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54 PHYSICS character of momentum has not been evident. ∆p ∆p Intheexamplessofar,momentumandchange F ∝ or F = k ∆t ∆t in momentum both have the same direction. But this is not always the case. Suppose a where k is a constant of proportionality. Taking stone is rotated with uniform speed in a ∆p the limit ∆t → 0, the term becomes the horizontal plane by means of a string, the ∆t magnitude of momentum is fixed, but its derivative or differential co-efficient of p with direction changes (Fig. 4.4). A force is needed to cause this change in momentum vector. respect to t, denoted by dp . Thus dt This force is provided by our hand through the string. Experience suggests that our hand F = kdp (4.2) needs to exert a greater force if the stone is dt rotated at greater speed or in a circle of For a body of fixed mass m, smaller radius, or both. This corresponds to greater acceleration or equivalently a greater dp = d m v = m dv = m a (4.3) rate of change in momentum vector. This dt dt ( ) dt suggests that the greater the rate of change in momentum vector the greater is the force i.e the Second Law can also be written as F = k m a (4.4) applied. which shows that force is proportional to the product of mass m and acceleration a. The unit of force has not been defined so far. In fact, we use Eq. (4.4) to define the unit of force. We, therefore, have the liberty to choose any constant value for k. For simplicity, we choose k = 1. The second law then is F = dp = m a (4.5) dt In SI unit force isone that causes an acceleration of 1 m s to a mass of 1 kg. This unit is known as newton : 1 N = 1 kg m s .2 Fig. 4.4 Forceisnecessary forchangingthedirection Letusnoteatthisstagesomeimportantpoints of momentum, even if its magnitude is about the second law : constant. We can feel this while rotating a1. Inthesecondlaw,F=0impliesa=0.Thesecond stoneinahorizontalcirclewithuniformspeed law is obviously consistent with the first law. by means of a string. 2. The second law of motion is a vector law. It is These qualitative observations lead to the equivalent to three equations, one for each second law of motion expressed by Newton as component of the vectors : follows: F = dp x = ma The rate of change of momentum of a body is x dt x directly proportional to the applied force and takes place in the direction in which the force dp y acts. Fy= dt = ma y Thus, if under the action of a force F for time Fz= dp z=m a z (4.6) interval ∆t, the velocity of a body of mass m dt changes fromv tov+∆vi.e. its initial momentum This means that if a force is not parallel to p = m v changes by ∆p = m∆v . According to the the velocity of the body, but makes some angle with it, it changes only the component of Second Law, velocity along the direction of force. The Reprint 2026-27