0.005 \, \text{s} \), then what is the force exerted on the ball due to each other?

A bag is gently dropped on a conveyor belt moving at a speed of \( 2 \, \text{m/s} \). The coefficient of friction between the conveyor belt and bag is \( 0.4 \). initially, the bag slips on the belt before it stops due to friction. The distance travelled by the bag on the belt during slipping motion, is:
[Take : \( g = 10 \, \text{m/s}^{-2} \) ]

A boy ties a stone of mass \( 100 \, \text{g} \) to the end of a \( 2 \, \text{m} \) long string and whirls it around in a horizontal plane. The string can withstand the maximum tension of \( 80 \, \text{N} \). If the maximum speed with which the stone can revolve is \( \frac{K}{\pi} \, \text{rev.}/\text{min} \). The value of \( K \) is : (Assume the string is massless and unstretchable)

Three forces \( F_1 = 10 \,\text{N} \), \( F_2 = 8 \,\text{N} \), \( F_3 = 6 \,\text{N} \) are acting on a particle of mass \( 5 \,\text{kg} \). The forces \( F_2 \) and \( F_3 \) are applied perpendicularly so that particle remains at rest. If the force \( F_1 \) is removed, then the acceleration of the particle is:

A block of mass \( 5 \, \text{kg} \) is moving on an inclined plane which makes an angle of \( 30^{\circ} \) with the horizontal. Friction coefficient between the block and inclined plane surface is \( \dfrac{\sqrt{3}}{2} \). The force to be applied on the block so that the block will move down without acceleration is________ \( \text{N} \). \( (g = 10 \, \text{m}/\text{s}^{2}) \).

A block ‘A’ takes \( 2 \, \text{s} \) to slide down a frictionless incline of \( 30^\circ \) and length '\( l \)', kept inside a lift going up with uniform velocity ‘\( v \)’. If the incline is changed to \( 45^\circ \), the time taken by the block, to slide down the incline, will be approximately.

If the radius of curvature of the path of two particles of same mass are in the ratio \( 3\!:\!4 \), then in order to have constant centripetal force, their velocities will be in the ratio of:

A disc with a flat small bottom beaker placed on it at a distance \( R \) from its center is revolving about an axis passing through the center and perpendicular to its plane with an angular velocity \( \omega \). The coefficient of static friction between the bottom of the beaker and the surface of the disc is \( \mu \). The beaker will revolve with the disc if :

A stone of mass \( 1 \, \text{kg} \) is tied to end of a massless string of length \( 1 \, \text{m} \). If the breaking tension of the string is \( 400 \, \text{N} \), then maximum linear velocity, the stone can have without breaking the string, while rotating in horizontal plane, is:

A lift of mass \( M = 500 \, \text{kg} \) is descending with speed of \( 2 \, \text{m s}^{-1} \). Its supporting cable begins to slip thus allowing it to fall with a constant acceleration of \( 2 \, \text{m s}^{-2} \). The kinetic energy of the lift at the end of fall through to a distance of \( 6 \, \text{m} \) will be ______\( \text{kJ} \).

A train is moving with a speed of \( 12 \, \text{m/s} \) on rails which are \( 1.5 \, \text{m} \) apart. To negotiate a curve of radius \( 400 \, \text{m} \), the height by which the outer rail should be raised with respect to the inner rail is:
(Given, \( g = 10 \, \text{m/s}^2 \))

A uniform chain of \( 6 \, \text{m} \) length is placed on a table such that a part of its length is hanging over the edge of the table. The system is at rest. The co-efficient of static friction between the chain and the surface of the table is \( 0.5 \), the maximum length of the chain hanging from the table is ________ \( \text{m} \).

The coefficient of static friction between two blocks is \( 0.5 \) and the table is smooth. The maximum horizontal force that can be applied to move the blocks together is ________ \( \text{N} \). \( \text{(take } g = 10 \, \text{m s}^{-2}\text{)} \).

A lift of mass \( 1600 \, \text{kg} \) is supported by thick iron wire. If the maximum stress which the wire can withstand is \( 4 \times 10^8 \, \text{N}/\text{m}^2 \) and its radius is \( 4 \, \text{mm} \), then maximum acceleration the lift can take is _____ \( \text{m}/\text{s}^2 \).

