Momentum and Newton’s Second Law of Motion Physics Notes
Momentum and Newton’s Second Law of Motion:
Linear momentum: Momentum of the body is the physical quantity of motion possessed by the body and mathematically, It is defined as the product of mass and velocity of the body.
As the linear momentum or simply momentum is equal to a scalar time a vector (velocity), it is
therefore a vector quantity and is denoted by p. The momentum of a body of mass m moving with velocity v is given by the relation:
\(\vec{p}=m \vec{v}\)
Dimensional formula = [M1L1T-1]
Unit = Kg m/s
Suppose that a ball of mass mi and a car of mass m2 (m2 > m1) are moving with the same velocity v. If p1 and p2 are momentum of ball and car respectively then:
\(\frac{p_{1}}{p_{2}}=\frac{m_{1} v}{m_{2} v}\)
or
\(\frac{p_{1}}{p_{2}}=\frac{m_{1}}{m_{2}}\)
As m2 > m1: It follows that p2 > p1. If a ball and a car are traveling with the same velocity, the momentum of the car will be greater than that of the ball. Similarly, we can show that if two objects of same masses are thrown at different velocities, the one moving with the greater velocity possesses greater momentum. Finally, if two objects of masses m1 and m2 are moving with velocities v1 and v2 possess equal momentum.
m1v1 = m2 v2
\(\frac{v_{1}}{v_{2}}=\frac{m_{2}}{m_{1}}\)
In case m2 > m1 then v2 < v1 i. e, two bodies of different masses possess same momentum, the lighter body possesses greater velocity.
The concept of momentum was introduced by Newton in order to measure the quantitative effect of force.
The momentum of body in terms of kinetic energy
Explanation of Newton’s Second Law:
According to Newton’s second law of motion, the rate of change of linear momentum of a body is directly proportional to the applied external force on the body, and this change takes place always in the direction of the applied force.
Let, m = mass of a body
v = velocity of the body
The linear momentum of the body
\(\vec{p}=m \vec{v}\) …(1)
let \(\vec{F}\) = External force applied on the body in the direction of motion of the body. .
Δ \(\vec{p}\) = a small change in linear momentum of the body in a small time Δt.
Rate of change of linear momentum of the body = \(\frac{\Delta \vec{p}}{\Delta t}\)
According to Newton’s second law
Where k is proportionality constant.
Taking the limit Δt → 0, the term \(\frac{\Delta \vec{p}}{\Delta t}\) becomes the derivative or differential coefficient of \(\vec{p}\) w.r.t. time t.
It is denoted by \(\frac{d \vec{p}}{d t}\)
\(\vec{F}=k \frac{d \vec{p}}{d t}\)
Where k = 1 in all the system
As acceleration is a vector quantity and mass is scalar, therefore force \(\vec{F}\) being the product of m and \(\vec{a}\) is a vector. The direction of is the same as the direction of \(\vec{a}\). Equation (4) represents the equation of motion of the body. We can rewrite equation (6) in scalar form as:
F = ma …(7)
Thus, magnitude of the force can be calculated by multiplying mass of the body and the acceleration produced in it. Hence, second law of motion gives a measure of force.
Important Facts:
1. If the applied force produces acceleration a, such that ax,ay and a2 are the magnitudes of the component of acceleration along the X-axis, Y-axis, X-axis respectively.
Then
\(\vec{F}\) = m(ax î + ay ĵ + az k̂) …(iv)
If Fx, Fy ,Fz are components of force along X-axis, Y-axis and X-axis respectively, then
\(\vec{F}\) = Fxî + Fy ĵ + Fz k̂ …….(v)
From the equation (4), (5) we have
The set of equation (8) expresses Newton’s second law of motion in component form. Three mutually perpendicular components of the force and the acceleration have to obey the set of equation (8).
As the force is equal to a scalar (mass) times a vector (acceleration), it is a vector quantity. The equation (8) is called the equation of motion of body.
In scalar form F = ma
If we know the values of m and a, the force F acting on the body can be calculated and hence second law gives a measure of the force.
2. Equal forces applied for equal time on different bodies change equal momentum.
3. Newton’s second law shows the relation between net external force and acceleration of the body.
4. Initially two bodies are in rest position, a constant force is applied for a definite time interval then the lighter body receives more speed than the heavy body because of change in momentum In both bodies are equal.
Thus, m1v1 = m2v2
If m1 < m2
then v1 > v2
Dimension and Unit of Force:
As F = ma
.-. F= [M1] [L1T-2 ] = [M1L1T-2]
This is the dimensional formula of force.
Unit of force:
The unit of force is Newton in the (M.K.S.) system and dynein the C.G.S. system and Poundal in (F.P.S.) system.
Definition of 1 Newton
F = ma
When m = 1kg and a = 1 m/s
then F = 1 Newton
The force which produces the acceleration of 1m/s-2 in the 1 kg body, is equal to the 1 Newton.
1 Newton = 1 kg m/ s2
In C.G.S. System
F = ma
If m = 1 g and a = 1 cm/s2
then F = 1 dyne
Thus 1 dyne is that force that produces 1 cm/s acceleration in the mass of 1g body.