Laws of Motion

A solid cylinder of radius $R$ rolls without slipping with a center of mass speed $v_0=\sqrt{\dfrac{gR}3}$ on a horizontal surface with a vertical edge, as shown in the figure. Here, $g$ is the acceleration due to the gravity. At the moment when the cylinder loses contact with the surface due to rotation around the corner, the speed of its center of mass is:

The position of an object having mass 0.1 kg as a function of time $t$ is given as $\vec r=\left(10t^2\hat i+5t^3\hat j\right)$ m. At $t=1$ s, which of the following statements are correct ?

A. The linear momentum $\vec p=\left(2\hat i+1.5\hat j\right)$ kg·m/s.

B. The force acting on the object $\vec F=\left(2\hat i+3\hat j\right)$ N.

C. The angular momentum of the object about its origin $\vec L=15\hat k$ J s.

D. The torque acting on the object about its origin $\vec\tau=20\hat k$ N m.

Choose the correct answer from the options given below :

A 0.5 kg mass is in contact against the inner wall of a cylindrical drum of radius 4 m rotating about its vertical axis. The minimum rotational speed of the drum to enable the mass to remain stuck to the wall (without falling) is 5 rad/s. The coefficient of friction between the drum's inner wall surface and mass is ________. (Take $g=10$ m/s²)

Two blocks of masses 2 kg and 1 kg respectively, are tied to the ends of a string which passes over a light frictionless pulley as shown in the figure below. The masses are held at rest at the same horizontal level and then released. The distance traversed by the centre of mass in 2 s is ________ m. (Take $g=10$ m/s²)

The time taken by a block of mass $m$ to slide down from the highest point to the lowest point on a rough inclined plane is 50 % more compared to the time taken by the same block on identical inclined smooth plane. Both inclined planes are at 45° with the horizontal. The coefficient of kinetic friction between the rough inclined surface and block is ________.

A body of mass 2 kg begins to move under the influence of time dependent force $\vec F=\left(2t\hat i+6t^2\hat j\right)$ N, where $\hat i$ and $\hat j$ are unit vectors along $x$ and $y$-axis respectively. The power produced by the force at $t=2$ s is ________ W.

At $t=0$, a body of mass 100 g starts moving under the influence of a force $\left(5\hat i+10\hat j\right)$ N. After 2 s its position is $\left(2x\hat i+5y\hat j\right)$ m. The ratio $x:y$ is ________.

Three masses $m_1=4$ kg, $m_2=4$ kg and $m_3=6$ kg are suspended from a fixed smooth frictionless pully as shown in the figure below. The value of $T_1/T_2$ is ________.
(take $g=10$ m/s²)

A wedge $Y$ with mass of 10 kg and all frictionless surfaces and the inclined surface making 37° with horizontal. A block $X$ with mass 2 kg is placed at the highest point of the wedge as shown in figure is at rest. At $t=0$ wedge ($Y$) is pulled toward right with constant force ($f$) of 24 N. Taking the block $X$ at rest at $t=0$, the time taken by it to slide down 8.8 m on the slope, while $Y$ is on the move, is ________ s.
(take $\tan(37^\circ)=3/4$ and $g=10$ m/s²)

A mass of 1 kg is kept on a inclined plane with 30° inclination with respect to horizontal plane and it is at rest initially. Then the whole assembly is moved up with constant velocity of 4 m/s. The work done by the frictional force in time 2 s is ________ J. (Take $g=10$ m/s²)

Two blocks ($P$ and $Q$) with respectively masses 2 kg and 1.5 kg are joined by a massless thread. These blocks are mounted on a frictionless pully which is fixed on the edge of a cube ($S$), as shown in the figure below. Block $P$ is positioned on the top surface which has no friction and block $Q$ is in contact with side-surface, having coefficient friction $\mu$. The cube ($S$) moves towards the right with acceleration of $\dfrac g2$, where $g$ is gravitational acceleration. During this movement the block $P$ and $Q$ remain stationary. The value of $\mu$ is ________.

(take $g=10$ m/s²)

A lift of mass 1600 kg is supported by thick iron wire. If the maximum stress which the wire can withstand is $4\times10^8$ N/m² and its radius is 4 mm, then maximum acceleration the lift can take is ________ m/s².

(take $g=10$ m/s² and $\pi=3.14$)

The velocity at which 6 kg mass (shown in figure) strikes the ground when it is released from a height of 6 m above the ground is ________ m/s. Assume pulley is massless and string is light and inextensible. (Take $g=10$ m/s²)

A block takes $t$ time to slide down a plane inclined at 45° to the horizontal. If the surface is made smooth (frictionless), the block takes time $\dfrac t2$ to slide down the plane. The coefficient of friction between the block and the inclined plane is $\left(\dfrac\alpha{100}\right)$. The value of $\alpha$ is ________.

A car moving with a speed of 54 km/h takes a turn of radius 20 m. A simple pendulum is suspended from the ceiling of the car. Determine the angle made by the string of the pendulum with the vertical during the turning. (Take $g=10$ m/s²)

A solid cylinder having radius $R$ and length $L$ is slipping on a rough horizontal plane. At time $t=0$ the cylinder has a translational velocity $v_o=49$ m/s, perpendicular to its axis and a rotational velocity $v_o/4R$ about the centre. The time taken by the cylinder to start rolling is ________ seconds. (coefficient of kinetic friction $\mu_K=0.25$ and $g=9.8$ m/s²)

A conducting square loop of side $L$, mass $M$ and resistance $R$ is moving in the $XY$ plane with its edges parallel to the $X$ and $Y$ axes. The region $y \ge 0$ has a uniform magnetic field, $\vec B = B_0\hat k$. The magnetic field is zero everywhere else. At time $t = 0$, the loop starts to enter the magnetic field with an initial velocity $v_0\hat j$ m/s, as shown in the figure. Considering the quantity $K = \dfrac{B_0^2L^2}{RM}$ in appropriate units, ignoring self-inductance of the loop and gravity, which of the following statements is/are correct:

A person sitting inside an elevator performs a weighing experiment with an object of mass 50 kg. Suppose that the variation of the height $y$ (in m) of the elevator, from the ground, with time $t$ (in s) is given by $y = 8\left[1 + \sin\left(\dfrac{2\pi t}{T}\right)\right]$, where $T = 40\pi$ s. Taking acceleration due to gravity, $g = 10$ m/s$^2$, the maximum variation of the object's weight (in N) as observed in the experiment is _____.

The left and right compartments of a thermally isolated container of length $L$ are separated by a thermally conducting, movable piston of area $A$. The left and right compartments are filled with $\dfrac32$ and 1 moles of an ideal gas, respectively. In the left compartment the piston is attached by a spring with spring constant $k$ and natural length $\dfrac{2L}{5}$. In thermodynamic equilibrium, the piston is at a distance $\dfrac{L}{2}$ from the left and right edges of the container as shown in the figure. Under the above conditions, if the pressure in the right compartment is $P = \dfrac{kL}{A}\alpha$, then the value of $\alpha$ is _____.

A particle of mass $m$ is moving in a circular orbit under the influence of the central force $F(r) = -kr$, corresponding to the potential energy $V(r) = kr^2/2$, where $k$ is a positive force constant and $r$ is the radial distance from the origin. According to the Bohr's quantization rule, the angular momentum of the particle is given by $L = n\hbar$, where $\hbar = h/(2\pi)$, $h$ is the Planck's constant, and $n$ a positive integer. If $v$ and $E$ are the speed and total energy of the particle, respectively, then which of the following expression(s) is(are) correct?