Laws of Motion

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 conical pendulum of length 1 m makes an angle θ = 45° w.r.t. Z-axis and moves in a circle in the XY plane. The radius of the circle is 0.4 m and its center is vertically below O. The speed of the pendulum, in its circular path, will be : (Take g = 10 ms$^{-2}$)

A uniform wooden stick of mass 1.6 kg and length $l$ rests in an inclined manner on a smooth, vertical wall of height $h\,(<l)$ such that a small portion of the stick extends beyond the wall. The reaction force of the wall on the stick is perpendicular to the stick. The stick makes an angle of 30° with the wall and the bottom of the stick is on a rough floor. The reaction of the wall on the stick is equal in magnitude to the reaction of the floor on the stick. The ratio $h/l$ and the frictional force $f$ at the bottom of the stick are $(g = 10\ \mathrm{m\,s^{-2}})$

The distance $r$ of the block at time $t$ is

The net reaction of the disc on the block is

A point particle of mass $m$, moves along the uniformly rough track PQR as shown in the figure. The coefficient of friction, between the particle and the rough track equals μ. The particle is released, from rest, from the point P and it comes to rest at a point R. The energies, lost by the ball, over the parts, PQ and QR, of the track, are equal to each other, and no energy is lost when particle changes direction from PQ to QR.

The values of the coefficient of friction μ and the distance $x$(=QR), are, respectively close to :

A rocket is fired vertically from the earth with an acceleration of 2g, where g is the gravitational acceleration. On an inclined plane inside the rocket, making an angle θ with the horizontal, a point object of mass m is kept. The minimum coefficient of friction $\mu_{min}$ between the mass and the inclined surface such that the mass does not move is :

A car of weight W is on an inclined road that rises by 100 m over a distance of 1 km and applies a constant frictional force $\dfrac W{20}$ on the car. While moving uphill on the road at a speed of 10 ms$^{-1}$, the car needs power P. If it needs power $\dfrac P2$ while moving downhill at speed $v$ then value of $v$ is :

A particle of mass M is moving in a circle of fixed radius R in such a way that its centripetal acceleration at time t is given by $n^2Rt^2$ where n is a constant. The power delivered to the particle by the force acting on it, is :

Concrete mixture is made by mixing cement, stone and sand in a rotating cylindrical drum. If the drum rotates too fast, the ingredients remain stuck to the wall of the drum and proper mixing of ingredients does not take place. The maximum rotational speed of the drum in revolutions per minute(rpm) to ensure proper mixing is close to :

(Take the radius of the drum to be 1.25 m and its axle to be horizontal) :

A bottle has an opening of radius a and length b. A cork of length b and radius (a + Δa) where (Δa << a) is compressed to fit into the opening completely (See figure). If the bulk modulus of cork is B and frictional coefficient between the bottle and cork is μ then the force needed to push the cork into the bottle is :

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

Given in the figure are two blocks A and B of weight 20 N and 100 N, respectively. These are being pressed against a wall by a force F as shown. If the coefficient of friction between the blocks is 0.1 and between block B and the wall is 0.15, the frictional force applied by the wall on block B is :

Two long current carrying thin wires, both with current I, are held by insulating threads of length L and are in equilibrium as shown in the figure, with threads making an angle 'θ' with the vertical. If wires have mass λ per unit length then the value of I is :

(g = gravitational acceleration)

A block of mass m = 10 kg rests on a horizontal table. The coefficient of friction between the block and the table is 0.05. When hit by a bullet of mass 50 g moving with speed $v$, that gets embedded in it, the block moves and comes to stop after moving a distance of 2 m on the table.

If a freely falling object were to acquire speed $\dfrac v{10}$ after being dropped from height H, then neglecting energy losses and taking $g=10$ ms$^{-2}$, the value of H is close to :

A large number (n) of identical beads, each of mass m and radius r are strung on a thin smooth rigid horizontal rod of length L (L>>r) and are at rest at random positions. The rod is mounted between two rigid supports (see figure). If one of the beads is now given a speed $v$, the average force experienced by each support after a long time is (assume all collisions are elastic) :

A particle is moving in a circle of radius r under the action of a force $F=\alpha r^2$ which is directed towards centre of the circle. Total mechanical energy (kinetic energy + potential energy) of the particle is (take potential energy = 0 for r = 0) :

In the figure, a ladder of mass $m$ is shown leaning against a wall. It is in static equilibrium making an angle $\theta$ with the horizontal floor. The coefficient of friction between the wall and the ladder is $\mu_1$ and that between the floor and the ladder is $\mu_2$. The normal reaction of the wall on the ladder is $N_1$ and that of the floor is $N_2$. If the ladder is about to slip, then

A rocket is moving in a gravity free space with a constant acceleration of $2\ \mathrm{m\,s^{-2}}$ along $+x$ direction (see figure). The length of a chamber inside the rocket is 4 m. A ball is thrown from the left end of the chamber in $+x$ direction with a speed of $0.3\ \mathrm{m\,s^{-1}}$ relative to the rocket. At the same time, another ball is thrown in $-x$ direction with a speed of $0.2\ \mathrm{m\,s^{-1}}$ from its right end relative to the rocket. The time in seconds when the two balls hit each other is