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JEE Mains | PYQ

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 ______.

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:

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

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:

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 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 \( (v) \)?

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)

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.

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 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 ________ \( \text{m} \).

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:

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 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. \( \text{(Given } g = 10 \; \text{ms}^{-2}\text{)} \).

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 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 \( g = 10 \, \text{m s}^{-2} \)]