System of particles practice · Medium

Two loops $P$ and $Q$ are made from a uniform wire. The radii of $P$ and $Q$ are $r_1$ and $r_2$ respectively, and their moments of inertia are $I_1$ and $I_2$ respectively. If $I_2=4 I_1$, then $\frac{r_2}{r_1}$ equals -

A particle of mass 1 kg is projected with an initial velocity 10 $\mathrm{m}{\mathrm{s}^{ - 1}}$  at an angle of projection ${45^0}$ with the horizontal. The average torque acting on the projectile, between the time at which it is projected and the time at which it strikes the ground, about the point of projection in newton-metre is

A particle of mass 1 kg is projected with an initial velocity 10 $\mathrm{m}{\mathrm{s}^{ - 1}}$  at an angle of projection ${45^0}$ with the horizontal. The average torque acting on the projectile, between the time at which it is projected and the time at which it strikes the ground, about the point of projection in newton-metre is

A uniform sphere of mass 250 g rolls without slipping on a plane horizontal surface with its centre moving at a speed of 5 cm/s. Its kinetic energy is

An ice stacker spins at $3\, \pi\, rad \, s^{-1}$with her arms extended. If her moment of inertia with arms folded is $75 \%$ of that with arms extended, her angular velocity when she folds her arms is

If a body of moment of inertia $2\, {~kg} \,{~m}^2$ revolves about its own axis making 2 rotations per second, then its angular momentum (in ${J} {s}$ ) is

A constant torque acting on a uniform circular wheel changes its angular momentum from ${A_0}$ to $4{A_0}$ in 4 seconds. The magnitude of this torque is

A fly-wheel of mass $25 \mathrm{~kg}$ has a radius of $0.2 \mathrm{~m}$. It is making $240 \mathrm{~rpm}$. What is the torque necessary to bring to rest in $20 \mathrm{~s}$?

The instantaneous angular position of a point on a rotation wheel is given by the equation $\theta (t) = 2{t^3} - 6{t^2}$. The torque on the wheel becomes zero at

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

Three identical sphere lie at rest along a line on a smooth horizontal surface. The separation between any two adjacent spheres is L. The first sphere is moved with a velocity u towards the second sphere at time t=0. The coefficient of restitution for collision between any two blocks is 1/3. Then choose the correct statement

Assertion: If linear momentum of a particle is constant, then its angular momentum about any point will also remain constant.

Reason: Linear momentum remains constant, if ${F}_{\text {net}}=0$ and angular momentum remains constant if $\tau_{\text {net}}=0$.

A hollow sphere and a solid sphere, of equal mass and equal radii roll down without slipping on an inclined plane. If the torque experienced by the hollow sphere and solid sphere are $\tau_{\mathrm{H}}$ and $\tau_{\mathrm{S}}$ respectively, then

Assertion: When a sphere is rolls freely on a horizontal table, it slows down and eventually stops.

Reason: When the sphere rolls on the table, both the sphere and the surface deform near the contact. As a result, the normal force does not pass through the centre and provide an angular deceleration.

We have two spheres, one of which is hollow and other is solid. They have identical mass and moment of inertia about its tangent. The ratio of their radii is given by

A hollow sphere of mass $M$ and radius $R$ is in pure rolling on the horizontal rough surface. The ratio of rotational kinetic energy to the total kinetic energy of the hollow sphere will be

When a ceiling fan is switches on it makes 10 revolutions in the first 3 seconds. Assuming a uniform angular acceleration, how many rotation it will make in the next 3 seconds.

Two particles of equal masses have velocities ${v}_1=4 \hat{i}{\mathrm{~ms}}^{-1}$ and ${v}_2=4 \hat{{j}} \mathrm{~ms}^{-1}.$ First particle has an acceleration $a_1=(2 \hat{i}+2 \hat{j}) \mathrm{ms}^{-2},$ while the acceleration of the other particle is zero. The centre of mass of the two particles moves in a path of:

A hollow sphere and solid sphere rolls on a horizontal surface without slipping with same linear velocity. If both has same mass and radius and total kinetic energy of hollow sphere is $15 \mathrm{~J}$ then total kinetic energy of the solid sphere is

The highest moment of inertia if all the objects have same mass and same radius is of

A body of mass 36 kg moves from one end to other end of a boat of mass 72 kg in still water. If the length of the boat is 2.7 m, the displacement of the boat is

A nonzero external force acts on a system of particles. The velocity and the acceleration of the centre of mass are found to be $v_0$ and $a_0$ at an instant $t$. It is possible that

S-I: Ratio of the distances of the centre of mass of a two particle system are in the inverse ration of their masses
S-II: The location of the centre of mass is independent of the reference frame used locate it.

When a body does not rotate or rotates with a constant angular velocity $(\omega).$ It is said to be in:

A solid cylinder of mass $M$ and radius $R$ rolls without slipping down an inclined plane of length $L$ and height $h$. What is the speed of its centre of mass when the cylinder reaches its bottom

Let $\mathrm{M}$ and $\mathrm{L}$ be the mass and length of thin uniform rod respectively. In $1^{\text {st }}$ case, axis of rotation is passing through centre and perpendicular to its length. In $2^{\text {nd }}$ case, axis of rotation is passing through one end and perpendicular to its length. The ratio of radius of gyration in first case to second case is

The centre of mass of a body is defined as the point at which the whole of its mass is supposed to be concentrated, while centre of gravity of a body is defined as the point at which whole of its weight is supposed to be concentrated then,

An equilateral triangle $A B C$ formed from a uniform wire has two small identical beads initially located at $A$. The triangle is set rotating about the vertical axis $A O$. Then the beads are released from rest simultaneously and allowed to slide down, one along $A B$ and the other along $A C$ as shown. Neglecting frictional effects, the quantities that are conserved as the beads slide down, are

The unit mass has ${r}=8 \hat{{i}}-4 \hat{{j}}$ and ${v}=8 \hat{{i}}+4 \hat{{j}}$. Its angular momentum is

Two uniform solid spheres having unequal masses and unequal radii are released from rest from the same height on a rough incline. If the spheres roll without slipping,