The gravitational force F between two masses m₁ and m₂, with their centers of mass separated by a distance r, can be written as:

F = Gm₁ m_{2/ r}² r^{^}, (Equation 1)

where r^{^} denotes that the direction of the force is pointed radially inward. The following description will investigate the gravitational force between the Earth and an object of mass m on its surface. Using Newton's second law, F = m a, the force on the mass m due to the Earth's gravity can be written as:

ma = Gm m_E / r² r^{^}, (Equation 2)

where G is a universal constant of proportionality that has been measured experimentally and m_E is the mass of the Earth. In this context, the acceleration vector is typically denoted as a scalar g, with an implied direction pointing radially inward, toward the center of the Earth. For people standing on the ground, this direction is simply referred to as "down." Canceling the mass m on both sides of the equation; substituting g for a; and noting that the distance between the objects' centers of mass is just the radius of the Earth, r_E, the magnitude of the downward force can be rewritten as:

g = G m_E / r²_E. (Equation 3)

In the famous example of the apple falling from a tree, the Earth is exerting a force on the apple to make it fall, and the apple is exerting an equal and opposite force on the earth, given by Equation 1. The reason that the Earth is essentially unaffected by the force of the apple on the Earth is that the mass of the Earth is so much larger than that of the apple. For larger objects, a larger force is required to make them accelerate. Thus, the apple falls toward the Earth, not the Earth toward the apple. Similarly, for people standing on the ground, the Earth is exerting an even larger force on them than on the apple. The people exert an equal and opposite force on the Earth. Again, because the Earth is so much more massive than a person, the gravitational force a person, or even many people, exert on the Earth essentially goes unnoticed.

This lab will demonstrate how to measure the acceleration g, given in Equation 3. Since all the quantities on the right-hand side of this equation are known, the measured value of g can be compared to their product. The values for g and G are known from experiments to be 9.8 m/s² and 6.67 x 10^-11 Nm²/kg².

For this lab, a ball will be dropped, and the time it takes for the ball to travel a known distance will be measured. From kinematics, the distance y can be written as:

y = y₀ + v₀t + ½ a t². (Equation 4)

If the ball is dropped from rest and the acceleration a is just the gravitational acceleration, this becomes:

y-y₀ = ½ g t². (Equation 5)

Equivalently:

g = 2d / t², (Equation 6)

where d = y - y₀ is the total distance traveled. G will now be experimentally determined.

The value of g measured from the experimental procedure is shown in Table 1. The freefall time from step 1.4 is recorded in the first column of Table 1. The measured value of g is then calculated using Equation 6. The accuracy of this value can be checked by comparing it to the value of g calculated from Equation 3 using the following values: G = 6.67 x 10^-11 m³kg^-1s^-2, m_E = 5.98 x 10²⁴ kg, and r_E = 6.38 x 10³ km. This comparison is also shown in Table 1 with a percent difference. The percent difference is calculated as:

| measured value - expected value | / expected value. (Equation 7)

A low percent difference indicates that Newton's law of universal gravitation is a very good description of gravity.

Table 1. Results.

The branch of mechanics that is concerned with the analysis of forces on objects that do not move is called statics. Engineers who construct building and bridges use statics to analyze the loads on the structures. The equation F = mg is used throughout this field, so an accurate measurement of g is extremely important in this case. Newton's law of universal gravitation is used by NASA to explore the solar system. When they send probes to Mars and beyond, they use the universal law of gravitation to calculate spacecraft trajectories to a very high level of accuracy. Some scientists are interested in doing experiments in zero-gravity environments. To achieve this, astronauts on the International Space Station perform experiments for them. The space station is in a stable orbit around the Earth because of our understanding of the universal law of gravitation.

In this experiment, the gravitational acceleration of an object on the surface of the Earth was measured. Using a ball with two timing gates attached to a meter stick, the time it took for the ball to travel 1 m from rest was measured. Using kinematic equations, the acceleration g was calculated and found to be very close to the accepted value of 9.8 m/s².