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The child is running arbitrarily on a trolley moving with velocity v. However, the running of the child will produce no effect on the velocity of the centre of mass of the trolley. This is because the force due to the boy’s motion is purely internal. Internal forces produce no effect on the motion of the bodies on which they act. Since no external force is involved in the boy–trolley system, the boy’s motion will produce no change in the velocity of the centre of mass of the trolley.

Angular momentum of planet.

False
Frictional force acts opposite to the direction of motion of the centre of mass of a body. In the case of rolling, the direction of motion of the centre of mass is backward. Hence, frictional force acts in the forward direction.
True
Rolling can be considered as the rotation of a body about an axis passing through the point of contact of the body with the ground. Hence, its instantaneous speed is zero.
False
When a body is rolling, its instantaneous acceleration is not equal to zero. It has some value.
True

When perfect rolling begins, the frictional force acting at the lowermost point becomes zero. Hence, the work done against friction is also zero.
True

The rolling of a body occurs when a frictional force acts between the body and the surface. This frictional force provides the torque necessary for rolling. In the absence of a frictional force, the body slips from the inclined plane under the effect of its own weight.

A rigid body is said to be in equilibrium if both the linear momentum and angular momentum of the rigid body remain constant with time. 

It must posses the following two equilibria simultaneously:

(i) Translation equilibrium: The resultant of all the external forces acting on the body must be zero. 

(ii) Rotational equilibrium: The resultant of all the torques due to all the forces acting on the body about any point must be zero.

A body is said to posses’ rotational motion if all its particles move along circles in parallel planes. The centers of these circles lie on a fixed line perpendicular to the parallel planes and is called the axis of rotation .

A rigid body is one whose constituent particles retain their relative positions even when they move under the action of an external force.

Yes, if body has no linear and angular acceleration. Hence a body in uniform straight line motion will be in equilibrium.

In this case the force which bring the vehicle to rest is friction, and it is an external force.

The moment of a couple is equal to the product of either of the forces and the perpendicular distance, called the arm of the couple, between their lines of action. It is independent of the choice of the fulcrum or the point of rotation. 

Moment of couple = Force x arm of the couple 

A couple can only be balanced by an equal and opposite couple. 

When a body is in rotational equilibrium, the sum of the clockwise moments about any point is equal to the sum of the anticlockwise moments about that point. Or, the algebraic sum of moments about any point is zero. 

Clockwise moment = Anticlockwise moment 

Or Load* load arm = Effort*effort arm

This is sometimes called the lever principle.

The turning effect of a force about the axis of rotation is called moment of force or torque due to the force. 

SI unit of torque is Nm.

If a cat is dropped they almost always tend to land on their feet because they use the conservation of angular momentum to change their orientation. When a cat falls, as you would expect, its centre of mass follows a parabolic path. The cat falls with a definite angular momentum about an axis through the cat centre of mass. When the cat is in the air, no net external torque acts on it about its centre of mass, so the angular momentum about the cats centre of mass cannot change. By pulling in its legs, the cat can considerably reduce it rotational inertia about the same axis and thus considerably increase its angular speed. Stretching out its legs increases its rotational inertia and thus slows the cat angular speed. The conservation of angular momentum allows the cat to rotate its body and slow its rate of rotation enough so that it lands on its feet safely.

It may be defined as the distance from the axis of rotation at which if whole mass of the body were supposed to be concentrated, the moment of inertia, would be same as with the actual distribution of mass. 

It states that the moment of inertia of a plane lamina about an axis perpendicular to its plane is equal to the sum of the moment of inertia of the lamina about any two mutually perpendicular axes in its plane and intersecting each other at the point, where the perpendicular axes pass through the lamina. Mathematically, 

Where X-and Y- axes lie in the plane of the lamina and Z- axis is perpendicular to its plane and passes through the point of intersection of X- and Y- axes. 

When vertical line through centre of gravity passes through the base of the body.

Potential Energy is minimum.

A rigid body is a body that can rotate with all the parts locked together and without any change in its shape. 

Centre of mass of a system is a point that moves as though all the mass were concentrated there and all external forces were applied there.

For A V_A = Rω₀ in forward direction 

For B = V_B = Rω₀ in backward direction

For CV_C = R/2ω₀ in forward direction disc will not roll.  

No, external torque acts on a body, will its angular velocity be constant

ω ∝ I/I

(i) Shape of body 

(ii) mass distribution

Torque Factors 

(i) Magnitude of force 

(ii) Perpendicular distance of force vector from axis of rotation.

Mass distribution.

(i) Mass of body 

(ii) Size and shape of body 

(iii) Mass distribution w.r.t. axis of rotation 

(iv) Position and orientation of rotational axis

The moment of inertia of a body depends on the following factors:  

(i) Mass of the body. 

(ii) Size and shape of the body. 

(iii) Distribution of mass about the axis of rotation. 

(iv) Position and orientation of the axis of rotation with respect to the body.

It will be about an axis passing through the centre of the cube and connecting the opposite corners.

It is the moment of linear momentum of a particle about the axis of rotation. 

If a particle of mass m moves with uniform speed v along a circle of radius r, then its angular moment is L= mvr 

SI unit of angular momentum is kgm² 

The linear momentum and velocity vector are always parallel to each other. But the angular momentum and the angular velocity are not necessarily parallel vectors. 

Moment of inertia : 1/2 MR²

The moment of inertia of a rigid body about an axis of rotation is the sum of the products of masses of the various particles and squares of their perpendicular distances from the axis of rotation.

SI unit of moment of inertia = kg m² 

The moment of inertia of a body can be regarded as the rotational inertia of a body. It is the rotational analogue of mass in linear motion. 

No, it changes with the position of axis of rotation.

Sphere of small density will have large moment of inertia.