Calculating Acceleration Due to Gravity: Formula & Concept

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  • 0:01 Formula for…
  • 0:40 Newton's Second Law of Motion
  • 2:46 Newton's Law of…
  • 4:42 Calculation of…
  • 5:19 Examples
  • 6:57 Lesson Summary
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Lesson Transcript
Instructor: Thomas Zesiger

Thomas has taught electronics and communications engineering, math, and physics and has a master's degree in electrical engineering.

You have no doubt seen gravity work in your life. After all, it is the force that keeps your feet on the ground! This lesson explores the acceleration a body experiences due to gravity and provides details of the mechanics behind it.

Formula for Acceleration Due to Gravity

The formula for the acceleration due to gravity is based on Newton's Second Law of Motion and Newton's Law of Universal Gravitation. These two laws lead to the most useful form of the formula for calculating acceleration due to gravity: g = G*M/R^2, where g is the acceleration due to gravity, G is the universal gravitational constant, M is mass, and R is distance. The remainder of this lesson develops this formula, provides further explanation of its meaning, and shows practical examples of its use in calculating acceleration due to gravity.

Newton's Second Law of Motion

Newton's Second Law of Motion states that an object is accelerated whenever a net external force acts on it and the net force equals the mass of the object times its acceleration. Mathematically, this is given by the formula F = m*a, where F is the net force acting on the object, m is the mass of the object, and a is the acceleration.

As the formula shows, mass is a measure of the resistance of an object to acceleration. Mass is also a measure of the quantity of matter in an object. Mass is often confused with weight. Weight is the force of gravity acting on an object. The weight of an object varies with the location of the object in the universe.

For example, if you were standing on the surface of the Moon, your weight would be approximately 1/6 your weight on the surface of the Earth. This is because the average acceleration due to gravity on the surface of the Moon is approximately 1/6 the average acceleration due to gravity on the surface of the Earth. If you want to lose a large amount of weight quickly, then just travel to the Moon!

Unfortunately, since mass is the amount of matter contained in your body, your mass is constant throughout the universe. If you're looking to lose weight, it is probably more accurate to say you are trying to lose mass. This is because what you are really trying to do is reduce the size of your body and not just to have a smaller number appear the scales.

When the acceleration is due to gravity, we replace a with g in Newton's Second Law of Motion, where g represents the acceleration due to gravity. As we stated previously, the force of gravity acting on a substance is defined as weight, so we replace F with W. The formula then becomes W = mg.

We could solve for g to get the formula g = W/m, but this form of the equation does not provide much practical use for determining the acceleration due to gravity. Newton's Second Law of Motion in the form W = mg is most useful to relate weight and mass when the acceleration due to gravity is already known.

Weight

Newton's Law of Universal Gravitation

Newton's Law of Universal Gravitation says that every object exerts a gravitational force on every other object. The force is proportional to the masses of both objects and inversely proportional to the square of the distance between their centers. Mathematically, this is given by F = G*(m1*m2/d^2), where F is the force, G is the universal gravitational constant, m1 is the mass of object 1, m2 is the mass of object 2, and d is the distance between their centers.

The universal gravitational constant was discovered experimentally in 1798 by English physicist Henry Cavendish. It is measured in Newton-square meters per square kilogram (N-m^2/kg^2) and is equal to 6.67 * 10^-11 N-m^2/kg^2.

Newton

Quite often when we use this formula, the mass of one object is much greater than the mass of the other object. For example, when we consider the force of gravity acting on our bodies, the two objects involved are our bodies and the Earth. Another example is the force of gravity acting on a ball as it free-falls to the ground after being thrown straight up.

Both the mass of our bodies and the mass of the ball are negligible compared to the mass of the Earth. In this case, we can replace m1 and m2 in the formula with m to represent the much smaller object and M to represent the much larger object. As the formula indicates, d is the distance between the centers of the two objects. The distance from the center of the Earth to the center of our bodies, or to the center of the ball, is essentially the same as the distance from the center of the Earth to the surface of the Earth. Therefore, we can also replace d with R, which is the average radius of the Earth. This gives us the formula F = G*(m*M/R^2).

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