Method 1
Gravitational Fields are regions surrounding a massive object in which other objects having mass will feel a gravitational force of attraction.
Gravitational fields are usually illustrated by using radial field vectors. In this vector diagram, the gravitational field is stronger where the field lines are closer together.
Radial fields always obey an inverse square relationship.
This means that when you compare two positions in a radial field (A and B) where
R_{B} = 2R_{A}, the gravitational field at B will be (1/2)^{2} or 1/4^{th} as strong as it is at A.
In the above diagram, the central mass, M, is surrounded by a radial, gravitational field which means that the field lines look like extensions of the radius of a circle. All field lines point to the center of M. Since the three red masses, m, are all placed the same distance from the center of M they would feel the same magnitude force of gravitational attraction.
where G = 6.67 x 10^{11} Nm^{2}/kg^{2}, m and M are measured in kilograms, and r is the radial distance from the center of m to the center of M measured in meters.
If we were to place a different mass at the same distance r from M (that is somewhere along the same dotted surface) if would feel a different magnitude of gravitational force.
If the unknown mass, m, was three times larger than m, then the gravitational force, F, would become 3F. If the unknown mass was 10 times larger than m, then the gravitational force would become 10F. That is, the gravitational force is directly proportional to the product of the two masses.
In our diagram, all of our masses are experiencing the same ratio of gravitational force to mass which is called the gravitational field strength. In our lab, we used a spring scale to measure the force of gravitational attraction between the Earth and the 8 differentsized balls. When purchased, the spring scale had been calibrated to recognize a given amount of weight (gravitational force) as a distinct stretch in the spring’s length. Since the divisions on the spring scale were delineated in multiples of 0.4 newtons, our measurements were restricted to one decimal place and involved a large degree of estimation.
We then measured the gravitational mass of each ball using a triple beam balance. Although we could measure each mass up to three decimal places our ultimate ratio of the gravitational field strength, F/m, was limited to only one decimal place because of the restrictions on our measurement of each ball’s weight. During one class period they obtained an average value of 11.24 N/kg, or 11.2 N/kg.
If we had used our formula and the values for the mass of the earth and the earth’s average radius, we would have obtained a value for the gravitational field strength at sea level to be So the percent error for their calculation was about 14%. Method 2 The second method of measurement to determine the strength of the earth’s gravitational field was to measure and record the slope of a velocitytime graph created by a motion detector tracking each ball’s descent.
When we examined the probe’s software we observed two graphs: one showing us position vs time and the second showing us velocity vs time.
In the equation to the left, the expression net F represents the sum of all the forces acting on an object. In our experiment, the only force acting on each object was the gravitational attraction between the ball and the earth. For air resistance to have had an impact on a ball's motion it would have had to have travelled much, much faster and have had a much larger surface area.
When the same group averaged all of its slopes from the 8 velocitytime graphs they got a value of 10.5 m/sec^{2}.
But what do the slopes of those velocity graphs represent? To answer this question we must define a new behavior called acceleration. Acceleration is the rate of change of velocity with respect to time. Recall that earlier in the year we defined inertia as a measure of the resistance an object has to being accelerated – or having its state of motion altered.
The equation a = net F/m is Newton’s Second Equation of Motion. This equation states that the acceleration an object experiences is directly proportional to the net force acting upon it and inversely proportional to the object’s mass. It represents a mathematical statement of what was happening to each released ball as it was being pulled downward by the attractive force of gravity.
Since this acceleration is so special it is called the acceleration due to gravity and is represented by the variable g. But notice that g is equal to the same expression as the gravitational field strength, F/m.
This relationship allows us to now calculate the percent error for the second part of the experiment again using 9.80 m/sec^{2} as the accepted answer. The error for their average slope was 7.1%.
SUMMARY
In summary, the average of our ratios of F/m and the average of our slopes for each velocitytime graph should have been the same, 9.80 m/sec^{2}. But be careful when working problems; this value changes with a planet’s mass and/or the object’s distance from the planet’s center.
