Impulse and momentum

A thin uniform rod of length $L$ and certain mass is kept on a frictionless horizontal table with a massless string of length $L$ fixed to one end (top view is shown in the figure). The other end of the string is pivoted to a point O. If a horizontal impulse $P$ is imparted to the rod at a distance $x = L/n$ from the mid-point of the rod (see figure), then the rod and string revolve together around the point O, with the rod remaining aligned with the string. In such a case, the value of $n$ is _____.

An annular disk of mass $M$, inner radius $a$ and outer radius $b$ is placed on a horizontal surface with coefficient of friction $\mu$, as shown in the figure. At some time, an impulse $\mathcal J_0\hat x$ is applied at a height $h$ above the center of the disk. If $h = h_m$ then the disk rolls without slipping along the $x$-axis. Which of the following statement(s) is(are) correct?

A batsman hits back a ball of mass 0.4 kg straight in the direction of the bowler without changing its initial speed of 15 ms$^{-1}$. The impulse imparted to the ball is ______ Ns.

A block of metal weighing 2 kg is resting on a frictionless plane (as shown in figure). It is struck by a jet releasing water at a rate of 1 kgs$^{-1}$ and at a speed of 10 ms$^{-1}$. Then, the initial acceleration of the block, in ms$^{-2}$, will be :

Two bodies A and B of masses 5 kg and 8 kg are moving such that the momentum of body B is twice that of the body A. The ratio of their kinetic energies will be :

Two billiard balls of mass 0.05 kg each moving in opposite directions with 10 ms$^{-1}$ collide and rebound with the same speed. If the time duration of contact is $t=0.005$ s, then what is the force exerted on the ball due to each other ?

A ball of mass 0.15 kg hits the wall with its initial speed of 12 ms$^{-1}$ and bounces back without changing its initial speed. If the force applied by the wall on the ball during the contact is 100 N, calculate the time duration of the contact of ball with the wall.

A body of mass 10 kg is projected at an angle of 45° with the horizontal. The trajectory of the body is observed to pass through a point (20, 10). If T is the time of flight, then its momentum vector, at time $t=\dfrac T{\sqrt2}$, is ______.

[Take g = 10 m/s$^2$]

In two different experiments, an object of mass 5 kg moving with a speed of 25 ms$^{-1}$ hits two different walls and comes to rest within (i) 3 second, (ii) 5 seconds, respectively. Choose the correct option out of the following :

A pressure-pump has a horizontal tube of cross sectional area 10 cm$^2$ for the outflow of water at a speed of 20 m/s. The force exerted on the vertical wall just in front of the tube which stops water horizontally flowing out of the tube, is :

[given: density of water = 1000 kg/m$^3$]

The value of $J$ is ____.

The value of $K$ is ____.

A solid horizontal surface is covered with a thin layer of oil. A rectangular block of mass $m = 0.4$ kg is at rest on this surface. An impulse of 1.0 N s is applied to the block at time $t = 0$ so that it starts moving along the $x$-axis with a velocity $v(t) = v_0 e^{-t/\tau}$, where $v_0$ is a constant and $\tau = 4$ s. The displacement of the block, in metres, at $t = \tau$ is __________. Take $e^{-1} = 0.37$.

In a photoelectric experiment a parallel beam of monochromatic light with power of 200 W is incident on a perfectly absorbing cathode of work function 6.25 eV. The frequency of light is just above the threshold frequency so that the photoelectrons are emitted with negligible kinetic energy. Assume that the photoelectron emission efficiency is 100%. A potential difference of 500 V is applied between the cathode and the anode. All the emitted electrons are incident normally on the anode and are absorbed. The anode experiences a force $F = n \times 10^{-4}$ N due to the impact of the electrons. The value of $n$ is __________. Mass of the electron $m_e = 9 \times 10^{-31}$ kg and $1.0\ \mathrm{eV} = 1.6 \times 10^{-19}$ J.

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 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) :

Water is flowing at a speed of 1.5 ms$^{-1}$ through a horizontal tube of cross-sectional area $10^{-2}$ m$^2$ and you are trying to stop the flow by your palm. Assuming that the water stops immediately after hitting the palm, the minimum force that you must exert should be (density of water $=10^3$ kgm$^{-3}$).

A block of mass 2 kg is free to move along the $x$-axis. It is at rest and from $t=0$ onwards it is subjected to a time-dependent force $F(t)$ in the $x$ direction. The force $F(t)$ varies with $t$ as shown in the figure. The kinetic energy of the block after 4.5 seconds is