Center of Mass

A ball falls from a height $h$ upon a fixed horizontal floor. The co-efficient of restitution for the collision between the ball and the floor is ' $e$ '. The total distance covered by the ball before coming to rest is [neglect the air resi...

The variation of density of a solid cylindrical rod of cross sectional area $\alpha$ and length $L$ is $\rho=\rho_0 \frac{x^2}{L^2}$, where $x$ is the distance from one end of the rod. The position of its centre of mass from one end $(x=0)$...

The position of the centre of mass of the uniform plate as shown in the figure is

A uniform rod \(\mathrm{AB}\) of length \(1 \mathrm{~m}\) and mass \(4 \mathrm{~kg}\) is sliding along two mutually perpendicular frictionless walls OX and OY. The velocity of the two ends of the \(\operatorname{rod} \mathrm{A}\) and $$\ma...

There are n elastic balls placed on a smooth horizontal plane. The masses of the balls are \(\mathrm{m}, \frac{\mathrm{m}}{2}, \frac{\mathrm{m}}{2^{2}}, \ldots \frac{\mathrm{m}}{2^{\mathrm{n}-1}}\) respectively. If the first ball hits the s...

A particle of mass m is projected at a velocity u, making an angle \(\theta\) with the horizontal (x-axis). If the angle of projection \(\theta\) is varied keeping all other parameters same, then magnitude of angular momentum (L) at its max...

Three particles, each of mass 'm' grams situated at the vertices of an equilateral \(\Delta\)ABC of side 'a' cm (as shown in the figure). The moment of inertia of the system about a line AX perpendicular to AB and in the plane of ABC in g-c...

A body of mass m is thrown with velocity u from the origin of a co-ordinate axes at an angle \(\theta\) with the horizon. The magnitude of the angular momentum of the particle about the origin at time t when it is at the maximum height of t...

A particle is moving in an elliptical orbit as shown in figure. If \(\overrightarrow p\), \(\overrightarrow L\) and \(\overrightarrow r\) denote the linear momentum, angular momentum and position vector of the particle (from focus O) re...

Centre of mass (CM) of three particles of masses 1 kg, 2 kg and 3 kg lies at the point (1, 2, 3) and CM of another system of particles of 3 kg and 2 kg lies at the point (\(-\)1, 3, \(-\)2). Where should we put a particle of mass 5 kg, so t...

A uniform thin rod of length L, mass m is lying on a smooth horizontal table. A horizontal impulse P is suddenly applied perpendicular to the rod at one end. The total energy of the rod after the impulse is

A solid spherical ball and a hollow spherical ball of two different materials of densities \(\rho\)1 and \(\rho\)2 respectively have same outer radii and same mass. What will be the ratio, the moment of inertia (about an axis passing thro...

Two particles A and B (both initially at rest) start moving towards each other under a mutual force of attraction. At the instant, when the speed of A is v and the speed of B is 2v, the speed of the centre of mass is

A small steel ball bounces on a steel plate held horizontally. On each bounce the speed of the ball arriving at the plate is reduced by a factor e (coefficient of restitution) in the rebound, so that Vupward = eVdownward. If the ball is ini...

Two bodies of masses m1 and m2 are separated by a distance R. The distance of the centre of mass of the bodies from the mass m1 is