PhysicsLAB: Kepler's 1st and 2nd Laws

Step 1: (Maxwell's method for constructing ellipses) Obtain a piece of string whose length is 55 cm so that you can tie a loop which is EXACTLY 44 cm in circumference. Fold your sheet of white cardboard into four equal quadrants and carefully mark the center. Next, measure carefully and place two straight pins into the cardboard at (-9,0) and (+9,0) along the x-axis. Loop the string over these pins. Bring the string loop into tension by placing a pencil inside the loop and stretching the string so that the pins and the pencil form a "triangle". Gently move the pencil around the paper and trace out an ellipse. Keep the string tight. This should form an ellipse with a major axis close to 27 cm and a minor axis close to 20 cm.

Step 2: Kepler's First Law states that a planet travels in an elliptical orbit with the sun at one foci. Therefore, label the left foci S for sun, the left endpoint of the major axis P for perihelion, and the right endpoint A for aphelion. Ignore the presence of the right focus.

Step 3: Mark off an arc at A along the curve of the ellipse that extends 3 cm below A [call this point N] and the 3 cm above A [call this point M]. Draw a line that connects S to M and then a second line that connects S to N.

Step 4: Carefully cut this wedge segment -- delineated by the radii SM and SN and the arc MN -- out of your ellipse. Keep the point at S sharp.

Measure and record the mass of this wedge in grams.

Re-measure its arc length, it should be close to 6 cm.

Step 5: Kepler's Second Law states that in equal intervals of time, a line from the planet to the Sun sweeps out equal areas of space. To verify this, CAREFULLY begin to cut another wedge shaped segment that starts at S, contains point P, and is symmetric about the radius PS. Stop trimming this wedge when it reaches the same mass as the wedge in Step 4. CAUTION - start with a large wedge which you can make smaller and NEVER trim away the Sun!

When you have finally produced a wedge that equals as close as possible the same mass as SMN, measure and record its mass in grams.

Now measure and record its total arc length in centimeters.

Conclusions

1. Calculate the eccentricity, e, of your ellipse where e equals the ratio of c / a.

c =

a =

e =

2. Does the wedge in Step 5 that is symmetric to the radius AP have the same area as the wedge SMN produced in Step 4? Defend your answer.

3. If this ellipse represented planet X's trip around the sun, does planet X have its greatest orbital speed at A or P? Support your answer.

4. Use the arc lengths measured for each wedge to calculate the ratio of the planet's speed at the perihelion to its speed at the aphelion. Remember that the time, t, taken by planet X to travel along each arc length is the same, so t cancels.

arc length_aphelion =

arc length_perihelion =

v_perihelion / v_aphelion =

Remember to put your names on each sector along with each sector's mass and arc length. Staple them along with your string to the back of your written law report (calculations) when you turn it in.