Question

Deduce the equation of a force using Newton’s second law of motion.

Answer 1

“The force acting on a body is directly proportional to the rate of change of linear momentum of the body and the change in momentum takes place in the direction of the force”.
Let, ‘m’ be the mass of a moving body, moving along a straight line with an initial speed ‘u’ After a time interval of ‘t's, the velocity of the body changes to ‘v’ due to the impact of an unbalanced external force F.
Initial momentum of the body (P_i) = mu
Final momentum of the body (P_f) = mv
Change in momentum ∆p = P_f – P_i = mv – mu
By Newton’s second law of motion,
Force, F ∝ rate of change of momentum
F ∝ change in momentum/time

F ∝ `("mu"-"mu")/"t"`

F = `("km"("v"-"u"))/"t"`

Here, k is the proportionality constant, k = 1 in all systems of units.

Hence, F = `("m"("v"-"u"))/"t"`

Since, acceleration = change in velocity / time,
a = (v – u)/t.
Hence, we have F = m × a
Force = mass × acceleration

• No external force is required to maintain the motion of a body moving with uniform velocity.
• When the net force acting on a body is not equal to zero, then definitely the velocity of the body will change.
• Thus, a change in momentum takes place in the direction of the force. The change may take place either in magnitude or in direction or in both.