Measuring Gravity: Thinking outside the box

Physics

Measuring Gravity: Thinking outside the box

Sir Isaac Newton told us how important gravity is, but left some gaps in the story. Today scientists are measuring the gravitational forces on individual atoms in an effort to plug those gaps. In this lesson you will investigate the following: ? What is "big G"? ? How do we measure it? ? What else determines the strength of gravity? ? What is "little g"? So stay grounded as we go back in time to follow the story of a universal constant.

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Introduction: Gravity (P1)

Over 300 years ago the famous English physicist, Sir Isaac Newton, had the incredible insight that gravity, which we're so familiar with on Earth, is the same force that holds the solar system together. Suddenly the orbits of the planets made sense, but a mystery still remained. To calculate the gravitational attraction between two objects Newton needed to know the overall strength of gravity. This is set by the universal gravitational constant "G" ? commonly known as "big G" ? and Newton wasn't able to calculate it. Part of the problem is that gravity is very weak ? compare the electric force, which is 1040 times stronger (that's a bigger difference than between the size of an atom and the whole universe!). Newton's work began a story to discover the value of G that continues to the present day. For around a century little progress was made. Then British scientist Henry Cavendish got a measurement from an experiment using 160 kg lead balls. It was astoundingly accurate. Since then scientists have continued in their attempts to measure G. Even today, despite all of our technological advances (we live in an age where we can manipulate individual electrons and measure things that take trillionths of a second), different experiments produce significantly different results. Experiments have always used masses that you might measure in kilograms, but in the latest attempt scientists used individual atoms. They launched atoms of rubidium (a metal) up a tube with a laser, tracking their motion as it was altered by heavy metal blocks around the tube. So, is it settled? Do we know big G? No, the scientists got a figure of 6.67191 x 10-11 m3 kg-1 s-2, too far from the current "official" value, 6.67384 x 10-11. While the new method marks an astounding change in tactics, the hunt goes on.

Read the full Cosmos magazine article here.

Reflecting upon why an apple falls in a straight line perpendicular to the ground, Newton had his epiphany about gravity and its application across the cosmos. Or so the story goes.

Question 1

Isolate: Gravity is a very weak force. If you were setting up an experiment to measure the gravitational attraction between two objects you would need to be sure that no other forces were interfering, or at least that their influence was minimised. What are some of the other forces you would have to consider? What sorts of steps could you take to ensure that these had no influence?

Gather: Gravity (P1)

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Acceleration

Newton said that much as an apple is attracted to the Earth and falls towards it, so too does the Moon. But clearly the Moon hasn't crashed into Earth and it doesn't look like it will. So how can it be falling? A force, like gravity, acting on a mass makes it accelerate. But acceleration is change in velocity, which has two components ? direction and speed. So you can accelerate a mass by:

1. changing its speed travelling in a straight line (e.g. apples falling), and/or 2. changing the direction it is moving, away from a straight line (e.g. the Moon in orbit).

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Newton had made incredible progress in understanding that the Moon was continually falling to Earth, but never reaching it. But he couldn't measure big G because he didn't have all the information he needed. The Earth was the only object with a gravitational

effect strong enough to measure but he didn't know its mass ? one of the values he needed for his formula. Still, he had other information:

The Earth's radius is 6.37 x 106 m (this came from the ancient Greek astronomer, Eratosthenes, 240 BC) The distance from the Earth's centre to the Moon is 3.84 x 108 m (this from another Greek, Hipparchus, 190 BC) All objects on Earth fall with the same acceleration ? 9.8 m/s2 (famous Italian astronomer Galileo demonstrated this in the late 16th century)

In addition, Newton had discovered the formula for the acceleration of an object moving in a circle:

a

=

4 2 r T2

where a = acceleration, r = radius of orbit and T = time for one revolution, called the period.

He couldn't calculate the gravitational force acting on the Moon, but he could work out its acceleration. Acceleration is directly proportional to the force that causes it (this is Newton's second law of motion), so he still had something to work with.

The Moon's acceleration ? worked example

The Moon's period, T, is 27 days, 7 hours and 43 minutes. What is its acceleration?

Note: We have to use standard units, i.e. metres and seconds, to get an answer in m/s2

Calculation

First convert the period T into seconds:

T = ((27 ? 24 + 7) ? 60 + 43) ? 60 = 2, 360, 580 seconds r = 3.84 ? 108 m

Substituting into the equation:

a=

4 2 r T2

=

4 ? 3.142 ? 3.84 ? 108 2, 360, 5802

= 0.00272 m/s2

Question 1

Calculate: As well as the moon there are many man-made satellites in orbit around the Earth. Some of these are "geostationary", meaning they circle the Earth once every 24 hours, moving in the same direction as the Earth's rotation. They stay fixed over the same point on the planet. Geostationary satellites orbit at an altitude of 36,000 km. What is the acceleration of a geostationary satellite? Calculate to three significant figures. You may be best to write your calculation on paper, photograph it, then upload as an image.

Hint: What is altitude?

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