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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:

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 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 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:

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 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:

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 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 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:

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 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 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:

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 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: