Laws of Motion — JEE Main PYQs

A 0.5 kg mass is in contact against the inner wall of a cylindrical drum of radius 4 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 $g = 10 \text{ m/s}^2$)

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 $45°$ with the horizontal. The coefficient of kinetic friction between the rough inclined surface and block is _______.

At $t = 0$, a body of mass $100 \text{ g}$ starts moving under the influence of a force $\left(5\hat{i} + 10\hat{j}\right) \text{N}$. After $2$ s its position is $\left(2x\hat{i} + 5y\hat{j}\right) \text{m}$. The ratio $x : y$ is _______.

Three masses $m_1 = 4$ kg, $m_2 = 4$ kg and $m_3 = 6$ kg are suspended from a fixed smooth frictionless pully as shown in the figure below. The value of $T_1/T_2$ is ______

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

[Figure: A pulley fixed to the ceiling. Over the pulley, tension $T_1$ on both sides; mass $m_1$ hangs on the left side, mass $m_2$ hangs on the right side, and mass $m_3$ hangs below $m_2$ connected by a string with tension $T_2$.]

A wedge $Y$ with mass of 10 kg and all frictionless surfaces and the inclined surface making $37°$ with horizontal. A block $X$ with mass 2 kg is placed at the highest point of the wedge as shown in figure is at rest. At $t = 0$ wedge ($Y$) is pulled toward right with constant force ($f$) of 24 N. Taking the block $X$ at rest at $t = 0$, the time taken by it to slide down 8.8 m on the slope, while $Y$ is on the move, is ______ s.

(take $\tan(37°) = 3/4$ and $g = 10 \text{ m/s}^2$)

[Figure: A wedge $Y$ with incline angle $37°$; block $X$ at the top of the incline; a horizontal force $f = 24$ N applied to the wedge toward the right.]

Two blocks ($P$ and $Q$) with respectively masses 2 kg and 1.5 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. Block $P$ 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 $g = 10\ \text{m/s}^2$)

[Figure: A cube $S$ with block $P$ (2 kg) on its top surface connected over a pulley at the top edge to block $Q$ (1.5 kg) hanging against the right side-surface of the cube]

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

[Figure: A pulley fixed to the ceiling with a string over it; a 6 kg block hangs on one side at a height of 6 m above the ground, and a 2 kg block rests on the ground on the other side.]

A block takes $t$ time to slide down a plane inclined at $45^\circ$ to the horizontal. If the surface is made smooth (frictionless), the block takes time $\dfrac{t}{2}$ to slide down the plane. The coefficient of friction between the block and the inclined plane is $\left(\dfrac{\alpha}{100}\right)$. The value of $\alpha$ is ________.

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

Two masses of $3.4\,\text{kg}$ and $2.5\,\text{kg}$ are accelerated from an initial speed of $5\,\text{m/s}$ and $12\,\text{m/s}$, respectively. The distances traversed by the masses in the $5^{\text{th}}$ second are $104\,\text{m}$ and $129\,\text{m}$, respectively. The ratio of their momenta after $10\,\text{s}$ is $\dfrac{x}{8}$. The value of $x$ is ________.

[Figure: a block of mass 1 kg hanging from a ceiling by two strings; string with tension $T_1$ makes $60°$ with the ceiling on the left and string with tension $T_2$ makes $30°$ with the ceiling on the right; weight mg acts downward]

A body of mass 1 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 N) :

(Take acceleration due to gravity $10~\mathrm{m/s^2}$)

A body of mass $m$ is suspended by two strings making angles $\theta_1$ and $\theta_2$ with the horizontal ceiling with tensions $\mathrm{T_1}$ and $\mathrm{T_2}$ simultaneously. $\mathrm{T_1}$ and $\mathrm{T_2}$ are related by $\mathrm{T_1} = \sqrt{3}\,\mathrm{T_2}$, the angles $\theta_1$ and $\theta_2$ are

A cubic block of mass m is sliding down on an inclined plane at $60°$ with an acceleration of $\frac{g}{2}$, the value of coefficient of kinetic friction is

An object with mass 500 g moves along $x$-axis with speed $v = 4\sqrt{x}$ m/s. The force acting on the object is :

A body of mass 2 kg moving with velocity of $\vec{v}_{\text{in}} = 3\hat{i} + 4\hat{j}\ \mathrm{ms}^{-1}$ enters into a constant force field of 6N directed along positive z–axis. If the body remains in the field for a period of $\frac{5}{3}$ seconds, then velocity of the body when it emerges from force field is.

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

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

[Figure: a string of length L fixed to the ceiling makes angle θ with the vertical; tension T acts along the string and the mass at its end moves in a horizontal circle of radius R]

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 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 :

Two forces $\vec{\mathrm{F}}_1$ and $\vec{\mathrm{F}}_2$ are acting on a body. One force has magnitude thrice that of the other force and the resultant of the two forces is equal to the force of larger magnitude. The angle between $\vec{\mathrm{F}}_1$ and $\vec{\mathrm{F}}_2$ is $\cos^{-1}\left(\frac{1}{n}\right)$. The value of $|n|$ is ________.

