Center of Mass and Rotational Mechanics

The concept of Center of mass of a system enables us to discuss overall motion of the system by replacing the system by an equivalent single point object.

Center of mass of a body or a system of bodies is defined as a point at which the entire mass of the entire/system of bodies, is supposed to be concentrated.

The center of mass of the given system is only a point defined mathematically for the sake of convenience. In certain cases, there may not be any mass present at the center of mass.

The concept of center of mass, motion of total mass of the system is describe under the effect of external forces only. The internal forces between the particles forming the system cancel out in pairs.

The position of the center of mass of a system is independent of the choice of co-ordinate system.

The position of the center of mass depends on the shape and size of the body and the distribution of its mass. Hence, it may lie within or outside the material of the body.

In symmetrical bodies with uniform distribution of mass, the center of mass coincides with the geometrical center of symmetry of the body.

The center of mass changes its potential only under the translatory motion. There is no effect of rotatory motion on center of mass of a body.

Velocity of center of mass is not affected by internal forces. Further, if resultant external force action on a system is zero, velocity of center of mass and hence linear momentum of the system remains constant.

The system, as a whole, may be changing shapes or orientations due to internal forces, but it will have no effecton the trajectroy of center of mass of isolated system.

The center of gravity of a body is a point where the whole weight of the body is supposed to be concentrated. for ordinary bodies, the center of mass and center of gravity coincide with each other.

The moment of force or torque due to a force gives us the turning effect on the force about the fixed point/axis. It is measured by the product of magnitude of force and perpendicular distance of the line of action of force from the axis of rotation.

Angular momentum of a body about a given axis is the product of linear momentum and perpendicular distance of line of linear momentum vector from the axis of rotation.

Conceptual points

- If a body is rotating, is it necessarily being acted upon by an external torque? No, torque is required only for producing angular acceleration. For uniform rotation, no torque is required.
- A child is sits stationary at one end of a long trolley moving uniformly with a speed v on a smooth horizontal floor. If the child gets up and runs about on the trolley in any manner, what is the speed of the c.m. of system.
- the speed of the center of mass of the system (trolley + child) shall remain unchanged when the child gest up and runs about on the trolley in any manner. This is because forces involved in the exercise are purely internal i.e. from within the system. No external force acts on the system and hence there is no change in velocity of the system.

- To produce a given turning moment, force required is smaller, when r is large. This is what happens when handle of the screw is made wide.
- Angular momentum (L)= I (inertia) *W (omega)
- The radial component of linear momentum makes no contribution to angular momentum.
- Moment of linear momentum= angular momentum
- Rate of change of angular momentum: Torque

The **Radius of Gyration (K) **of a body about a given axis is the perpendicular distance of a point P from the axis, where if whole mass of the body were concentrated, the body shall have the same moment of Inertia as it has with the actual distribution of mass.

Torque =I * alpha = dL\dt

Principle of conservation of angular momentum: according to this principle, when no external torque acts on a system of particles, then the total angular momentum of the system remains always a constant.

We know that in the absence of an external torque, angular momentum vector remains constant. Hence, the direction of L is also remains unchanged. This suggests a fixed plane for the orbit of every planet. Hence, we conclude that every planet revolves around the sun in a fixed elliptical orbit i.e. plane of the orbit of a planet can never change on its own.

The angular velocity of a planet around the sun in an elliptical orbit increases, when the planet come closer to the sun and vice-versa. This is because, when the planet comes closer to the sun, its moment of inertia (I) about the sun decreases. As L= I*W is constant, therefore, its angular velocity (omega) increase.

Examples:

- A circus acrobat performs feats involving spin by bringing his arms and legs closer to his body or vice-versa.
- On the same basis, an ice skater or a ballet dancer perform the feats.
- The angular speed of inner layers of the whirl wind (tornado) is alarmingly high. This is because the moment of Inertia of inner layer is much smaller.
- All helicopters are provided with two propellers.
- A spacecraft orientation.
- The incredible shrinking of star. (in all the examples, angular momentum is conserved but K.E. of rotation is not conserved).

Theorems on Moment of Inertia

- Theorem of parallel axis => I (ab) = I (kl) +Mh^2 (M is the mass and h is the perpendicular distance between two axes)
- Theorem of perpendicular axis=> I (z) = I (x)+ I (y)

Moment of Inertia of

- Uniform rod of length l = (1/12) Ml^2
- Uniform circular ring of radius R = MR^2 = Hollow cylinder of radius R
- Uniform circular disc of radius R = 1/2 MR^2 = Solid cylinder of radius R
- Hollow sphere of radius R = 2/3 M*R^2
- Solid Sphere of radius R = 2/5 M*R^2

Equilibrium of Rigid Bodies

**First condition of equilibrium**: a rigid body is said to be in translational equilibrium, if it remains at rest or moving with a constant velocity in a particular direction. For this, the net external force or the vector sum of all the external forces acting on the body must be zero.**Second condition of equilibrium:**a rigid body is said to be in rotational equilibrium, if the body does not rotate or rotates with constant angular velocity. For this, the net external torque or the vector sum of all the torques acting on the body is zero.

Two stars that are bound to each other by gravity and orbit about a common center of mass are called binary stars.

Moment of inertia of ring is greater than disc because entire mass of ring is at its periphery i.e. at maximum distance from the center.

A body in translatory motion can have angular momentum, unless the point lies on the line of motion.