Derive an expression for kinetic energy of an object

The energy that an object has as a consequence of its motion is known as kinetic energy.

We must apply force to an object if we want it to accelerate. Applying force requires effort. After the work is completed, energy is transferred to the item, which then moves at a new steady speed. Kinetic energy is the amount of energy transmitted and is determined by the mass and speed achieved.

Kinetic energy is a type of energy that may be exchanged between objects and converted into other forms of energy. A flying squirrel, for example, could crash with a motionless chipmunk. Some of the squirrel’s initial kinetic energy may have been transferred to the chipmunk or converted to another kind of energy after the collision.

Kinetic Energy

It is an object’s ability to perform work as a result of its motion. Wind, for example, has the kinetic energy to rotate the blades of a windmill and so produce electricity. The kinetic energy of an item is stated as, where K is the object’s kinetic energy in joules (J), m is the object’s mass in kilograms, and v is the object’s velocity:

 Therefore it is represented as:

   K = 1/2mv²

Expression for kinetic energy of an object

As we all Know that the energy of a body due to its motion is known as kinetic energy.

Derivation:

Let us consider we have an object of mass ‘m’ which is at rest on a plane. Let a force, F acts on the object and let us consider that the object moves from point A to point B and covers a displacement S.

Now the work done is given by:

W=F×S……..(1)  

Work done = Force × Displacement

Now from the third law of Motion, we know that

V²-U²=2aS

S=(V²-U²)/ 2a………(2)

By Newton’s Second Law of Motion:

F=ma

Now, from the equation 1 and 2 we get

W=m×a×V²-U²/2a

As we assume that the object is at rest, Hence u=0

W=m×V²/2

Therefore, the Work done appears as kinetic energy of the body.

Hence,

Kinetic Energy=K.E=(1/2mv²)

Properties of Kinetic Energy

There are following properties of kinetic energy:

• Kinetic energy is related to the object’s velocity squared. This indicates that as an object’s velocity doubles, so does its kinetic energy. 
• An automobile going at 60 miles per hour has four times the kinetic energy of a car travelling at 30 miles per hour, resulting in four times the potential for death and damage in the case of a collision.
• Kinetic energy must always have a positive or zero value. Regardless of whether velocity is positive or negative, velocity squared is always positive.
• Kinetic energy is a type of energy that is not a vector. A tennis ball thrown right with a velocity of 5 m/s has the same kinetic energy as a tennis ball thrown down with the same velocity.

Time period of a Simple Pendulum

A heavy point mass (called bob) is linked to one end of a completely inextensible, flexible, and weightless rope in an ideal simple pendulum. In actuality, we make it by threading a tiny cotton stitching thread through a metallic spherical bob.

A Simple Pendulum’s Length

The length of the simple pendulum is the distance between the point of suspension of the pendulum and it’s Center of Gravity (C.G.), which is the C.G. of the bob. 

A Simple Pendulum’s Time Period

The time taken by the bob of a simple pendulum to complete one complete oscillation is called the time period. The letter T is used to signify it.

The pendulum begins a periodic motion when the bob is shifted to a particular angle, and the periodic motion is a simple harmonic motion with angular displacement of the bob for small values of the displacement angle.

F=mgsinθ

And 

a=gsinθ

Let us Consider is small for the small oscillation

Therefore,

sinθ=θ=x/l

After Putting this value in a, we get

a=gθ

a=gx/l

  Hence, the Angular Frequency is 

ω=√g/l 

And as we know that

T=2π/ω

Now after Substitution we get:

T=2π√l/g

So, based on the equation for the simple pendulum’s time period, we may deduce that the time period is independent of the pendulum’s mass and does not change with changes in the small amplitude of oscillation. It is entirely determined by the length of the string and the rate of acceleration. It has been commonly used to track a fixed interval of time because of this property.

Conclusion

In this article we are studying kinetic energy and derive an expression for it and also derive the time period of a simple pendulum. A simple pendulum is composed of a mass m suspended from a string with a length L and a pivot point P. When the pendulum is shifted to an initial angle and released, it will swing back and forth in a regular pattern and this is the working principle of a simple pendulum.