Work Energy theorem

The work-energy theorem is a fundamental concept in physics that describes the relationship between the work done on an object and its resulting change in kinetic energy. In simple terms, it states that the work done on an object is equal to the change in its kinetic energy.

Table of Contents:

• Work Energy theorem
• Derivation
• Applications of Work-Energy Theorem
• Work done by a constant force
• Work done by Non-Uniform force
• FAQs

Work Energy theorem

The work-energy theorem is a fundamental concept in physics that describes the relationship between the work done on an object and its resulting change in kinetic energy. In simple terms, it states that the work done on an object is equal to the change in its kinetic energy.

Mathematically, the work-energy theorem can be expressed as:

W = ΔKE

where W represents the work done on the object, and ΔKE represents the change in its kinetic energy. The work done on an object is equal to the force applied to it multiplied by the distance over which the force is applied. The change in kinetic energy is equal to the final kinetic energy of the object minus its initial kinetic energy.

The work-energy theorem is a powerful tool that allows us to analyze the motion of objects and predict their behavior based on the work done on them. It is used in many areas of physics, including mechanics, thermodynamics, and electromagnetism, among others.

Derivation

To derive the work-energy theorem, we start with the definition of work done on an object by a force:

W = Fd cosθ

where W is the work done, F is the force applied, d is the distance over which the force is applied, and θ is the angle between the force and the direction of motion of the object.

Next, we use the equation for kinetic energy:

KE = (1/2)mv^2

where KE is the kinetic energy of the object, m is its mass, and v is its velocity.

We can differentiate this equation with respect to time to obtain the rate of change of kinetic energy:

dKE/dt = mv(dv/dt)

Using Newton’s second law of motion, F = ma, we can substitute for acceleration (a) as:

a = dv/dt

So, we have:

F = ma = m(dv/dt)

Substituting this expression for force into the work equation, we get:

W = mad cosθ

Using the expression for acceleration, we can rewrite this as:

W = m(dv/dt)(d cosθ)

We can integrate both sides of this equation with respect to time from the initial time t1 to the final time t2, and from the initial position x1 to the final position x2, to obtain:

∫(t1 to t2) W dt = ∫(x1 to x2) mad cosθ dx

We can then substitute the expression for acceleration and integrate with respect to velocity to obtain:

∫(t1 to t2) W dt = ∫(v1 to v2) mvdv

Evaluating the integral on the right-hand side, we get:

∫(v1 to v2) mvdv = (1/2)mv2 – (1/2)mv1 = ΔKE

Substituting this expression for ΔKE into the left-hand side of the equation, we get:

∫(t1 to t2) W dt = ΔKE

Thus, we have derived the work-energy theorem:

W = ΔKE

which states that the work done on an object is equal to the change in its kinetic energy.

Applications of Work-Energy Theorem

The work-energy theorem states that the net work done on an object is equal to its change in kinetic energy. This theorem has several applications in various fields such as physics, engineering, and even in everyday life. Here are some examples:

Physics: In physics, the work-energy theorem is used to calculate the work done on an object when a force is applied to it. It is also used to calculate the kinetic energy of an object and to analyze the motion of an object in terms of its energy.

Engineering: In engineering, the work-energy theorem is used to design machines that convert energy from one form to another. For example, a car engine converts the chemical energy in gasoline to the kinetic energy of the car.

Sports: The work-energy theorem is used to understand the performance of athletes in sports such as gymnastics, diving, and skiing. It helps in analyzing the movements of the athletes and the energy they need to perform certain maneuvers.

Rollercoaster: The work-energy theorem can be applied to design roller coasters. The kinetic energy of the coaster is converted into potential energy when it is lifted up the first hill. The coaster then gains kinetic energy as it moves down the hill, and the cycle continues.

Pendulum: The work-energy theorem can also be applied to analyze the motion of a pendulum. The potential energy of the pendulum is at its maximum when it is at the highest point, and its kinetic energy is at its maximum when it is at the lowest point.

In summary, the work-energy theorem has various applications in physics, engineering, sports, and everyday life.

Work done by a constant force

Let’s consider a constant force, F, acting on an object that moves a distance, d, in the direction of the force. The work done by the force can be calculated as follows:

The work done, W, is defined as the product of the force, F, and the distance, d, over which the force acts:

W = Fd

We can also express the force as the product of the object’s mass, m, and its acceleration, a, as given by Newton’s second law:

F = ma

If the object starts from rest and reaches a final velocity, v, after moving a distance, d, with a constant acceleration, a, we can use the equations of motion to express the distance and acceleration in terms of the final velocity and time, t:

d = (1/2)at^2
v = at

Solving for t and a in terms of d and v, we get:

t = √(2d/a)
a = v^2/(2d)

Substituting the expressions for F and a in terms of m and v, we get:

F = ma = m(v^2/(2d))

Substituting the expression for F in the equation for work, we get:

W = Fd = m(v^2/(2d))d = (1/2)mv^2

This is the final result, which shows that the work done by a constant force on an object that moves a distance, d, in the direction of the force, is equal to the change in the object’s kinetic energy, (1/2)mv^2.

Work done by Non-Uniform force

The work done by a non-uniform force can be calculated using integration.

Let’s consider a non-uniform force, F(x), acting on an object that moves a distance, x, in the direction of the force. To calculate the work done by this force, we can divide the distance, x, into small intervals, dx, and approximate the force as constant within each interval.

Then, the work done by each interval of force can be calculated as the product of the force and the distance it acts over, which is dx. Summing up the work done by all the intervals of force over the entire distance, x, gives us the total work done by the non-uniform force.

The work done by a non-uniform force, F(x), over a distance, x, is given by the following integral:

W = ∫ F(x) dx

where the integral is taken over the entire distance, x, for which the force acts.

To evaluate this integral, we need to know the function that describes the non-uniform force, F(x). Once we know this function, we can substitute it into the integral and integrate it over the given distance.

For example, let’s consider a spring with a spring constant, k, that is stretched or compressed by a distance, x, from its equilibrium position. The force exerted by the spring on the object is given by Hooke’s law as:

F(x) = -kx

where the negative sign indicates that the force is directed in the opposite direction to the displacement.

Substituting this function into the integral for work, we get:

W = ∫ -kx dx

Integrating this expression with respect to x, we get:

W = -(1/2)kx^2 + C

where C is the constant of integration. The limits of integration for this integral are the initial position, x1, and the final position, x2. Therefore, the total work done by the spring force over the distance, x2 – x1, is given by:

W = -(1/2)k(x2^2 – x1^2)

This formula shows that the work done by the non-uniform force, F(x) = -kx, depends on the square of the distance moved by the object and the spring constant, k.

Work Energy theorem FAQS