Laws of Motion

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$^2$]

A block of mass M slides down on a rough inclined plane with constant velocity. The angle made by the incline plane with horizontal is θ. The magnitude of the contact force will be :

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$^2$)

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 :

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$^2$}

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$^3$]

A uniform metal chain of mass m and length 'L' passes over a massless and frictionless pulley. It is released from rest with a part of its length '$l$' is hanging on one side and rest of its length 'L − $l$' is hanging on the other side of the pully. At a certain point of time, when $l=\dfrac Lx$, the acceleration of the chain is $\dfrac g2$. The value of $x$ is ______.

A smooth circular groove has a smooth vertical wall as shown in figure. A block of mass m moves against the wall with a speed $v$. Which of the following curve represents the correct relation between the normal reaction on the block by the wall (N) and speed of the block ($v$) ?

The time period of oscillation of a simple pendulum of length L suspended from the roof of a vehicle, which moves without friction down an inclined plane of inclination α, is given by :

Two bodies of masses m$_1$ = 5 kg and m$_2$ = 3 kg are connected by a light string going over a smooth light pulley on a smooth inclined plane as shown in the figure. The system is at rest. The force exerted by the inclined plane on the body of mass m$_1$ will be :[Take g = 10 m s$^{-2}$]

A tube of length 50 cm is filled completely with an incompressible liquid of mass 250 g and closed at both ends. The tube is then rotated in horizontal plane about one of its ends with a uniform angular velocity $x\sqrt F$ rad s$^{-1}$. If F be the force exerted by the liquid at the other end then the value of $x$ will be _______.

A metal wire of length 0.5 m and cross-sectional area $10^{-4}$ m$^2$ has breaking stress $5\times10^8$ N m$^{-2}$. A block of 10 kg is attached at one end of the string and is rotating in a horizontal circle. The maximum linear velocity of block will be _______ m s$^{-1}$.

A mass of 0.5 kg is attached to one end of a massless spring whose natural length is 1 m and spring constant is 800 N/m. The other end of the spring is fixed while the mass moves in a circular path in a horizontal plane with an angular speed of 20 rad/s. The extension in the length of the spring will be :

An oscillating simple pendulum of time period T$_o$ is placed in a lift which is accelerating upwards with 2.5 m/s$^2$. The time period of pendulum in the lift will be (if g = 10 m/s$^2$)

A horizontal force $F$ is applied at the center of mass of a cylindrical object of mass $m$ and radius $R$, perpendicular to its axis as shown in the figure. The coefficient of friction between the object and the ground is $\mu$. The center of mass of the object has an acceleration $a$. The acceleration due to gravity is $g$. Given that the object rolls without slipping, which of the following statement(s) is(are) correct?

A cylindrical tube, with its base as shown in the figure, is filled with water. It is moving down with a constant acceleration $a$ along a fixed inclined plane with angle $\theta = 45^\circ$. $P_1$ and $P_2$ are pressures at points 1 and 2, respectively, located at the base of the tube. Let $\beta = (P_1 - P_2)/(\rho g d)$, where $\rho$ is density of water, $d$ is the inner diameter of the tube and $g$ is the acceleration due to gravity. Which of the following statement(s) is(are) correct?

The value of $J$ is ____.

The value of $K$ is ____.

Four identical particles of equal masses 1 kg made to move along the circumference of a circle of radius 1 m under the action of their own mutual gravitational attraction. The speed of each particle will be :