Gravitational Potential Energy

FIELDS OF STUDY: Classical Mechanics

ABSTRACT: One of the many forms of energy that an object can possess is gravitational potential energy. This energy has the ability to do work and move objects that are separated by a distance. When objects fall down to Earth, it is due to the effect of gravitational forces and gravitational potential energies.

Principal Terms

• center of mass: the weighted average position of the distribution of the mass that makes up an object.
• displacement: the shortest distance between the initial position of an object and its final position.
• gravitational acceleration: the change in velocity of a falling object caused by Earth’s gravitational pull.
• kinetic energy: the energy an object gains due to its motion.
• mass: the amount of matter that makes up an object.
• potential energy: the energy stored in an object due to its position.
• weight: the force exerted on an object due to gravity.

Exchange of Energy

Everything on Earth’s surface is being pulled down toward its center. The planet exerts a gravitational pull on all objects on around it, near and far. This property is not unique to Earth; every person on the planet is exerting a gravitational influence on all surrounding objects as well. This is due to the fact that Earth and everyone on it have mass. The concept of mass is difficult to measure, and physicists have long strived to develop a good system for doing so. In the International System of Units (SI), mass is measured in kilograms, where one kilogram is equal to the mass of the international prototype kilogram, also known as the Big K.

Mass is directly related to the weight of an object. An object’s weight is equal to the force of the gravitational pull it experiences. On or near Earth’s surface, the mass of the planet pulls on objects with a strength of approximately 9.81 meters per second per second, or meters per second squared, for every unit of mass. This rate, known as the gravitational acceleration, is the rate at which objects accelerate as they fall toward Earth.

From Forces to Energies

When a person holds his or her phone at head height in order to speak to somebody, that phone has stored energy due to the fact that it is away from Earth’s surface. This stored energy is called gravitational potential energy. Potential energy is energy that can be converted into other forms of energy in order to do work. Near Earth’s surface, the gravitational potential energy (U_g) can be defined as the product of the mass of an object (m), the gravitational acceleration (g), and the object’s displacement from its resting position—that is, its height above the ground (h):

U_g = mgh

The value of U_g is given in joules (J), an SI derived unit of energy or work. One joule is equal to the energy used to apply a force of one newton (N) over a distance of one meter (m). The newton is also an SI derived unit; it is defined as the force required to accelerate a mass of one kilogram (kg) at a rate of one meter per second squared (m/s²). Thus, one joule is equal to one newton-meter (N·m), and one newton is equal to one kilogram-meter per second squared (kg·m/s²).

Now imagine the person holding the phone accidentally drops it. As the phone starts to fall, it starts to lose gravitational potential energy as its displacement from the ground decreases. That gravitational potential energy is being turned into kinetic energy (K), causing the phone to fall faster and faster. An object’s kinetic energy is determined by its mass (m) and its velocity (v):

When a pendulum oscillates, it loses kinetic energy and gains potential energy as it swings up, which causes it to slow. When all its kinetic energy has turned into potential energy, the pendulum stops for an instant at the top of its swing. Then it drops back down, losing potential energy and gaining kinetic energy.

The same thing can be said of a ball kicked into the air. The force of the initial kick gives the ball kinetic energy. As the ball rises, it loses some of that kinetic energy and gains potential energy. Eventually it achieves its highest point, having gained all the potential energy it can. Then it drops down, converting that potential energy back to kinetic energy and moving faster and faster as it falls.

A Universal Law

Gravitational potential energy does not apply only to Earth or other planets. The equation given above for U_g is a simplification of the equation for the gravitational potential energy between any two objects. The gravitational potential energy possessed by any object due to the influence of any other object can be calculated, given the universal gravitational constant (G), the masses of both objects (m₁ and m₂), and the distance between their centers of mass (r):

The value of G is 6.67384 × 10⁻¹¹ m³/kg·s². Because G is a constant, its value does not change.

From this equation, physicists have obtained the universal law of gravitation. This law defines the gravitational force (F_g) exerted in newtons by one object with a mass of m₁ kilograms on a second object that has a mass of m₂ kilograms and is a distance of r meters away:

Sample Problem

Earth has a mass of approximately 5.97 × 10²⁴ kilograms. Its moon has a mass of approximately 7.34 × 10²² kilograms. The average distance between Earth’s center of mass and the moon’s is about 3.84 × 10⁸ meters. What is the gravitational force in newtons that the moon exerts on Earth?

Answer:

Calculate the gravitational force (F_g) using the universal law of gravitation:

Use the commutative property of multiplication to rearrange the terms.

Simplify the exponents and divide.

Recall that 1 N is equal to 1 kg·m/s². Therefore, the gravitational force exerted on Earth by the moon is approximately 19.83 × 10¹⁹ N (rounded), or, in proper scientific notation, 1.983 × 10²⁰ N.

Strength in Gravitation

In order to calculate the force Earth exerts on the moon, one would use the same equation with all the same values as when calculating the force the moon exerts on Earth. The values of m₁ and m₂ would be switched, but because they are being multiplied and multiplication is commutative, the answer would be the same. As it turns out, Earth pulls on the moon with the same force that the moon pulls on Earth.

While this may seem unlikely, it is a direct result of Isaac Newton’s (1642–1727) third law of motion. Newton’s third law states that for every action, there is an equal and opposite reaction. This applies to any two objects, regardless of the difference in their masses. A single person pulls on Earth with the same force that Earth pulls on the person. So why do objects (and people) fall down to Earth but Earth does not "fall up"? This is where the difference in mass comes in. An object’s acceleration is equal to the accelerating force divided by the mass of the object. Because the mass of a person is so much smaller than that of Earth, the same amount of force will cause the person to accelerate at a much, much greater rate than the planet.

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