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
-
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.
-
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
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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.
-
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.
