Newton's laws

Two bodies of masses m$_1$ = 5 kg and m$_2$ = 3 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 m s$^{-2}$]

A thin and uniform rod of mass $M$ and length $L$ is held vertical on a floor with large friction. The rod is released from rest so that it falls by rotating about its contact-point with the floor without slipping. Which of the following statement(s) is/are correct, when the rod makes an angle 60° with vertical?

[$g$ is the acceleration due to gravity]

Consider a body of mass 1.0 kg at rest at the origin at time $t = 0$. A force $\vec F = (\alpha t\,\hat i + \beta\,\hat j)$ is applied on the body, where $\alpha = 1.0\ \mathrm{N\,s^{-1}}$ and $\beta = 1.0$ N. The torque acting on the body about the origin at time $t = 1.0$ s is $\vec\tau$. Which of the following statements is (are) true?

A flat plate is moving normal to its plane through a gas under the action of a constant force $F$. The gas is kept at a very low pressure. The speed of the plate $v$ is much less than the average speed $u$ of the gas molecules. Which of the following options is/are true?

A body of mass $m=10^{-2}$ kg is moving in a medium and experiences a frictional force $F=-kv^2$. Its initial speed is $v_0=10$ ms$^{-1}$. If, after 10 s, its energy is $\dfrac18mv_0^2$, the value of k will be :

A particle of mass m is acted upon by a force F given by the empirical law $F=\dfrac R{t^2}v(t)$. If this law is to be tested experimentally by observing the motion starting from rest, the best way is to plot :

A large spherical mass $M$ is fixed at one position and two identical point masses $m$ are kept on a line passing through the centre of $M$ (see figure). The point masses are connected by a rigid massless rod of length $l$ and this assembly is free to move along the line connecting them. All three masses interact only through their mutual gravitational interaction. When the point mass nearer to $M$ is at a distance $r=3l$ from $M$, the tension in the rod is zero for $m=k\left(\dfrac{M}{288}\right)$. The value of $k$ is

A block A of mass 4 kg is placed on another block B of mass 5 kg, and the block B rests on a smooth horizontal table. If the minimum force that can be applied on A so that both the blocks move together is 12 N, the maximum force that can be applied on B for the blocks to move together will be :

A body of mass 5 kg under the action of constant force $\vec F=F_x\hat i+F_y\hat j$ has velocity at $t=0$ s as $\vec v=\left(6\hat i-2\hat j\right)$ m/s and at $t=10$ s as $\vec v=+6\hat j$ m/s. The force $\vec F$ is :

Column II shows five systems in which two objects are labelled as X and Y. Also in each case a point P is shown. Column I gives some statements about X and/or Y. Match these statements to the appropriate system(s) from Column II.

Two particles of mass $m$ each are tied at the ends of a light string of length $2a$. The whole system is kept on a frictionless horizontal surface with the string held tight so that each mass is at a distance '$a$' from the center P (as shown in the figure). Now, the mid-point of the string is pulled vertically upwards with a small but constant force $F$. As a result, the particles move towards each other on the surface. The magnitude of acceleration, when the separation between them becomes $2x$, is

A particle moves in the X-Y plane under the influence of a force such that its linear momentum is $\vec p(t)=A\left[\hat i\cos(kt)-\hat j\sin(kt)\right]$, where $A$ and $k$ are constants. The angle between the force and the momentum is

STATEMENT-1: A cloth covers a table. Some dishes are kept on it. The cloth can be pulled out without dislodging the dishes from the table.

because

STATEMENT-2: For every action there is an equal and opposite reaction.