Mechanical Energy Formula: Definition, Equation, Solved Examples 

Mechanical Energy Formula: Mechanical energy is the sum of kinetic energy (KE) and potential energy (PE) in a system. Kinetic energy is associated with the motion of an object, while potential energy is linked to its position or configuration. The formula for mechanical energy is:

Mechanical Energy (ME) = Kinetic Energy (KE) + Potential Energy (PE)

Mechanical Energy Formula Equations

1. Kinetic Energy (KE): The kinetic energy of an object can be calculated using the formula:

KE = 0.5 * m * v ²

Where:

KE is the kinetic energy.

m is the mass of the object.

v is the velocity of the object.

2. Potential Energy (PE): Potential energy comes in various forms, including gravitational potential energy and elastic potential energy. Here are the equations for each:

Gravitational Potential Energy (PE _gravity ):

PE _gravity = m * g * h

Where:

PE _gravity is the gravitational potential energy.

m is the mass of the object.

g is the acceleration due to gravity (approximately 9.81 m/s²).

h is the height or vertical position of the object.

Elastic Potential Energy (PE _elastic ):

PE _elastic = 0.5 * k * x ²

Where:

PE _elastic is the elastic potential energy.

k is the spring constant.

x is the displacement of the spring from its equilibrium position.

Also Check – Voltage Drop Formula

Solved Examples of Mechanical Energy Formula

1. Example 1: Calculating Kinetic Energy

Suppose a car with a mass of 1000 kg is moving at a velocity of 20 m/s. Calculate its kinetic energy.

Solution:

Using the formula for kinetic energy:

KE = 0.5 * m * v ²

KE = 0.5 * 1000 kg * (20 m/s) ² = 200,000 J

The kinetic energy of the car is 200,000 joules.

2. Example 2: Determining Gravitational Potential Energy

An object with a mass of 5 kg is lifted to a height of 10 meters above the ground. Calculate its gravitational potential energy.

Solution:

Using the formula for gravitational potential energy:

PE _gravity = m * g * h

PE _gravity = 5 kg * 9.81 m/s² * 10 m = 490.5 J

The gravitational potential energy of the object is 490.5 joules.

Also Check – Transformer Formula

Example 3: Combining Kinetic and Potential Energy

Suppose a rocket with a mass of 500 kg is launched into the air with an initial velocity of 150 m/s and reaches a height of 500 meters. Calculate its mechanical energy at its highest point.

Solution:

First, we calculate the rocket's kinetic energy using the formula for kinetic energy:

KE = 0.5 * m * v ²

KE = 0.5 * 500 kg * (150 m/s) ² = 5,625,000 J

Next, we calculate the rocket's gravitational potential energy at its highest point using the formula for gravitational potential energy:

PE _gravity = m * g * h

PE _gravity = 500 kg * 9.81 m/s² * 500 m = 2,452,500 J

Now, we can find the total mechanical energy by adding kinetic and potential energy:

Mechanical Energy (ME) = KE + PE_gravity

ME = 5,625,000 J + 2,452,500 J = 8,077,500 J

The rocket's mechanical energy at its highest point is 8,077,500 joules.

Example 4: Elastic Potential Energy

A spring with a spring constant (k) of 200 N/m is compressed by 0.2 meters. Calculate the elastic potential energy stored in the spring.

Solution:

Using the formula for elastic potential energy:

PE_elastic = 0.5 * k * x^2

PE_elastic = 0.5 * 200 N/m * (0.2 m)^2 = 4 J

The elastic potential energy stored in the spring is 4 joules.

Example 5: Kinetic Energy in a Moving Vehicle

A bicycle with a mass of 15 kg is moving at a speed of 8 m/s. Calculate the kinetic energy of the bicycle.

Solution:

Using the formula for kinetic energy:

KE = 0.5 * m * v^2

KE = 0.5 * 15 kg * (8 m/s)^2 = 480 J

The kinetic energy of the bicycle is 480 joules.

Also Check – Specific Gravity Formula

Example 6: Potential Energy on a Ski Slope

A skier with a mass of 60 kg descends a ski slope with a vertical drop of 150 meters. Calculate the skier's gravitational potential energy at the bottom of the slope.

Solution:

Using the formula for gravitational potential energy:

PE _gravity = m * g * h

PE _gravity = 60 kg * 9.81 m/s² * 150 m = 88,110 J

The skier's gravitational potential energy at the bottom of the slope is 88,110 joules.

Example 7: Combining Potential Energy

A 2 kg object is lifted 5 meters above the ground, and then it is further raised to a height of 10 meters. Calculate the total gravitational potential energy of the object.

Solution:

First, calculate the potential energy at the initial height (5 meters):

PE1 = m * g * h1

PE1 = 2 kg * 9.81 m/s² * 5 m = 98.1 J

Next, calculate the potential energy at the final height (10 meters):

PE2 = m * g * h2

PE2 = 2 kg * 9.81 m/s² * 10 m = 196.2 J

Now, find the total gravitational potential energy by adding both:

Total PE = PE1 + PE2

Total PE = 98.1 J + 196.2 J = 294.3 J

The total gravitational potential energy of the object is 294.3 joules.