Newton’s Laws of Motion

A plank is rotating in a vertical plane about one of its ends with a constant angular velocity $\omega =\sqrt{2}$ rad/s. A block of mass m = 2 kg is placed at a distance I = 1m from its end A (see diagram) which is hinged. The block starts sliding down when the plank makes an angle $\theta =30^{\circ }$ with the horizontal. If coefficient of friction between the plank and the block is $\mu$ and given that ${\mu }^2=\text{k/25}$. The value of k is

Two spherical bodies of mass $M$ and undefined and radii $R$ and $2\mathit{ }R$, respectively are released in free space with initial separation between their centres equal to$12 R$ . If they attract each other due to gravitational force only, then the distance covered by the smaller body just before the collision is

A trolley of mass $8 \mathrm{kg}$ is standing on a frictionless surface inside which an object of mass $2 \mathrm{kg}$ is suspended. A constant force $F$ starts acting on the trolley as a result of which the string stood at an angle of $37°$ from the vertical (bob at rest relative to trolley). Then:

A light string is wrapped round a cylindrical log of wood which is placed on a horizontal surface with its axis vertical and it is pulled with a constant force $F$ as shown in the figure. (Friction is absent everywhere)

In the system shown in the figure $m_1>m_2$. System is held at rest by thread $BC$. Just after the thread $BC$ is burnt:

In the arrangement shown in figure, the mass of the body $A$ is $n=4$ times that of body $B$. The height $h=20 \mathrm{cm}$. At a certain instant, the body $B$ is released and the system is set in motion. What is the maximum height (in $\mathrm{cm}$) the body $B$ will go up? Assume enough space above $B$ and $A$ sticks to ground. ($A$ and $B$ are of small size) $(g=10 \mathrm{m} {\mathrm{s}}^{-2})$

The acceleration of the block $B$ in the figure, assuming the surfaces and the pulleys $P_1$ and $P_2$ are all smooth and pulleys and string are light is $\frac{3F}{xm}$ then value of $x$ is.

A uniform rope of length $L$ and mass $M$ is placed on a smooth fixed wedge as shown. Both ends of rope are at same horizontal level. The rope is initially released from rest, then the magnitude of initial acceleration of rope is

Consider a cart being pulled by a horse with a constant velocity. The horse exerts a force on the cart. (The subscript indicates the force on the cart due to the horse). The first subscript denotes the body on which force acts and the second due to which it acts.

Choose the correct statement(s).

Two blocks of mass $M$ and $m$ are used to compress two different massless springs, as shown. The left spring is compressed by, $3 \mathrm{cm}$ while the right spring is compressed by an unknown amount. The system is at rest and all surfaces are fixed and smooth. Which of the following statements are true?

In the figure shown, all the surfaces are smooth. All the blocks $A$, $B$ and $C$ are movable, $x$-axis is horizontal and $y$-axis is vertical, as shown. Just after the system is released from the position as shown,

Two masses $m$ and $M$ are attached with strings as shown in the figure. For the system to be in equilibrium, we have

Three blocks are suspended as shown in the figure. If all the strings and pulleys are ideal and friction is absent everywhere, then the acceleration of the $500 \mathrm{g}$ block is

A machine gun has a mass of $5 \mathrm{kg}$. It fires $50 \mathrm{gram}$ bullets at a rate of $30$ bullets per minute at a speed of $400 \mathrm{m }{\mathrm{s}}^{-1}$. What force (in $\mathrm{N}$) is required to keep the gun in position?

A metallic square plate EFGH is suspended vertically with a pair of sides horizontal by a light inelastic string (shown in the figure). A beaker of the water is brought below the plate and raised till the plate is completely immersed and the level of the water is above the plate. If the point of support is slowly raised vertically at a constant velocity, the graph of tension T in the string against the displacement s of the point support is represented by


       

An 80 kg person is parachuting and is experiencing a downward acceleration of$\mathrm{2.8 }{\mathrm{m }\mathrm{s}}^{-2}$. The mass of the parachute is 5 kg. The upward force on the open parachute is (Take $g=\mathrm{9.8 }{\mathrm{m }\mathrm{s}}^{-2})$