System of Particles and Rotational Motion

• Centre of mass of a two-particle system

• momentum conservation and centre of mass motion

• Centre of mass of a rigid body

• Centre of mass of a uniform rod

• Moment of a force

• Torque

• angular momentum

• laws of conservation of angular momentum and its applications

• Equilibrium of rigid bodies

• rigid body rotation and equations of rotational motion

• comparison of linear and rotational motions

• Moment of inertia

• radius of gyration

• Values of moments of inertia, for simple geometrical objects (no derivation)

• Statement of parallel and perpendicular axes theorems and their applications

SUMMARY

1. Ideally, a rigid body is one for which the distances between different particles of the body do not change, even though there are forces on them.

2. A rigid body fixed at one point or along a line can have only rotational motion. A rigid body not fixed in some way can have either pure translational motion or a combination of translational and rotational motions.

3. In rotation about a fixed axis, every particle of the rigid body moves in a circle which lies in a plane perpendicular to the axis and has its centre on the axis. Every Point in the rotating rigid body has the same angular velocity at any instant of time.

4. In pure translation, every particle of the body moves with the same velocity at any instant of time.

5. Angular velocity is a vector. Its magnitude is ω = dθ/dt and it is directed along the axis of rotation. For rotation about a fixed axis, this vector ω has a fixed direction.

6. The vector or cross product of two vector a and b is a vector written as a×b. The magnitude of this vector is absinθ and its direction is given by the right handed screw or the right hand rule.

7. The linear velocity of a particle of a rigid body rotating about a fixed axis is given by v = ω × r, where r is the position vector of the particle with respect to an origin along the fixed axis. The relation applies even to more general rotation of a rigid body with one point fixed. In that case r is the position vector of the particle with respect to the fixed point taken as the origin.

8. Velocity of the centre of mass of a system of particles is given by V = P/M, where P is the linear momentum of the system. The centre of mass moves as if all the mass of the system is concentrated at this point and all the external forces act at it. If the total external force on the system is zero, then the total linear momentum of the system is constant.

9. A rigid body is in mechanical equilibrium if (1) it is in translational equilibrium, i.e., the total external force on it is zero and (2) it is in rotational equilibrium, i.e. the total external torque on it is zero.

10.The centre of gravity of an extended body is that point where the total gravitational torque on the body is zero.

11.Rotation about a fixed axis is directly analogous to linear motion in respect of kinematics and dynamics.

12. For a rigid body rotating about a fixed axis (say, z-axis) of rotation, Lz = Iω, where I is the moment of inertia about z-axis. In general, the angular momentum L for such a body is not along the axis of rotation. Only if the body is symmetric about the axis of rotation, L is along the axis of rotation. In that case, | L |= Lz = Iω . The angular acceleration of a rigid body rotating about a fixed axis is given by Iα = τ. If the external torque τ acting on the body is zero, the component of angular momentum about the fixed axis (say, z-axis), Lz (=Iω) of such a rotating body is constant.