According to this law, "Rate of change of momentum of a moving body is directly proportional to force applied on the body and in same direction in which force is applied.
Suppose the momentum of a moving body is \(\vec{p}\), then rate of change of momentum \(=\frac{d\vec p}{dt}\)
∴ According to second law of motion, the force acting on the body,
\(\vec F\propto\frac{d\vec p}{dt}\)
or \(\vec F = K\frac{d\vec p}{dt}\ ...(1)\)
where K is constant of proportionality. The value of K depends on the system of chosen unit. We chose the units such that
K = 1
\(\therefore\ \vec F = \frac{d\vec p}{dt}\ ..(2)\)
Second law of motion is known as real law.
Therefore the force applied on a body is defined by the product of mass of body and acceleration produced.
Case (ii) When Velocity is constant, then
\(\frac{dv}{dt} = 0\)
\(\therefore\ \vec F = \vec v\frac{dm}{dt}\)
Unit of force : ∵ F = ma
∴ In M.K.S. system, unit of F = kg.ms-2 = Newton (N)
In C.G.S. system, unit of F= g.cm.s-2 = dyne
Dimensions: ∵ F = ma
∴ Dimensional formula of F = [M1L1T-2]
Relation between N and dyne:
\(1\ N = 1\ kg\times\frac{1m}{1s^2} = \frac{1000g\times100cm}{s^2}\)
= 105 g.cm.s-2
or 1 N = 105 dyne
Definition of N: ∵ F= ma
If m = 1 kg; a = 1 ms-2; then F = 1 N
Thus, “1 N is the force which can produce an acceleration of 1 ms-2 in an object of mass 1 kg.”
Impulse:
Total effect of force on the motion of a body is called impulse or impulse of force. If a constant force is applied on a body then product of force and time for which the force is applied provides the impulse i.e.,
Impulse = Force × time interval = F ∆t
Impulse is denoted by I
∴ I = F ∆t
or I = Force × time = \(\frac{\text{change}\ in\ \text{momentum}}{time}\times \text{time}\)
or I = Change in momentum
or I = ∆p
Impulse is a vector quantity and its direction is the same as that of change in momentum.
∴ In vector form,
\(\vec I = \Delta\vec p\)
∴ I = F ∆t = ∆p
Therefore for certain change in momentum, if ∆t decreases, the force F increases accordingly i.e., we have to obtain a certain change in momentum in less time, then more force will have to be applied. If I increases, then force F decreases. This is why when a cricketer catches the ball, he draws his hands back to increase the time interval ∆t.
If a force acts on a body for short time interval dt, then impulse of this force
dI = F dt .....(1)
If force F remains acting for time from t1 to t2, then total impulse
\(I = \int _{t_1}^{t_2} \ Fdt\)
If force F is not the function of time, then it will remain constant.
\(I = F\int_{t_1}^{t_2} \ dt= F[t]_{t_1}^{t_2}\)
or I =F(t1 - t2)
or I = F ∆t
Unit of impulse : ∵ I = F ∆t
∴ Unit of I in M.K.S. system = N.s
and unit of I in C.G.S. system = dyne s
Dimensions of impulse: ∵ I = F ∆t
∴ Dimensional formula of I = M1L1T-2T1
= [M1L1T-2]
Impulse of a force:
I = F ∆t = ∆p
If change in momentum is constant, then
F ∆t = constant
If two forces F1 and F2 acting on a body produce same change in momentum, then
F1 ∆t1 = F2 ∆t2
Thus it is clear that if the time duration of an impulse is increased, the force exerted will be decreased. The following practical examples will make this concept more clear.
(i) Buffers are provided between the bogies of a train : Buffers increase the time of jerk during shunting the train. This decreases the force of impact between the bogies. The bogies are thus prevented from receiving severe jerks. Thus bogies receive slow jerks.
(ii) Automobiles are provided with shockers : When a vehicle, bike, car, bus etc, move on bumpy road, it receives jerk. The shockers increase the time of jerk and hence reduce their force. This makes journey comfortable and saves the automobile from damage due to bumps.
(iii) China wares are packed in straw papers: The straw paper between the china wares increases the time of experiencing the jerk during transportation. Hence, they strike against each other with a lesser force and are less likely to be damaged.
(iv) A person receive more injuries on falling cemented floor than on a heap of sand : In both the cases, the impulse or total change in momentum is same. On the cemented road floor,the person is stopped abruptly. Due to small duration, cemented floor exerts a greater force of reaction causing him severe injuries. When person falls on a heap of sand, the sandy fields gets depressed under his weight, so the person takes longer time to stop. To make the impulse constant, the force exerted by the sandy floor on the person decreases.
(v) A cricket player lowers his hands while catching a ball : When a player catches the ball, he draws his hands backwards to increase the time of catch. Hence the force exerted on his hands lowers and it does not hurt him.
