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A body of mass \( 50 \, \text{kg} \) is lifted to a height of \( 20 \, \text{m} \) from the ground in the two different ways as shown in the figures. The ratio of work done against the gravity in both the respective cases, will be:

An automobile of mass '\( m \)' accelerates starting from origin and initially at rest, while the engine supplies constant power \( P \). The position is given as a function of time by:

Given below is the plot of a potential energy function \( U(x) \) for a system, in which a particle is in one dimensional motion, while a conservative force \( F(x) \) acts on it. Suppose that \( E_{\text{mech}} = 8 \, \text{J} \), the incorrect statement for this system is:

[where K.E. = kinetic energy]

A nucleus disintegrates into two nuclear parts, in such a way that ratio of their nuclear sizes is \( 1 : 2^{1/3} \). Their respective speed have a ratio of \( n : 1 \). The value of \( n \) is ______.

If the maximum load carried by an elevator is \( 1400 \, \text{kg} \) (\( 600 \, \text{kg} \) – passengers + \( 800 \, \text{kg} \) – elevator), which is moving up with a uniform speed of \( 3 \, \text{m s}^{-1} \) and the frictional force acting on it is \( 2000 \, \text{N} \), then the maximum power used by the motor is ______ \( \text{kW} \) (\( g = 10 \, \text{m/s}^2 \)).

A coin placed on a rotating table just slips when it is placed at a distance of \( 1 \, \text{cm} \) from the center. If the angular velocity of the table in halved, it will just slip when placed at a distance of _______ from the centre.

A block is fastened to a horizontal spring. The block is pulled to a distance \( x = 10 \,\text{cm} \) from its equilibrium position (at \( x = 0 \)) on a frictionless surface from rest. The energy of the block at \( x = 5 \,\text{cm} \) is \( 0.25 \,\text{J} \). The spring constant of the spring is _______ \( \text{Nm}^{-1} \).

A block of mass \( 1 \, \text{kg} \), moving along \( x \) with speed \( v_i = 10 \, \text{m/s} \) enters a rough region ranging from \( x = 0.1 \, \text{m} \) to \( x = 1.9 \, \text{m} \). The retarding force acting on the block in this range is \( F_r = -k x \, \text{N} \) with \( k = 10 \, \text{N/m} \). Then the final speed of the block as it crosses rough region is:

A solid steel ball of diameter \( 3.6 \, \text{mm} \) acquired terminal velocity \( 2.45 \times 10^{-2} \, \text{m}/s \) while falling under gravity through an oil of density \( 925 \, \text{kg m}^{-3} \). Take density of steel as \( 7825 \, \text{kg m}^{-3} \) and \( g \) as \( 9.8 \, \text{m/ s}^{2} \). The viscosity of the oil in SI unit is:

A block of mass \( 5 \, \text{kg} \) starting from rest pulled up on a smooth incline plane making an angle of \( 30^\circ \) with horizontal with an affective acceleration of \( 1 \, \text{m s}^{-2} \). The power delivered by the pulling force at \( t = 10 \, \text{s} \) from the start is ________ \( \text{W} \) [use \( g = 10 \, \text{m s}^{-2} \)] (calculate the nearest integer value)

Four forces are acting at a point \( P \) in equilibrium as shown in figure. The ratio of force \( F_{1} \) to \( F_{2} \) is \( 1:x \) where \( x = \) ______.

A passenger sitting in a train A moving at \( 90 \, \text{km/h} \) observes another train B moving in the opposite direction for \( 8 \, \text{s} \). If the velocity of the train B is \( 54 \, \text{km/h} \), then length of train B is:

A small block slides down from the top of hemisphere of radius \( R = 3 \, \text{m} \) shown in the figure. The height ‘\( h \)’ at which the block will lose contact with the surface of the sphere is ________ \( \text{m} \).
(Assume there is no friction between the block and the hemisphere)

A body of mass \( 2 \, \text{kg} \) is initially at rest. It starts moving unidirectionally under the influence of a source of constant power \( P \). Its displacement in \( 4 \, \text{s} \) is \( \dfrac{1}{3} \alpha^{2} \sqrt{P} \, \text{m} \). The value of \( \alpha \) will be _____.