(take \( g = 10 \, \text{m}/\text{s}^2 \) and \( \pi = 3.14 \))

Two mass \( M_1 \) and \( M_2 \) are tied together at the two ends of a light inextensible string that passes over a frictionless pulley. When the mass \( M_2 \) is twice that of \( M_1 \), the acceleration of the system is \( a_1 \). When the mass \( M_2 \) is thrice that of \( M_1 \), the acceleration of the system is \( a_2 \). Then ratio \( \dfrac{a_1}{a_2} \) will be:

The position of an object having mass \( 0.1 \,\text{kg} \) as a function of time \( t \) is given as \( \vec{r}=(10t^{2}\hat{\imath}+5t^{3}\hat{\jmath})\,\text{m} \). At \( t=1\,\text{s} \), which of the following statements are correct?
A. The linear momentum \( \vec{p}=(2\hat{\imath}+1.5\hat{\jmath})\,\text{kg}\cdot\text{m}/\text{s} \)
B. The force acting on the object \( \vec{F}=(2\hat{\imath}+3\hat{\jmath})\,\text{N} \)
C. The angular momentum of the object about its origin \( \vec{L}=15\hat{k}\,\text{J}\,\text{s} \)
D. The torque acting on the object about its origin \( \vec{\tau}=20\hat{k}\,\text{N}\,\text{m} \)
Choose the correct answer from the options given below:

At any instant the velocity of a particle of mass \( 500 \, \text{g} \) is \( (2t\hat{i} + 3t^{2}\hat{j}) \, \text{ms}^{-1} \). If the force acting on the particle at \( t = 1 \, \text{s} \) is \( (\hat{i} + x\hat{j}) \, \text{N} \). Then the value of \( x \) will be:

A steel block of \( 10 \, \text{kg} \) rests on a horizontal floor as shown. When three iron cylinders are placed on it as shown, the block and cylinders go down with an acceleration \( 0.2 \, \text{m/s}^2 \). The normal reaction \( R \)' by the floor if mass of the iron cylinders are equal and of \( 20 \, \text{KG} \) each, is ________ \( \text{N} \). \([\text{Take } g = 10 \, \text{m/s}^2 \text{ and } \mu_s = 0.2] \)

A body of mass '\( m \)' is launched up on a rough inclined plane making an angle of \( 30^\circ \) with the horizontal. The coefficient of friction between the body and plane is \( \dfrac{\sqrt{x}}{5} \) if the time of ascent is half of the time of descent. The value of \( x \) is \( \_\_\_\_\_\_ \).

A particle of mass \( m \) is suspended from a ceiling through a string of length \( L \). The particle moves in a horizontal circle of radius \( r \) such that \( r = \frac{L}{\sqrt{2}} \). The speed of particle will be:

A light string passing over a smooth light fixed pulley connects two blocks of masses \( m_1 \) and \( m_2 \). If the acceleration of the system is \( {g}/{8} \), then the ratio masses is:

An object of mass \( m \) is being moved with a constant velocity under the action of an applied force of \( 2 \, \text{N} \) along a frictionless surface with following surface profile.

The correct applied force vs distance graph will be:

Sand is being dropped from a stationary dropper at a rate of \( 0.5 \, \text{kg s}^{-1} \) on a conveyor belt moving with a velocity of \( 5 \, \text{m s}^{-1} \). The power needed to keep the belt moving with the same velocity will be:

A ball of mass \( 0.15 \, \text{kg} \) hits the wall with its initial speed of \( 12 \, \text{ms}^{-1} \) and bounces back without changing its initial speed. If the force applied by the wall on the ball during the contract is \( 100 \, \text{N} \), calculate the time duration of the contract of ball with the wall:

A car is moving on a plane inclined at \( 30^\circ \) to the horizontal with an acceleration of \( 10 \, \text{m s}^{-2} \) parallel to the plane upward. A bob is suspended by a string from the roof of the car. The angle in degrees which the string makes with the vertical is ________.
(Take \( g = 10 \, \text{m s}^{-2} \) )

A block of mass \( 5 \ \text{kg} \) is placed at rest on a table of rough surface. Now, if a force of \( 30 \ \text{N} \) is applied in the direction parallel to surface of the table, the block slides through a distance of \( 50 \ \text{m} \) in an interval of time \( 10 \ \text{s} \). Coefficient of kinetic friction is: (given \( g = 10 \ \text{m} / \text{s}^{2} \) )

A string of length \( L \) is fixed at one end and carries a mass of \( M \) at the other end. The mass makes \(\displaystyle \left(\frac{3}{\pi}\right) \) rotations per second about the vertical axis passing through end of the string as shown. The tension in the string is __________ \( ML \).

A block of mass \( 40 \; \text{kg} \) slides over a surface, when a mass of \( 4 \; \text{kg} \) is suspended through an inextensible massless string passing over frictionless pulley as shown below:
The coefficient of kinetic friction between the surface and block is \( 0.02 \). The acceleration of block is. \( \text{(Given } g = 10 \; \text{ms}^{-2}\text{)} \).

Three blocks \( M_1, M_2, M_3 \) having masses \( 4 \, \text{kg} \), \( 6 \, \text{kg} \) and \( 10 \, \text{kg} \) respectively are hanging from a smooth pully using rope \( 1 \), \( 2 \) and \( 3 \) as shown in figure. The tension in the rope \( 1 \), \( T_1 \) when they are moving upward with acceleration of \( 2 \, \text{m s}^{-2} \) is __________ \( \text{N} \) (if \( g = 10 \, \text{m/s}^2 \)).

A cricket player catches a ball of mass \( 120 \, \text{g} \) moving with \( 25 \, \text{m/s} \) speed. If the catching process is completed in \( 0.1 \, \text{s} \) then the magnitude of force exerted by the ball on the hand of player will be:
(in SI unit):

A small block of mass \( 100 \, \text{g} \) is tied to a spring of spring constant \( 7.5 \, \text{N/m} \) and length \( 20 \, \text{cm} \). The other end of spring is fixed at a particular point A. If the block moves in a circular path on a smooth horizontal surface with constant angular velocity \( 5 \, \text{rad/s} \) about point A, then tension in the spring is:

A balloon and its content having mass \( M \) is moving up with an acceleration \( 'a '\). The mass that must be released from the content so that the balloon starts moving up with an acceleration \( '3a '\) will be 
(Take \( 'g' \) as acceleration due to gravity)

A monkey of mass \( 50 \, \text{kg} \) climbs on a rope which can withstand the tension \( (T) \) of \( 350 \, \text{N} \). If monkey initially climbs down with an acceleration of \( 4 \, \text{m}/\text{s}^2 \) and then climbs up with an acceleration of \( 5 \, \text{m}/\text{s}^2 \). choose the correct option \( (g = 10 \, \text{m}/\text{s}^2) \).

A cubic block of mass \( m \) is sliding down on an inclined plane at \( 60^\circ \) with an acceleration of \( \dfrac{g}{2} \), the value of coefficient of kinetic friction is:

A \( 0.4 \, \text{kg} \) mass takes \( 8 \, \text{s} \) to reach ground when dropped from a certain height ‘\( P \)’ above surface of earth. The loss of potential energy in the last second of fall is ______ \( \text{J} \). (Take \( g = 10 \, \text{m/s}^{2} \))

A wooden block of mass \( 5 \, \text{kg} \) rests on a soft horizontal floor. When an iron cylinder of mass \( 25 \, \text{kg} \) is placed on the top of the block, the floor yields and the block and the cylinder together go down with an acceleration of \( 0.1 \, \text{m s}^{-2} \). The action force of the system on the floor is equal to:

A car is moving with a constant speed of \( 20 \, \text{m/s} \) in a circular horizontal track of radius \( 40 \, \text{m} \). a bon is suspended from the roof of the car by a massless string. The angle made by the string with the vertical will be: (Take \( g = 10 \, \text{m/s}^2 \))

A ball of mass \( 0.5 \, \text{kg} \) is attached to a string of length \( 50 \, \text{cm} \). The ball is rotated on a horizontal circular path about its vertical axis. The maximum tension that the string can bear is \( 400 \, \text{N} \). The maximum possible value of angular velocity of the ball in \( \text{rad}/\text{s} \) is:

A body of mass \( 500 \, \text{g} \) moves along x-axis such that it’s velocity varies with displacement \( x \) according to the relation \( v = 10\sqrt{x} \, \text{m/s} \) the force acting on the body is :

Consider a block and trolley system as shown in figure. If the coefficient of kinetic friction between the trolley and the surface is \( 0.04 \), the acceleration of the system in \( \text{m}\,\text{s}^{-2} \) is: (Consider that the string is massless and unstretchable and the pulley is also massless and frictionless).

A curved in a level road has radius \( 75 \, \text{m} \). The maximum speed of a car turning this curved road can be \( 30 \, \text{m/s} \) without skidding. If radius of curved road is changed to \( 48 \, \text{m} \) and the coefficient of friction between the tyres and the road remains same, then maximum allowed speed would be __________ \( \text{m/s} \).

A hanging mass \( M \) is connected to a four times bigger mass by using a string pulley arrangement, as shown in the figure. The bigger mass is placed on a horizontal ice-slab and being pulled by \( 2 \, Mg \) force. In this situation, tension in the string is \( \dfrac{x}{5} \, Mg \) for \( x = \) ______. Neglect mass of the string and friction of the block (bigger mass) with ice slab.
(Given \( g = \) acceleration due to gravity)

Two bodies of masses \( m_1 = 5 \, \text{kg} \) and \( m_2 = 3 \, \text{kg} \) are connected by a light string going over a smooth light pulley on a smooth inclined plane as shown in the figure. The system is at rest.
The force exerted by the inclined plane on the body of mass \( m_1 \) will be: [Take \( g = 10 \, \text{m s}^{-2} \)]
 

A heavy box of mass \( 50 \, \text{kg} \) is moving on a horizontal surface. If co-efficient of kinetic friction between the box and horizontal surface is \( 0.3 \) then force of kinetic friction is: