Newton's second law of motion: Derivation of F=ma, Force,...

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INTRODUCTION OF NEWTON'S SECOND LAW OF MOTION

Newton’s second law of motion, unlike the first, applies to the behaviour of objects for whom all existing forces are imbalanced. The second rule of motion is more quantitative and is widely used to predict what happens in situations involving a force. This article goes over Newton’s second law in detail.

SECOND LAW OF MOTION

The rate of change of momentum of an object is proportional to the force applied on the object. Acceleration of the object also occurs in the direction in which the force acts.

So from second law of motion, F ∝ (mv-mu)

EQUATION F = ma & MEASUREMENT OF FORCE

Consider an outward constant net force ‘F’ acting on an object of constant mass ‘m’. As a result, the object will have an acceleration.

Hence by second law of motion,

F=ma …….(I)

i.e., Force = Mass x Acceleration.

This F = ma equation is recognized as the basis of kinetics.

Definition of unit force: The amount of force acting on unit mass which can cause unit acceleration is called unit force.
The form of the equation F=ma according to calculus is: Let the force F acting on an object of constant mass m and the resulting acceleration a=dv/dt. Hence by second law of motion, F=m · dv/dt …….(ii)
Units of Force in Different Systems of Units: A. Unit ‘dyn’ of ball in CGS method. A force acting on a mass of 1 gm causing an acceleration of 1 cm/s² is called a 1 dyn force. B. Unit ‘poundal’ of force in FPS system. A force acting on a mass of 1 lb causing an acceleration of 1 ft/s² is called 1 poundal force. C. SI unit of force is ‘newton’ or ‘N’. A force acting on a mass of 1 kg causing an acceleration of 1 m/s² is called a force of 1 N.
So, 1N = 1 kg X 1 m/s²
Dimension of Force: [F] = [ma] = M X LT ^-2 = MLT ^-2
Relation between units of force and units of momentum:
As we know, Force = Mass X Acceleration = Mass X (Change in Velocity / Time) = Change in momentum / Time
So, Change in Momentum = Mass X Time
As the unit of momentum and change in momentum is same, that’s why Unit of Momentum = Force X Time.
The unit of momentum in CGS dyn · sec
The unit of momentum in FPS poundal · sec
The unit of momentum in SI Newton · sec
Relation of units of force:
1 N = 1 kg X 1 m/s² = 1000 g X 100 cm/s² = 10⁵ dyn
1 poundal = 1 lb X 1 ft/s² = 453.6 g X 30.48 cm/s² = 13825.7 dyn
1N = 1 kg X 1 m/s² = 2.2 lb X 3.28 ft/s² = 7.2 poundal

Direction of ball: Newton’s second law of motion states not only the value of the ball, but also the direction of the ball. According to this formula,

Direction of applied force = Direction of acceleration (since mass is a scalar).

That is, in which direction the acceleration of the object is taking place, it is understood in which direction the force is being applied on the object. In fact, both sides of the equation F=ma are vectors; Hence it is a vector equation, i.e. →F = m→a. This equation also shows that the direction of the object’s acceleration is the direction of the applied force. Obviously, when a force is applied to a moving object in the direction opposite to its motion, the object accelerates in the direction opposite to its motion. That is, the velocity of the object gradually decreases or the object decelerates.

WEIGHT OF A BODY

Definition: The force with which the earth attracts an object is called the weight of that object.

Earth’s attraction is known as gravitational force. The acceleration of an object due to this force is the gravitational acceleration ‘g’ and this acceleration is always downward. So, if an object of mass ‘m’ has a weight ‘W’, applying the equation F=ma we can write,

W=mg

i.e., Weight = Mass X Gravitational Acceleration.

Because the value of ‘g’ is different at different places on the surface or at different heights above the surface, the weight of the same object can be different at different places.

INERTIAL MASS

Suppose a small piece of wood and a large piece of iron are in the presence of the floor. Our experience is that applying the same amount of force to both pieces will produce more acceleration in the piece of wood. Conversely, a greater force must be exerted on the piece of iron to produce the same acceleration. From this, the value of inertia of rest of the piece of iron is higher than that of the piece of wood. The mass of an object indicates its value of inertia of rest.

This discussion applies not only to inertia of rest but also to inertia of motion. A greater force must be applied to the lorry to stop a bicycle and a lorry moving at the same speed in the same distance, i.e. to produce the same deceleration in both cases. Hence, the inertia of motion of a lorry is higher than that of a bicycle. In this case also the mass of two objects indicates the value of their inertia of motion.

So, it can be said that the mass of an object determines its value of inertia. For this reason, this mass is also called inertial mass. Obviously, the mass of the object indicated in the equation F=ma is the inertial mass of the object.

IMPULSE OF FORCE

Definition: When a force of constant value acts on an object for some time, the product of the value of the force and the time of action of the force is called the impulse of the force.

Let us assume that, a constant force ‘F’ acts on an object of mass ‘m’ for a time ‘t’. Therefore, Impulse of Fore,

J= F X t.

Now when the velocity of the object changes from ‘u’ to ‘v’ under the action of the force, the acceleration of the object is,

a = (v-u)/t

Hence, J = F X t = ma X t = m(v-u)/t X t = m(v-u) = mv-mu = Change in momentum ….(I)

So, Impulse of force = change in momentum of object.

That is, the change in momentum of an object is equal to the square of the force exerted on the object. Since force is a vector quantity, the square of force is also a vector quantity. Hence equation (I) is indeed a vector equation. Obviously, the velocity-vector acts in the direction of the force-vector.

Unit & Dimension:

From definition, Unit of Impulse of Force = Unit of Force X Unit of Time

So, CGS — dyn · s

FPS — poundal · s

SI — N · s

Equation (I) shows that unit of the Impulse of Force and the unit of momentum are the same. Accordingly, the Impulse of Force and momentum has same dimension which is, MLT ^-1.

IMPULSIVE FORCE

Definition: If a large force acts on an object for a short period of time, that force is called impulsive force.

Examples of Impulsive Force:

When a nail is struck with a hammer, the force exerted is great but transient. So, the force exerted by the hammer while hammering a nail is an impulsive force. An impulse of force is created in the nail to apply this force. This is why the nail gets stuck in the wall.
Kicking a soccer ball involves striking the foot very hard, but the contact time between the foot and the football is very short. In this case, the foot exerts force and results in a ball strike in football.
If a person jumps onto the surface from some height, the surface exerts an impulsive force on the person. Because the impact of that ball caused an impulse of force to the man and the man came to a stable position in a very short time.

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Newton’s second law of motion: Derivation of F=ma, Force, Mass, Acceleration, Impulse of Force, Impulsive Force….

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