Gravitational potential energy is a form of energy that an object possesses due to its position within a system. Essentially, it is the energy that an object holds because of the force of gravity acting upon it. But how exactly does an object generate this type of energy? To comprehend this, we must first understand the concept of potential.
When we talk about potential, we are referring to the capacity for something to achieve greatness. Similarly, potential energy is the stored energy within an object as a result of its position within a system. This energy can originate from various sources such as electricity, gravity, or elasticity.
In this discussion, our focus will be on gravitational potential energy. We will delve into the mathematical principles governing it and explore a few illustrative examples. Let's begin!
Definition of Gravitational Potential Energy
Have you ever pondered why a rock, when dropped from a significant height into a pool, creates a larger splash compared to one dropped from just above the water's surface? The explanation lies in the concept of gravitational potential energy (GPE). When an object is raised within a gravitational field, it acquires GPE.
Consequently, the rock dropped from a greater elevation possesses more GPE than the one released from just above the water's surface. GPE represents a form of stored energy that transforms into kinetic energy as the object descends.
The quantity of GPE that an object possesses is contingent upon its elevation, the intensity of the gravitational field, and its mass. If two objects are elevated to the same height above the surfaces of the earth and the moon, the object on earth will exhibit greater GPE due to the stronger gravitational pull.
As an object's altitude increases, so does its GPE. When the object descends, its potential energy transitions into kinetic energy, while the total energy remains constant, in accordance with the principle of energy conservation. This principle asserts that energy cannot be generated or annihilated; it can only be converted from one form to another.
Hence, the next time you witness a substantial splash in a pool, you will understand that it is a result of the rock possessing more gravitational potential energy from being dropped from a greater height.
Did you know that water stored on top of a dam can generate electricity? When the water flows from a height, its gravitational potential energy is transformed into kinetic energy, which can then be harnessed to turn hydroelectric turbines and produce electrical energy.
Just like other forms of potential energy, the gravitational potential energy of the water is a stored form of energy. When the dam is opened, this energy is released and transformed into a different form.
Next time you see a dam, remember that it's not just holding back water - it's also storing potential energy that can be used to power homes and businesses.
Gravitational Potential Energy Formula
The work done to raise an object to a height is equal to the increase in gravitational potential energy. This is due to the principle of energy conservation, which states that energy can be converted from one form to another. The equation for the gravitational potential energy gained by an object of mass m when lifted to a height h in a gravitational field of g is:
PEg = mgh
Where PEg is the gravitational potential energy in Joules, m is the mass of the object in kilograms, h is the height in meters, and g is the gravitational field strength on Earth (9.8 m/s²).
Gravitational Potential Energy Examples
Let’s illustrate the concept of gravitational potential energy through a few examples:
Example 1: Dropping a rock from a height
Suppose a rock with a mass of 2 kg is dropped from a height of 10 meters. Calculate its gravitational potential energy at the point of release.
Using the formula:
#PE_g = m \times g \times h#
#PE_g = 2 \, \text{kg} \times 9.8 \, \text{m/s}^2 \times 10 \, \text{m}#
#PE_g = 196 \, \text{Joules}#
So, the rock has 196 Joules of gravitational potential energy at the point of release.
Example 2: Climbing stairs
If a person with a mass of 70 kg climbs a flight of stairs reaching a height of 5 meters, calculate:
(i) Their increase in gravitational potential energy.
(ii) The work done by the person to climb the stairs.
Using the formula:
#PE_g = m \times g \times h#
#PE_g = 70 \, \text{kg} \times 9.8 \, \text{m/s}^2 \times 5 \, \text{m}#
#PE_g = 3,430 \, \text{Joules}#
(i) The increase in gravitational potential energy is 3,430 Joules.
(ii) The work done by the person to climb the stairs is also 3,430 Joules, as work done equals the increase in potential energy in this case.
Example 3: Free fall of an apple
Calculate the speed of an apple when it hits the ground after falling from a tree 10 meters high:
First, compute time using the formula:
#t = \sqrt{\frac{2h}{g}}#
#t = \sqrt{\frac{2 \times 10 \, \text{m}}{9.8 \, \text{m/s}^2}} = 1.43 \, \text{s}#
Then, calculate the speed at impact:
#v = v_0 + gt#
#v = 0 + 9.8 \, \text{m/s}^2 \times 1.43 \, \text{s} = 14.01 \, \text{m/s}#
The apple hits the ground at a speed of approximately 14.01 m/s.
Example 4: A frog's leap
If a frog weighing 0.05 kg jumps vertically to a height of 0.3 m, determine:
The change in potential energy for the frog can be found using:
#\Delta PE = m \times g \times h = 0.05 \, \text{kg} \times 9.8 \, \text{m/s}^2 \times 0.3 \, \text{m} = 0.147 \, \text{J}#
So, the change in gravitational potential energy for the frog is 0.147 J. Using conservation of energy principles, the frog’s initial vertical velocity can be found as:
#v = \sqrt{2 \times g \times h} = \sqrt{2 \times 9.8 \, \text{m/s}^2 \times 0.3 \, \text{m}} = 2.42 \, \text{m/s}#
The frog jumps with an initial vertical speed of 2.42 m/s.
Gravitational Potential Energy - Key takeaways
- Work done to raise an object against gravity is equal to the gravitational potential energy gained by the object, measured in joules.
- Gravitational potential energy is transformed into kinetic energy when an object falls from a height.
- The potential energy is at a maximum at the highest point and it keeps reducing as the object falls.
- The potential energy is zero when the object is at ground level.
FAQs
What is gravitational potential energy?
Gravitational potential energy is the energy gained when an object is raised by a certain height against an external gravitational field.
What are some examples of gravitational potential energy?
An apple falling from the tree, the working of a hydroelectric dam, and the change in speed of a rollercoaster as it goes up and down inclines are a few examples of how gravitational potential energy is converted to velocity as the height of an object changes.
How is gravitational potential energy calculated?
The gravitational potential energy can be calculated using the formula: Egpe = mgh
How to find the derivation of gravitational potential energy?
As we know, gravitational potential energy is equal to the work done to raise an object in a gravitational field. Work done is equal to force multiplied by distance (W = F x s). This can be rewritten in terms of height, mass, and gravitational field, such that h = s and F = mg. Therefore, EGPE = W = F x s = mgh.
What is the gravitational potential energy formula?
The gravitational potential energy is given by the formula: Egpe = mgh