CENTRE OF MASS AND ROTATIONAL MOTION
For describing the motion of rigid bodies, we shall introduce the key concept of ‘centre of mass’. This concept enables us to understand how we can apply justifiably the Newton’s laws of motion, in essentially the same form to objects of large size including even the astronomical objects like the planets and the stars. KINDS OF MOTION OF A RIGID BODY-
A rigid body may have three kinds of motion-
(1) Pure Translation Motion- in such a motion, every particle of the body has the same velocity at a particular instant of time. For e.g. when a rectangular block slides down an inclined plane, any point like P1,P2 of the block, at any instant of time moves with the same velocity. This is because the block is a rigid body and all the particles of the body are moving together, with the same velocity. (2)Pure Rotational Motion- in such a motion, a rigid body rotates about a fixed axis. Every particle of the body moves in a circle, which lies in a plane perpendicular to the axis, and has its centre on the axis. For e.g., in an oscillating table fan or pedestal fan, the axis of rotation is horizontal. This axis has an oscillating sideways movement in a horizontal plane about the vertical through the point at which the axis is pivoted. (3) Combination Of Translational And Rotational Motion- for e.g., when a cylinder rolls down an inclined plane, its motion is a combination of rotation about a fixed axis and translation. As the cylinder shifts from top to the bottom of inclined plane, the points P1, P2, P3, P4 on the rolling cylinder have different velocities at a particular instant of time. If the cylinder were to roll without slipping, the velocity of the point of contact P3 would be 0, at any instant of time. From the above discussion we conclude that-
(i)The motion of a rigid body, which is not provided or fixed in some way is either a pure translation or a combination of translation and rotation. (ii)The motion of a rigid body, which is pivoted or fixed in some way is rotation. The rotation may be about an axis which is fixed e.g., in a ceiling fan or about an axis which is moving e.g., in an oscillating table fan. We shall confine ourselves to rotation about a fixed axis. CONCEPT OF CENTRE OF MASS-
The concept of centre of mass of a system enables us to discuss overall motion of the system by replacing the system by an equivalent single point object. We may define centre of mass of a body or a system of bodies as a point at which the entire mass of the body/system of bodies, is supposed to be concentrated. If all the external forces acting on the body/system of bodies were to be applied at the centre of mass, the state of rest/motion of the body/system of bodies, shall remain unaffected. It is not at all necessary that the total mass of the system be actually present at the centre of mass. CENTRE OF MASS OF TWO PARTICLE SYSTEM-
Suppose two particles of masses m1 and m2 are separated by a distance d. If we take the origin at the point mass m1 and the line joining the two particles as x-axis, then the co-ordinates of m1 and m2 are (0,0) and (d,0) respectively. m1 M m2 0 d1 d2 x d
so, xcm = (m1*0 + m2d)/(m1+m2) =(m2d)/(m1+m2) ycm=(m1*0+m2*0)/(m1+ m2)=0 zcm=0 Thus the centre of mass lies on the line joining the two particles at a distance of (m2d)/(m1+m2) from the mass m1. i.e. , d1=m2d/(m1+m2) and d2=m1d/(m1+m2) or, d1/d2=m2/m1...