Derive newton's second law of motion i.e. f=ma?

Question

Answer

Let us derive the relation of force F = ma from Newton’s second law:

According to the Newton’s 2nd Law of motion, the rate of change of linear momentum of a body is directly proportional to the applied external force and in the direction of force.

It means that the linear momentum will change faster when a bigger force is applied.

Consider a body of mass ‘m’ moving with velocity v.

The linear momentum of a body is given by:

p = mv

Now According to Newton’s 2nd Law of Motion:

Force is directly proportional to rate of change of momnetum, that is

F α dp/dt

F = k dp/dt

F = k d(mv)/dt

F = k md(v)/dt

F = k ma

Experimentally k =1

F = k ma

Which is the required equation of force.

Answer

Force=final momentum(mv)- initial momentum(mu)/t
mv-mu/t
m(v-u)/t
a=v-u/t
f=ma
force = mass × acceleration

Answer

Introduction:
Newton's second law of motion, also known as the law of acceleration, states that the force acting on an object is directly proportional to the mass of the object and the acceleration produced. It can be mathematically represented as F = ma, where F is the force, m is the mass of the object, and a is the acceleration produced.

Derivation of Newton's Second Law:

1. Initial Observations:
Before deriving Newton's second law, let's consider some initial observations:
- The acceleration of an object depends on the net force acting on it.
- The acceleration of an object is inversely proportional to its mass.
- The acceleration is directly proportional to the applied force.

2. Defining Force:
Force can be defined as the push or pull exerted on an object to change its state of motion. Mathematically, force is defined as the rate of change of momentum. Momentum is the product of mass and velocity (p = mv).

3. Change in Momentum:
When a force is applied to an object, it causes a change in its momentum. This change in momentum can be expressed as the rate of change of momentum, which is given by the equation:
Δp = mΔv

4. Relation between Force and Acceleration:
Acceleration is the rate of change of velocity (a = Δv/Δt). By substituting the equation for change in momentum, we can relate force and acceleration:
F = mΔv/Δt

5. Introducing Time:
To simplify the equation, let's consider a small time interval Δt approaching zero. This gives us:
F = m(dv/dt)

6. Instantaneous Acceleration:
As the time interval approaches zero, the average acceleration becomes the instantaneous acceleration. So, we can rewrite the equation as:
F = madv/dx

7. Integrating the Equation:
To obtain a general equation, we integrate both sides of the equation with respect to x (position):
∫F dx = ∫m dv

8. Defining Acceleration:
The integral of force with respect to position gives us the work done (W) and the integral of mass with respect to velocity gives us the change in momentum (Δp):
W = Δp

9. Newton's Second Law:
Combining the above equations, we get:
W = Δp = mv - mu, where u is the initial velocity and v is the final velocity.

10. Final Equation:
Since work done (W) is given by the product of force (F) and displacement (Δx), we can write:
FΔx = m(v - u)

Dividing both sides by Δx, we get:
F = ma

Conclusion:
By following the above steps, we have derived Newton's second law of motion, which states that the force acting on an object is equal to the product of its mass and acceleration (F = ma). This law provides a fundamental understanding of the relationship between force, mass, and acceleration, and is widely used in various fields of science and engineering.

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