### Overview

Source: Ketron Mitchell-Wynne, PhD, Asantha Cooray, PhD, Department of Physics & Astronomy, School of Physical Sciences, University of California, Irvine, CA

Legend states that Isaac Newton saw an apple fall from a tree. He noticed the acceleration of the apple and deduced that there must have been a force acting upon the apple. He then surmised that if gravity can act at the top of the tree, it can also act at even larger distances. He observed the motion of the moon and the orbits of the planets and eventually formulated the universal law of gravitation. The law states that every particle in the universe attracts every other particle with a force that is proportional to the product of their masses and inversely proportional to the square of the distance between them. This force acts along the line joining the two particles.

Gravitational acceleration *g, *which is the acceleration an object on the surface of the Earth experiences due to the Earth's gravitational force, will be measured in this lab. Accurately knowing this value is extremely important, as it describes the magnitude of the gravitational force on an object at the surface of the Earth.

### Principles

The gravitational force **F** between two masses *m _{1} *and

*m*, with their centers of mass separated by a distance

_{2}**r**, can be written as:

**F** = *Gm _{1 }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:

*m a* =

*Gm m*r

_{E }/^{2}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

*; and noting that the distance between the objects' centers of mass is just the radius of the Earth,*

**a***r*, the magnitude of the downward force can be rewritten as:

_{E}*g* = *G m _{E }*/ r

^{2}

_{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/s^{2} and 6.67 x 10^{-11} Nm^{2}/kg^{2}.

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 = y_{0} + v_{0}t + ½ * a *t

^{2}. (

**Equation 4)**

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

y-y_{0} = ½ *g *t^{2}. (**Equation 5)**

Equivalently:

*g* = 2d / t^{2}, (**Equation 6)**

where d = y - y_{0} is the total distance traveled. *G *will now be experimentally determined.

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### Procedure

1. Measure the acceleration of gravity at the Earth's surface.

- Obtain a ball, a meter stick, two timing gates, and three clamps.
- Use one clamp to attach the meter stick to a table or another sturdy surface slightly off the ground.
- Use the other two clamps to connect the timing gates to the top and bottom of the meter stick. Make sure that each sensor is lined up with the end of the meter stick. This way, d is known to be 1 m in
**Equation 6**. - Once it has been verified that the timing gates are working properly, drop the ball through the two timing gates and record the time. Make sure that the ball is dropped from rest; otherwise,
**Equation 6**is no longer valid. - Repeat step 1.4 five times and take the average time.
- Use the average value of
*t*to calculate*g*. Compare this to the value obtained when using the mass and radius of Earth in**Equation 3**.

The Law of Universal Gravitation was the culmination of years of effort by Isaac Newton to understand the force of attraction between masses.

According to legend, when Newton saw an apple dropping from a tree he deduced that a force must draw the apple to the Earth. If this force could act at the top of a tree, it could act at even greater distances. At the time, he was studying the orbits of the moon and planets and eventually formulated the law of universal gravitation to explain their motion.

Newton's law of universal gravitation states that every particle in the universe attracts every other particle with a force proportional to the product of their masses and inversely proportional to the square of the distance between them.

This video will show how to experimentally measure the acceleration due to gravity and compare it to the theoretical value from the equation defining gravitational force.

Before delving into the experiment, let's examine the principles behind the Law of Universal Gravitation. The gravitational force of the Earth on the moon is equal in magnitude and opposite in direction to the force of the moon on the Earth. This force FG acts along the line joining their centers of mass.

According to the law of gravity, FG equals G - the *universal gravitational constant*, times the product of the two masses, divided by the square of *r*, which is the distance between their centers of mass.

With this expression, it is possible to calculate the gravitational force Earth exerts on an object at any distance, including near or at its surface. In the case of the apple falling from a tree, let's say that the apple's mass is *m*, the Earth's mass is *mE* and the radius is *r*E.

Newton's second law of motion states that force equals mass times acceleration. If we combine this equation, applied to the apple, with the law of gravity, we can cancel the apple's mass *m* from both sides**. **In this context, acceleration is typically denoted by the letter g

Now, the gravitational force on the apple is given by the Law of Universal Gravitation, but from the second law of motion, this force may also be expressed as *mg*. As we saw earlier with the Earth and moon example, the force of the Earth on the apple is the same as the force of the apple on the Earth. But why do we only see the apple fall toward the Earth? Why do we not see the Earth move toward the apple?

If we look back at Newton's second law of motion, we can rearrange it to show that acceleration is equal to force divided by mass. That is, for a given force acceleration is inversely proportional to mass. Because the Earth is so much more massive than the apple, the acceleration of the Earth toward the apple is insignificant and essentially undetectable**. **And that's why the apple falls from the tree.

Going back to the gravitation equation for *g*, since all the values on the right hand side - the *universal gravitational constant, *the mass of the earth and the radius of the earth -- are known for an object close to earth's surface, the magnitude of *g *is also standard value, which is 9.8 meters per second squared.

However, this value can be calculated experimentally simply by dropping a ball from a known height and applying the kinematical equations. And we will demonstrate how to do that in the following sections.

This experiment uses a metal ball, a meter stick, one sensor from which the ball will be suspended, another sensor on which the ball will land, one timer connected to both sensors, one clamp, and one rod-stand. First, use the clamp to attach the ball sensor to the rod, at least 0.5 meters above the surface of the table. Then, place the second sensor directly below the first sensor.

Next, measure the distance between the top and bottom sensors. The distance should be measured with respect to the bottom of the ball.

Now, release the ball from the sensor so it falls onto the lower sensor and record the time.

Repeat this procedure five times and then calculate the average fall time

From the kinematics video in this collection, we know that this formula describes position in one-dimensional motion of an object with constant acceleration.

Since we are dealing with Earth's gravitation, the acceleration in this case is the acceleration due to gravity, or *g*. And the initial velocity is zero, since the ball was at rest before the drop. So if we move the initial position to other side of the equation, the left side becomes *y* minus *y0*, which is nothing but *d* - the distance between the initial and final measure point. Now we can rearrange the equation for *g*.

For this experiment, *d* was 0.72 meters and the average free fall time was 0.382 seconds. The resulting experimental gravitational acceleration is 9.9 meters per second squared. Experiment and theory differ only by about 1%, which indicates that Newton's Law of Universal Gravitation is a very good description of gravitational attraction.

The Universal Law of Gravitation is involved in calculations performed by different branches of engineering.

The branch of mechanical engineering called *statics *is concerned with the forces on stationary objects, like bridges. Engineers designing bridges use statics, and especially the equation *F = mg,* throughout their work to analyze structural loads.

A NASA gravity-mapping mission uses two identical satellites-one leading, another trailing-orbiting Earth together. When the leading satellite passes over an ice cap or other mass concentration, it accelerates due to relatively larger force of attraction. The trailing satellite experiences similar acceleration when it passes over the same area.

A ranging system measures how and where the distance changes between them, providing information about the distribution of mass concentrations around the Earth.

You've just watched JoVE's introduction to Newton's law of universal gravitation. You should now know how to determine the gravitational force between two masses, and understand how to calculate the acceleration due to the force of gravity at the Earth's surface. Thanks for watching!

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### Results

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} m^{3}kg^{-1}s^{-2}, *m _{E }= *5.98 x 10

^{24}kg, and r

_{E}= 6.38 x 10

^{3}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.**

Free Fall Time (s) | Measured g |
Calculated g |
% difference |

0.45 | 9.88 | 9.79 | 0.9 |

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### Applications and Summary

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/s^{2}.

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