You replace a with g in Newton’s

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.

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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).

Calculation of Acceleration Due to Gravity

To arrive at a formula that is useful in calculating the acceleration due to gravity, we must use both of Newton’s laws previously developed in this lesson. Since both laws define force, we can set each of them equal to each other.

In mathematical terms, we start with m*g = G*(m*M/R^2). The mass of the smaller object, m, divides out on each side, and we’re left with g = G*M/R^2. This is the most usable form of the equation to calculate acceleration due to gravity.

Examples

Let’s take a look at some examples.Example 1Calculate the acceleration due to gravity on the surface of the Earth. The mass of the Earth is 5.979 * 10^24 kg and the average radius of the Earth is 6.

376 * 10^6 m. Plugging that into the formula, we end up with 9.8 m/s^2.Example 2Calculate the acceleration due to gravity on the surface of the Moon.

The mass of the Moon is 7.35 * 10^22 kg and the average radius of the Moon is 1.74 * 10^6 m. Plugging that into the formula, we get 1.

6 m/s^2.Example 3A spacecraft is sent to Mars to explore the planet. Calculate the acceleration due to gravity when the spacecraft is 160,000 meters above the planet and compare this to the acceleration due to gravity near the surface of the planet.

The mass of Mars is 639 * 10^21 kg and the average radius of Mars is 3,390,000 meters. When the spacecraft is this far above the surface of the planet, we need to add the spacecraft’s altitude to the radius of the planet. Therefore, we have 3,390,000 + 160,000 = 3,550,000 m, or 3.55 * 10^6 m.

Plugging into the formula, we end up with 3.4 m/s^2.Near the surface of the planet, we end up with 3.

7 m/s^2.

Mars spacecraft

Lesson Summary

This lesson defined Newton’s Second Law of Motion as F = ma and Newton’s Law of Universal Gravitation as F = G*(m1*m2/d^2). 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.We then used both of these laws to derive the formula g = G*M/R^2, which defines the acceleration due to gravity when we have a massive object exerting a gravitational force on another object of relatively negligible mass. An example of this situation is the gravitational force of the Earth on our own bodies.

Highlighted Vocabulary

Gravity ; Newton
newtonsgravity
  • Newton’s Second Law of Motion: 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
  • Mass: a measure of the resistance of an object to acceleration; a measure of the quantity of matter in an object
  • Weight: the force of gravity acting on an object
  • Newton’s Law of Universal Gravitation: every object exerts a gravitational force on every other object; it’s proportional to the masses of both objects and inversely proportional to the square of the distance between their centers

Learning Outcomes

This lesson aims to shed light on gravity-related acceleration in order to help you to:

  • Discuss the basis for the formula associated with acceleration due to gravity
  • Distinguish between mass and weight
  • State Newton’s Laws
  • Calculate acceleration using Newton’s formulas
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