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# What Keeps Planets and Satellites in their Orbits? (page 3)

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Author: Jerry Silver

### Expected Results

This project leads to the following conclusions:

For a 12-gram rubber stopper, the expected results are shown in Figure 13-4, which shows an inverse relationship between force and string length.

1. The faster the rotation (or the shorter the period of rotation), the greater the centripetal force needed to maintain circular motion.
2. For a 12-gram rubber stopper, the expected results are shown in Figure 13-3. This shows the relationship is not linear, but that it increases more rapidly as the velocity increases.
3. The greater the mass, the greater the force needed to keep the rubber stopper going at a given speed at a particular radius. This result is expected to be linear.
4. For a given rotational speed, the shorter the string, the greater the force needed.

### Why It Works

The "string" that keeps an object going around in a circle is provided by a centripetal force. In this case, it is literally a string. In the case of a satellite or planet, the "string " is the gravitational force.

The faster the object goes (for a given radius), the greater the force, according to the equation:

where Fc is the centripetal force, m is the mass of the spinning object (the washer in our case), v is the velocity of the washer, and r is the radius of the circle.

### Finding the mathematical model

Given the data shown in Figure 13-3, we can determine that force increases with the square of the rubber stopper's velocity in one of two ways:

1. Use a curve-fitting program, such as Excel. From a scatter plot, with the data selected, go to the Chart menu, select Add Trendline, and then select a power fit option. Select Add Equation to the Chart from the Options tab. This displays the mathematical model for your data. The expected result is for this to be the form y = x2 or close to it.
2. Either using Excel or plotting by hand makes a graph of force versus velocity squared. If the relationship is of the form expected, that graph should be a straight line. This is shown in Figure 13-5.

Given the data previously shown in Figure 13-4, we can determine that force varies inversely with the radius (string length) using the same techniques.

1. Have Excel determine the trendline for the expected data, as shown on the graph for the previous Figure 13-4.
2. Plotting force versus the reciprocal of radius (1/r) results in a straight line, as shown in Figure 13-6.

### Sources of error

This project works reasonably well and enables you to find the model for centripetal force using very simple equipment. The following are potential sources of errors that may impact your results:

1. Friction between the sting and the tube overstates the required force.
2. Air resistance results in a slightly slower value of velocity.
3. At slower speeds, the circle may not be perfectly horizontal and may have a complicating effect from gravity.

### Determining the accuracy of the model you found

For any of the points you measured, compare the force you measured (by either the spring scale or the hanging mass) with the expected value for the centripetal force given by:

### The Point

Centripetal force keeps an object rotating in a circle. The centripetal force equals the mass of the object times the velocity squared divided by the radius.

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