Newton's Law of Universal Gravitation
Source: Ketron Mitchell-Wynne, PhD,Asantha Cooray, PhD, Department of Physics & Astronomy, School of Physical Sciences, University of California, Irvi…
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 Make sure that the ball is dropped from rest; otherwise, Equation 6 is no longer valid.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 rE.
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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Q1: What does Newton's Law of Universal Gravitation state?
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 force acts along the line joining the centers of mass of the two particles, explaining both terrestrial and celestial motion.
Q2: How does the gravitational force between Earth and the moon relate to Newton's second law?
The gravitational force between Earth and the moon is equal in magnitude and opposite in direction, following Newton's third law. When combined with Newton's second law of motion, which relates force, mass, and acceleration, we can derive that gravitational acceleration depends on the gravitational constant, Earth's mass, and distance—independent of the object's mass.
Q3: Why does an apple fall toward Earth rather than Earth moving toward the apple?
According to Newton's second law, acceleration equals force divided by mass. Although Earth and the apple exert equal gravitational forces on each other, Earth's enormous mass causes its acceleration toward the apple to be insignificant and undetectable, while the apple's smaller mass results in observable acceleration toward Earth.
Q4: What is the standard value of gravitational acceleration at Earth's surface?
The standard value of gravitational acceleration at Earth's surface is 9.8 meters per second squared. This value is derived from the gravitational constant, Earth's mass, and Earth's radius. It can be verified experimentally by dropping an object from a known height and measuring the fall time using kinematic equations.
Q5: How can you experimentally measure gravitational acceleration using a free fall experiment?
Drop a ball from a measured height between two sensors and record the fall time. Repeat five times and calculate the average time. Using the kinematic equation for constant acceleration with zero initial velocity, rearrange to solve for gravitational acceleration. Comparing experimental results to the theoretical value of 9.8 m/s² validates Newton's Law of Universal Gravitation.
Q6: What is the universal gravitational constant and why is it important?
The universal gravitational constant, denoted G, is a fundamental value used in Newton's gravitational force equation to calculate the attractive force between any two masses. Combined with the masses and distance between objects, G allows precise calculation of gravitational forces across the universe, from falling apples to orbiting planets.
Q7: How do engineers apply Newton's Law of Universal Gravitation in structural design?
Engineers use the gravitational force equation F = mg to analyze structural loads on stationary objects like bridges. The branch of mechanical engineering called statics relies on this principle to ensure structures can safely support the gravitational forces acting on them, making accurate gravitational calculations essential for safe infrastructure design.