Laws of Motion

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 _____.

A particle of mass $m$ is under the influence of the gravitational field of a body of mass $M\ (\gg m)$. The particle is moving in a circular orbit of radius $r_0$ with time period $T_0$ around the mass $M$. Then, the particle is subjected to an additional central force, corresponding to the potential energy $V_c(r) = m\alpha/r^3$, where $\alpha$ is a positive constant of suitable dimensions and $r$ is the distance from the center of the orbit. If the particle moves in the same circular orbit of radius $r_0$ in the combined gravitational potential due to $M$ and $V_c(r)$, but with a new time period $T_1$, then $(T_1^2 - T_0^2)/T_1^2$ is given by

[$G$ is the gravitational constant.]

A thin conducting rod MN of mass 20 gm, length 25 cm and resistance $10\,\Omega$ is held on frictionless, long, perfectly conducting vertical rails as shown in the figure. There is a uniform magnetic field $B_0 = 4$ T directed perpendicular to the plane of the rod-rail arrangement. The rod is released from rest at time $t = 0$ and it moves down along the rails. Assume air drag is negligible. Match each quantity in List-I with an appropriate value from List-II, and choose the correct option.

[Given: The acceleration due to gravity $g = 10\ \mathrm{m\,s^{-2}}$ and $e^{-1} = 0.4$]

A particle of mass $m$ is moving in the $xy$-plane such that its velocity at a point $(x, y)$ is given as $\vec v = \alpha(y\hat x + 2x\hat y)$, where $\alpha$ is a non-zero constant. What is the force $\vec F$ acting on the particle?

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 block of mass 10 kg starts sliding on a surface with an initial velocity of $9.8\ \mathrm{ms^{-1}}$. The coefficient of friction between the surface and block is 0.5. The distance covered by the block before coming to rest is :

[use $g=9.8\ \mathrm{ms^{-2}}$]

A boy ties a stone of mass 100 g to the end of a 2 m long string and whirls it around in a horizontal plane. The string can withstand the maximum tension of 80 N. If the maximum speed with which the stone can revolve is $\dfrac K\pi$ rev./min. The value of $K$ is :

(Assume the string is massless and unstretchable)

A stone of mass m, tied to a string is being whirled in a vertical circle with a uniform speed. The tension in the string is

Two identical charged particles each having a mass 10 g and charge $2.0\times10^{-7}$ C are placed on a horizontal table with a separation of L between them such that they stay in limited equilibrium. If the coefficient of friction between each particle and the table is 0.25, find the value of L. [Use $g=10\ \mathrm{ms^{-2}}$]

An object of mass 5 kg is thrown vertically upwards from the ground. The air resistance produces a constant retarding force of 10 N throughout the motion. The ratio of time of ascent to the time of descent will be equal to : [Use $g=10\ \mathrm{ms^{-2}}$].

A long cylindrical volume contains a uniformly distributed charge of density $\rho$. The radius of cylindrical volume is $R$. A charge particle ($q$) revolves around the cylinder in a circular path. The kinetic energy of the particle is :

A uniform chain of 6 m length is placed on a table such that a part of its length is hanging over the edge of the table. The system is at rest. The co-efficient of static friction between the chain and the surface of the table is 0.5, the maximum length of the chain hanging from the table is ______ m.

A force on an object of mass 100 g is $\left(10\hat i+5\hat j\right)$ N. The position of that object at t = 2 s is $\left(a\hat i+b\hat j\right)$ m after starting from rest. The value of $\dfrac ab$ will be ______.

A disc with a flat small bottom beaker placed on it at a distance $R$ from its center is revolving about an axis passing through the center and perpendicular to its plane with an angular velocity $\omega$. The coefficient of static friction between the bottom of the beaker and the surface of the disc is $\mu$. The beaker will revolve with the disc if :

A curved in a level road has a radius 75 m. The maximum speed of a car turning this curved road can be 30 m/s without skidding. If radius of curved road is changed to 48 m and the coefficient of friction between the tyres and the road remains same, then maximum allowed speed would be ______ m/s.

A block of mass 200 g is kept stationary on a smooth inclined plane by applying a minimum horizontal force $F=\sqrt x$ N as shown in figure.

The value of $x=$ ______.

A person is standing in an elevator. In which situation, he experiences weight loss ?

A ball is released from rest from point P of a smooth semi-spherical vessel as shown in figure. The ratio of the centripetal force and normal reaction on the ball at point Q is A while angular position of point Q is $\alpha$ with respect to point P. Which of the following graphs represent the correct relation between A and $\alpha$ when ball goes from Q to R ?

Time period of a simple pendulum in a stationary lift is 'T'. If the lift accelerates with $\dfrac g6$ vertically upwards then the time period will be :

(Where g = acceleration due to gravity)

In the arrangement shown in figure $a_1$, $a_2$, $a_3$ and $a_4$ are the accelerations of masses $m_1$, $m_2$, $m_3$ and $m_4$ respectively. Which of the following relation is true for this arrangement ?