A 2 kg brick begins to slide over a surface which is inclined at an angle of $45°$ with respect to horizontal axis. The co-efficient of static friction between their surfaces is:

A wooden block of mass 5 kg rests on a soft horizontal floor. When an iron cylinder of mass 25 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 $0.1\ \mathrm{ms}^{-2}$. The action force of the system on the floor is equal to :

Three blocks $M_1$, $M_2$, $M_3$ having masses 4 kg, 6 kg and 10 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 $2\,\mathrm{ms}^{-2}$ is ________ N(if $g = 10\ \mathrm{m/s^2}$).

[Figure: three blocks $M_1$ (4 kg), $M_2$ (6 kg), $M_3$ (10 kg) hanging vertically one below the other from a pulley, connected by ropes 1, 2 and 3 with tensions $T_1$, $T_2$, $T_3$, the whole system accelerating at 2 m/s²]

A heavy box of mass 50 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 :

A particle moves in $x$-$y$ plane under the influence of a force $\vec{F}$ such that its linear momentum is $\vec{p}(t) = \hat{i}\cos(kt) - \hat{j}\sin(kt)$. If $k$ is constant, the angle between $\vec{F}$ and $\vec{P}$ 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 car of 800 kg is taking turn on a banked road of radius 300 m and angle of banking $30°$. If coefficient of static friction is 0.2 then the maximum speed with which car can negotiate the turn safely : $(g = 10\ \mathrm{m/s^2},\ \sqrt{3} = 1.73)$

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

A player caught a cricket ball of mass 150 g moving at a speed of 20 m/s. If the catching process is completed in 0.1 s, the magnitude of force exerted by the ball on the hand of the player is:

A given object takes n times the time to slide down $45°$ rough inclined plane as it takes the time to slide down an identical perfectly smooth $45°$ inclined plane. The coefficient of kinetic friction between the object and the surface of inclined plane is :

A circular table is rotating with an angular velocity of $\omega$ rad/s about its axis (see figure). There is a smooth groove along a radial direction on the table. A steel ball is gently placed at a distance of 1 m on the groove. All the surfaces are smooth. If the radius of the table is 3 m, the radial velocity of the ball w.r.t. the table at the time ball leaves the table is $x\sqrt{2}\,\omega$ m/s, where the value of $x$ is __________.

[Figure: a circular table of radius 3 m rotating about a vertical axis with angular velocity $\omega$; a radial groove holds a ball placed 1 m from the centre]

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

A nucleus at rest disintegrates into two smaller nuclei with their masses in the ratio of 2:1. After disintegration they will move :

A 1 kg mass is suspended from the ceiling by a rope of length 4m. A horizontal force '$F$' is applied at the mid point of the rope so that the rope makes an angle of $45°$ with respect to the vertical axis as shown in figure. The magnitude of $F$ is :

(Assume that the system is in equilibrium and $g = 10~\mathrm{m/s^2}$)

[Figure: a rope hangs from the ceiling; at its midpoint a horizontal force $\vec{F}$ acts to the left, the upper segment $T_1$ makes $\theta = 45°$ with the vertical, and the lower segment $T_2$ carries a 1 kg block]

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 $\mathrm{ms^{-2}}$ is :

(Consider that the string is massless and unstretchable and the pulley is also massless and frictionless) :

[Figure: a 20 kg trolley on a horizontal surface (kinetic friction $f_k$ acting on it) connected by a string over a pulley at the table edge to a hanging 6 kg block with a 60 N downward force marked]

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

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

A body of mass 4 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 :

A body of mass 1000 kg is moving horizontally with a velocity 6 m/s. If 200 kg extra mass is added, the final velocity (in m/s) is :

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

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 spherical body of mass 100 g is dropped from a height of 10 m from the ground. After hitting the ground, the body rebounds to a height of 5 m. The impulse of force imparted by the ground to the body is given by : (given, $\mathrm{g} = 9.8~\mathrm{m/s^2}$)

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 kg is :

[Figure: a 2 kg block on a $30°$ incline connected by a string over pulleys to a hanging 4 kg block; a movable pulley hangs from the ceiling]

Three blocks $A$, $B$ and $C$ are pulled on a horizontal smooth surface by a force of $80N$ as shown in figure

[Figure: blocks A (5 kg), B (3 kg) and C (2 kg) in a row connected by strings with tensions $T_1$ (between A and B) and $T_2$ (between B and C); force $F = 80N$ applied to C towards the right]

The tensions $T_1$ and $T_2$ in the string are respectively:

A block of mass $m$ is placed on a surface having vertical crossection given by $y = x^2/4$. If coefficient of friction is $0.5$, the maximum height above the ground at which block can be placed without slipping is:

A 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 :

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 $M = 10$ kg will move down with an acceleration of $2\ \mathrm{m/s^2}$, is: (take $g = 10\ \mathrm{m/s^2}$ and $\tan 37° = 3/4$)

[Figure: double inclined plane with block M on the 53° incline and block m on the 37° incline, connected by a string over a pulley at the apex]

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 of masses is:

[Figure: a fixed pulley attached to the ceiling with masses $m_1$ and $m_2$ hanging on either side of the string]

[Figure: a 5 kg block on an inclined plane of angle $30°$ with coefficient of friction $\mu = 0.1$]

A block of mass 5 kg is placed on a rough inclined surface as shown in the figure. If $\vec{F_1}$ is the force required to just move the block up the inclined plane and $\vec{F_2}$ is the force required to just prevent the block from sliding down, then the value of $\left|\vec{F_1}\right| - \left|\vec{F_2}\right|$ is: [Use g = 10 m/s$^2$]