Mechanics Questions

One end of string of length $l$ is connected to a particle of mass ‘$m$ ’ and the other end is connected to a small peg on a smooth horizontal table. If the particle moves in circle with speed ‘$v$ ’, the net force on the particle (directed towards center) will be ($T$ represents the tension in the string):

The $x$ and $y$ coordinates of the particle at any time are $x=5t-2t^2$ and $y=10t$ respectively, where $x$ and $y$ are in meters and $t$ in seconds. The acceleration of the particle at $t=2\text{s}$ is:

A particle executes linear simple harmonic motion with an amplitude of $3\text{cm}$. When the particle is at $2\text{cm}$ from the mean position, the magnitude of its velocity is equal to that of its acceleration. Then its time period in seconds is:

A spring of force constant $k$ is cut into lengths of ratio $1:2:3$. They are connected in series and the new force constant is $k^{\prime }$. Then they are connected in parallel and force constant is $k^{\prime \prime }$. Then $k^{\prime }:k^{\prime \prime }$ is:

The acceleration due to gravity at a height $1\text{km}$ above the earth is the same as at a depth $d$ below the surface of earth. Then:

Which of the following statements are correct?(a) Centre of mass of a body always coincides with the centre of gravity of the body.(b) Centre of mass of a body is the point at which the total gravitational torque on the body is zero.(c) A couple on a body produce both translational and rotational motion in a body.(d) Mechanical advantage greater than one means that small effort can be used to lift a large load.

Two astronauts are floating in gravitational free space after having lost contact with their spaceship. The two will:

Consider a drop of rain water having mass $1\text{g}$ falling from a height of $1\text{km}$. It hits the ground with a speed of $50\text{m/s}$. Take $g$ constant with a value $10{\text{m/s}}^2$. The work done by the (i) gravitational force and the (ii) resistive force of air is:

Two blocks $A$ and $B$ of masses $3m$ and $m$ respectively are connected by a massless and inextensible string. The whole system is suspended by a massless spring. The magnitudes of acceleration of $A$ and $B$ immediately after the string is cut, are respectively:

Two discs of same moment of inertia rotating about their regular axis passing through centre and perpendicular to the plane of disc with angular velocities ${\omega }_1$ and ${\omega }_2$. They are brought into contact face to face coinciding the axis of rotation. The expression for loss of energy during this process is:

Preeti reached the metro station and found that the escalator was not working. She walked up the stationary escalator in time $t_1$. On other days, if she remains stationary on the moving escalator, then the escalator takes her up in time $t_2$. The time taken by her to walk up on the moving escalator will be:

A rope is wound around a hollow cylinder of mass $3\text{kg}$ and radius $40\text{cm}$. What is the angular acceleration of the cylinder if the rope is pulled with a force of $30\text{N}$?