Two projectiles are projected at \( 30^\circ \) and \( 60^\circ \) with the horizontal with the same speed. The ratio of the maximum height attained by the two projectiles respectively is:

As shown in the figure a block of mass \( 10 \, \text{kg} \) lying on a horizontal surface is pulled by a force \( F \) acting at an angle \( 30^\circ \), with horizontal. For \( \mu_s = 0.25 \), the block will just start to move for the value of \( F \):
[Given \( g = 10 \, \text{m s}^{-2} \)]

Identify the correct statements from the following:
(A) Work done by a man in lifting a bucket out of a well by means of a rope tied to the bucket is negative
(B) Work done by gravitational force in lifting a bucket out of a well by a rope tied to the bucket is negative
(C) Work done by friction on a body sliding down an inclined plane is positive
(D) Work done by an applied force on a body moving on a rough horizontal plane

The figure represents the momentum time \( (p - t) \) curve for a particle moving along an axis under the influence of the force. Identify the regions on the graph where the magnitude of the force is maximum and minimum respectively? If \( (t_{3} - t_{2}) < t_{1} \).

Figure (a), (b), (c) and (d) show variation of force with time:

The impulse is highest in figure:

An engine is attached to a wagon through a shock absorbed of length \( 1.5\,\text{m} \). The system with a total mass of \( 40{,}000\,\text{kg} \) is moving with a speed of \( 72\,\text{km}\,\text{h}^{-1} \) when the brakes are applied to bring it to rest. In the process of the system being through to rest, the spring of the shock absorber gets compressed by \( 1.0\,\text{m} \). If \( 90\% \) of energy of the wagon is lost due to friction, the spring constant is ______ \( \times 10^5\,\text{N/m} \).

A pendulum bob has a speed of \( 3 \, \text{m/s} \) at its lowest position. The pendulum is \( 50 \, \text{cm} \) long. The speed of bob, when the length makes an angle of \( 60^\circ \) to the vertical will be \( (g = 10 \, \text{m/s}^2) \), ____________ \( \text{m/s} \).

A stone tied to \( 180 \, \text{cm} \) long string at its end is making \( 28 \) revolutions in horizontal circle in every minute. The magnitude of acceleration of stone is \( \dfrac{1936}{x} \, \text{m s}^{-2} \). The value of \( x \) ___. \( \left( \text{Take } \pi = \dfrac{22}{7} \right) \)

Figure below shows a liquid being pushed out of the tube by a piston having area of cross section \( 2.0 \, \text{cm}^2 \). The area of cross section at the outlet is \( 10 \, \text{mm}^2 \). If the piston is pushed at a speed of \( 4 \, \text{cm s}^{-1} \), the speed of outgoing fluid is ______ \( \text{cm s}^{-1} \).

A body of mass \( m \) dropped from a height \( h \) reaches the ground with a speed of \( 0.8\sqrt{g h} \). The value of work done by the air-friction is:

A block of mass \( 200 \, \text{g} \) is kept stationary on a smooth inclined plane by applying a minimum horizontal force \( F = \sqrt{x} \, \text{N} \) as shown in figure.

The value of \( x = \) _______________.

A block of \( \sqrt{3} \, \text{kg} \) is attached to a string whose other end is attached to the wall. An unknown force \( F \) is applied so that the string makes an angle of \( 30^\circ \) with the wall. The tension \( T \) is: ( Given \( g = 10 \, \text{m s}^{-2} \) )

Two cylindrical vessels of equal cross sectional area of \( 2 \, \text{m}^{2} \) contain water up to heights \( 10 \, \text{m} \) and \( 6 \, \text{m} \), respectively. If the vessels are connected at their bottom then the work done by the force of gravity is:
(Density of water is \( 10^{3} \, \text{kg/m}^{3} \) and \( g = 10 \, \text{m/s}^{2} \) )

A projectile fired at \( 30^\circ \) to the ground is observed to be at same height at time \( 3 \, \text{s} \) and \( 5 \, \text{s} \) after projection, during its flight. The speed of projection of the projectile is _________ \( \text{m s}^{-1} \).
(Given \( g = 10 \, \text{m s}^{-2} \))

Statement I: A cyclist is moving on an unbanked road with a speed of \( 7 \, \text{km h}^{-1} \) and takes a sharp circular turn along a path of radius of \( 2 \, \text{m} \) without reducing the speed. The static friction coefficient is \( 0.2 \). The cyclist will not slip and pass the curve. \( (g = 9.8 \, \text{m s}^{-2}) \)
Statement II: If the road is banked at angle of \( 45^\circ \), cyclist can cross the curve of \( 2 \, \text{m} \) radius with the speed of \( 18.5 \, \text{km h}^{-1} \) without slipping.
In the light of the above statements, choose the correct answer from the options given below.

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JEE Main 2021 27th July Evening Shift

An automobile of mass '\( m \)' accelerates starting from origin and initially at rest, while the engine supplies constant power \( P \). The position is given as a function of time by: