Virtual Science Ltd
Measuring the Acceleration Due to Gravity
Introduction
Isaac Newton was the first person to realize that one force unites the movements of objects on Earth and out in space. His law of gravitation states that all objects in the universe exert an attractive force on all other objects in the universe, and that this force is proportional to the mass of each object and is inversely proportional to the square of the distance between them. That is:
F = GMm/r^2
F is the force, G is the gravitational constant, M and m are the masses of the objects, and r is the distance between them. The starting point for the discovery of this law was in divining how the planets moved. Tycho Brahe laid the foundation for this discovery by making accurate observations of the planets’ motions, which enabled Johannes Kepler to discover his three Laws of Planetary Motion:
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The planets move in elliptical orbits with the Sun at one focus.
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The line joining the planet to the Sun sweeps out equal areas in equal times.
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The square of the orbital period, T, is proportional to the cube of the mean distance from the Sun, or T^2 ~ r^3.
A conker (horse chestnut) being swung on a string does not fly off at a tangent because a force towards the center is being supplied by the swinger. The conker is constantly accelerating towards the center of the circle. Its direction is constantly being bent towards the middle of the circle by the tension in the string.
Around the same time as Kepler was formulating his laws, Galileo was making progress on the movement of bodies. He discovered the principle of inertia, which is if a body is moving in a straight line and has no forces acting on it, it just carries on moving in the same straight line at a uniform speed. Newton added to this, saying that if a body is to speed up (accelerate), then a force must be applied in the direction of motion and that the heavier the body, the bigger the force must be to get the same effect. The equation (Newton’s second law) is:
Force = Mass x Acceleration.
Or
F = Ma.
So, when Newton saw an apple fall downwards, he knew that there had to be a force causing this motion, and he wondered if that same force was acting on the Moon. This was an extraordinary, insightful thought.
Eureka! But Newton still needed a formula for calculating the force due to gravity. He realized that a consequence of Kepler’s third law is that the force must be weaker farther away from the Sun and directly in proportion to the square of the distance.
Additionally, it seemed clear that heavier objects exerted larger forces, and each body acted on the other. Therefore, the force should be proportional to the product of the two masses. Hence:
F = GMm/r^2
Where G is the gravitational constant, M and m are the two masses, and r is the distance.
Thus, the Universal Theory of Gravitation was born, which on Earth causes all bodies to accelerate towards the center of the Earth until something gets in the way. The rate of this acceleration is denoted by g and is the subject of this experiment.
Newton was never entirely happy that there was no obvious mechanism by which gravity worked. He wrote:
That one body may act upon another at a distance through a vacuum, without the mediation of anything else, by and through which their action and force may be conveyed from one to another, is to me so great an absurdity, that I believe no man, who has in philosophical matters a competent faculty of thinking, can ever fall into it.
However, Newton’s theory has stood the test of time. Einstein’s General Theory of Relativity has very successfully replaced the action at a distance problem with a geometrical solution – matter bends space, and objects follow straight paths in this curved space. However, it is still not clear whether Newton’s philosophical objections have quite gone away. How is space told how to bend, and how does this propagate? There are also several gravitational anomalies that have yet to be explained.
The objective
To find the acceleration on Earth due to gravity. In the e-practical, we can actually repeat the experiment on the Moon and on Mars.
The apparatus
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An electromagnet
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A gantry with a trap door switch
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An electronic timer
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A power supply
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A two-pole switch
The apparatus must be wired so that the two-pole switch simultaneously starts the timer and breaks the circuit to the electromagnet.
The variables
There is one dependent variable in this experiment: the time.
The Physics
For an accelerating body, the distance traveled is given by the formula:
s = ut + ft^2/2
Where u is the initial speed, which in this case is zero (the ball bearing starts at rest, being held by the electromagnet). f is the acceleration, which in this case is the unknown called g, and t is the time. We can rearrange this expression to give us:
g = 2s/t^2
We can measure the height fallen, that is a given, and t is the time of descent as recorded on the timer.
The method
With the switch in the closed position, electricity will flow from the power supply to the electromagnet at the top of the apparatus and keep the ball bearing at this position.
Measure and take note of the distance of the ball bearing to the trap beneath it.
Open the two-pole switch; the ball bearing will drop and the timer will start.
When the ball bearing falls through the trap at the bottom of the apparatus, it breaks the timer circuit and the timer stops. This will now indicate the time spent between the ball bearing being released by the electromagnet and breaking the connection at the trap. This gives you a timing for the descent. You need to measure the distance of the drop. You should take at least four readings and average the result.
The Video
Watch a video of the Acceleration due to Gravity e-practical here.
This shows how to use it and how to collect the data.
The e-Practical
Perform the experiment yourself, collect your own data, make mistakes and be able to correct them. The e-practical requires that your browser can run WebGL 2 (usually found on Windows browsers, safari on iOS, and various Mobile browsers, test with https://get.webgl.org/webgl2/). This link is for students and evaluation only, schools should purchase a site licence.
The timer can be reset by clicking on the central red button.
The circuit breaker can be moved by placing the cursor over the red handle and using the mouse wheel.
You can replace the ball bearing on the electromagnet by clicking on the ball bearing. If you have not put the switch back to its ‘on’ position, the ball bearing will fall right away.
The vertical distance fallen is 0.67 m.
You can repeat the experiment on the Moon by clicking on the Moon icon at the top left of the screen and on Mars by clicking on the Mars icon at the top left of the screen.
The Results
You should prepare a table for your results that shows the timings taken and the average. You can then compute the rate of acceleration for the Earth or for the other worlds using the formula:
g = 2s/t^2
Where s is the distance, g is the acceleration (the value you are looking for), and t is the time of descent.
If you are using the e-practical, you can repeat this experiment for the Moon and Mars.
Further Discussion
Look up the masses of these worlds on the internet. You could try plotting the masses against the values for f that you found. Then answer the following questions:
• What is the relationship between the mass of a planet and its acceleration due to its gravity?
• Why is the graph not a straight line?
• What other factor might influence the value of the acceleration (hint: are all of these worlds the same size?)?
Is it possible to show that the Moon falls towards the Earth the distance we would expect when compared to an apple at the surface of the Earth? In fact, this is surprisingly easy to show.
Here on Earth, the apple falls 4.9 meters in the first second. The Moon is 60 times farther away from the center of the Earth than the apple, so we should expect it to fall 1/60×60; i.e., 1/3600 of this, i.e., about 1.3 mm.
The angle the Moon moves through in 1 second is given by:
ɵ= 2 x pi/(28 × 24 x 60 × 60) radians
The denominator is the number of seconds in a lunar month, which corresponds to a single orbit around the Earth.
Where s is the distance, g is the acceleration (the value you are looking for), and t is the time of descent.
If you are using the e-practical, you can repeat this experiment for the Moon and Mars.
This works out at 2.59722E-06 radians. Given that the radius of the Moon’s orbit is approximately 384,400,000 m, the distance moved by the Moon in 1 second is:
D = 2.59722E-06 × 384,400,000 m = 998.37 m
From the earlier diagram:
R – h = Rcos(ɵ)
Therefore, h, the amount that the Moon has dropped towards us, is given by:
h = R – Rcos(ɵ)
Which works out at around 1.3 mm.
This section is adapted from material developed by Dr Robert Lucas and is related to the book High School and Undergraduate Physics Practicals, published by CRC Press.