Newton's laws

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 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 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 $\pi=3.14$)

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:

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 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 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 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 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 system to 10 balls each of mass 2 kg are connected via massless and unstretchable string. The system is allowed to slip over the edge of a smooth table as shown in figure. Tension on the string between the 7$^{\rm th}$ and 8$^{\rm th}$ ball is ______ N when 6$^{\rm th}$ ball just leaves the table.

A hanging mass M is connected to a four times bigger mass by using a string-pulley arrangement, as shown in the figure. The bigger mass is placed on a horizontal ice-slab and being pulled by 2 Mg force. In this situation, tension in the string is $\frac x5$ Mg for x = ______. Neglect mass of the string and friction of the block (bigger mass) with ice slab.

(Given g = acceleration due to gravity)

A block of mass M placed inside a box descends vertically with acceleration 'a'. The block exerts a force equal to one-fourth of its weight on the floor of the box. The value of 'a' will be

For a free body diagram shown in the figure, the four forces are applied in the '$x$' and '$y$' directions. What additional force must be applied and at what angle with positive x-axis so that the net acceleration of body is zero ?

A monkey of mass 50 kg climbs on a rope which can withstand the tension (T) of 350 N. If monkey initially climbs down with an acceleration of 4 m/s$^2$ and then climbs up with an acceleration of 5 m/s$^2$. Choose the correct option ($g=10$ m/s$^2$).

A block 'A' takes 2 s to slide down a frictionless incline of 30° and length '$l$', kept inside a lift going up with uniform velocity '$v$'. If the incline is changed to 45°, the time taken by the block, to slide down the incline, will be approximately :

Two inclined planes are placed as shown in figure. A block is projected from the Point A of inclined plane AB along its surface with a velocity just sufficient to carry it to the top Point B at a height 10 m. After reaching the Point B the block slides down on inclined plane BC. Time it takes to reach to the point C from point A is $t\left(\sqrt2+1\right)$ s. The value of t is ______.

(use g = 10 m/s$^2$)

As shown in the figure, a metallic rod of linear density 0.45 kg m$^{-1}$ is lying horizontally on a smooth inclined plane which makes an angle of 45° with the horizontal. The minimum current flowing in the rod required to keep it stationary, when 0.15 T magnetic field is acting on it in the vertical upward direction, will be :

{Use g = 10 m/s$^2$}