System of particles practice · Medium

The moment of inertia $(I)$ of a sphere of radius $R$ and mass $M$ is given by

Statement-1: If the centre of mass of a system is the origin then the total mass to the right of origin is same as total mass to left of origin.
Statement-2: $x_{\mathrm{cm}}=\frac{\Sigma_{m_i x_i}}{\Sigma_{m_i}}$

A particle performing uniform circular motion has

Which of the following relation is correct? (Where symbol have their usual meanings)

The two persons of masses 55 kg and 65 kg respectively, are at at opposite ends of a boat. The length of the boat is 30 m and weight 100 kg. The 55 kg man walkup to the 65 kg man and its with him. If the boat is in still water the centre of mass of the system shifts by

An automobile engine develops ${10^5}\,\mathrm{W}$ power when rotating at an angular speed of 900 rad/minute. The instantaneous torque delivered by the engine is

A disc of radius '$r$' is removed from the disc of radius '$R$' then
a) The minimum shift in centre of mass is zero
b) The maximum shift in centre of mass cannot be greater than $r^2 /(R+r)$
c) Centre of mass must lie where mass exists
d) The shift in centre of mass is $r^2 /(R+r)$

A solid sphere of radius ${R}$ is placed on a smooth horizontal surface. A horizontal force $F$ is applied at height $h$ from the lowest point. For the maximum acceleration of centre of mass, which is correct?

What remains constant in the field of central force

The linear acceleration of the centre of gravity of a solid sphere which rolls down without slipping over an inclined plane with an angle of inclination ${30^0}$ is

STATEMENT-1: If there is no external torque on a body about its center of mass, then the velocity of the center of mass remains constant.

STATEMENT-2: The linear momentum of an isolated system remains constant.

A particle of mass m is circulating on a circle of radius r having angular momentum L, then the centripetal force will be

In a bicycle the radius of rear wheel is twice the radius of front wheel. If ${r_F}\,and\,{r_r}$ are the radius, ${v_F}\,and\,{v_r}$, are speeds of top most points of wheel, then

The radius vector and linear momentum are respectively given by vector $\mathrm{2\hat i} + 2\mathrm{\hat j} + \mathrm{\hat k \,\,and\,\, 2\hat i} - 2\mathrm{\hat j} + \mathrm{\hat k}$  Their angular momentum is

In an elastic head on collision between two particles

The centre of mass of a body

One circular ring and one circular disc, both are having the same mass and radius. The ratio of their moments of inertia about the axes passing through their centres and perpendicular to their planes, will be

If a force $10\hat i + 15\hat j + 25\hat k$ acts on a system and gives an acceleration $2\hat i + 3\hat j - 5\hat k$ to the centre of mass of the system, the mass of the system is

A small object of mass m is attached to a light string which passes through a hollow tube. The tube is hold by one hand and the string by the other. The object is set into rotation in a circle of radius R and velocity v. The string is then pulled down, shortening the radius of path of $r$. What is conserved

A body of moment of inertia ${4 kgm^2}$ rotating with an angular speed of 4 rad/s has the 4 x J of kinetic energy. Determine the value of x.

Which of the following statements are incorrect about centre of mass?
I) Centre of mass can coincide with geometrical centre of a body
II) Centre of mass of a system of two particles does not always lie on the line joining the particles
III) Centre of mass should always lie on the body

A fan is making 600 revolutions per minute. If after some time it makes 1200 revolutions per minute, then increase in its angular velocity is

A disk and a sphere of same radius but different masses roll off on two inclined planes of the same altitude and length. Which one of the two objects gets to the bottom of the plane first?

A curved road of diameter 1.8 km is banked so that no friction is required at a speed of $30\;\mathrm{m}{\mathrm{s}^{ - 1}}$. What is the banking angle?

A bomb travelling in a parabolic path under gravity, explodes in mid air. The centre of mass of fragments will move

There are two identical balls of same material, one being solid and the other being hollow. How will you distinguish them without weighing?

The centre of mass of a system cannot change its state of motion, unless there is external force acting on it. Yet the internal force of the brakes can bring a car to rest. Then

A stone is swinging in a horizontal circle 0.8 m in diameter, at 30rev/min. A distant light causes a shadow of the stone to be formed on a nearby wall. What is the amplitude of the motion of the shadow? What is the frequency?

A uniform thin rod of mass $M$ and length $L$ is standing vertically along the $Y$-axis on a smooth horizontal surface with its lower end at origin $(0,0)$. A slight disturbance at $t=0$ causes the lower end to slip on smooth surface along the positive X-axis and the rod starts falling. The path followed by the centre of mass of the rod during its fall is a/an.

A non-zero external force acts on a systems of particles. The velocity and acceleration of the centre of mass are found to be ${v_0}\,and\,{a_0}$ at an instant t. It is possible that