Orbits Lab Revisited
R
Prerequisites:
Circumference of circle with radius R: C = 2πR
360 ° = 2π radians

Orbits Lab Revisited x
R
θ x
Suppose a lady bug walks around the entire circumference C of a circle with a radius R …
How far (arc length x ) has she walked? x = C = ___ R
What angle θ has she covered?
θ = ___ radians
So, x = 2πR = θ • R
Now suppose the lady bug walks some fraction f of the circumference .
How far has she walked?
x = f • C = __ • 2πR
What angle has she covered?
θ = f • ___ radians
So, x = f • 2πR = ( f • 2π ) R = θ • R

Orbits Lab Revisited: Basics
θ
R x
So, no matter what angle
θ (in radians) the ladybug covers, the distance she walks is: x = θ • R
If the ladybug walks that distance in a time t , then her “ tangential ” velocity
(her velocity along the circumference) is:
v = x / t
v = ( θ • R ) / t (by substitution for x) v = ( θ / t ) R (by re-arranging)
By definition: ω = ( θ / t ) where ω
(Greek letter lower-case “ omega ” ) is called the angular velocity. So: v = ω R
BTW: IF the ladybug completes the whole circumference in a time T , then v = 2πR/T , too.

Orbits Lab Revisited: Basics
EXERCISE: Verify the distance-angle relationship x = θ • R using the CD marked at various angles, a meter stick and a protractor.
• Choose one of the “ rays ” on the disk as the reference.
• Measure the angles on the disk as shown in the sketch.
• Roll the disk along the meter stick, and measure the distances traveled for each angle rolled.
• Plot the distance traveled (in cm) versus the angle rolled (in RADIANS – remember 180 ° = π radians).
• Note that the relationship x = θ • R (or x = R • θ ) represents a STRAIGHT
LINE of the form y = mx + b with a slope of R and an intercept of zero. Do a manual best fit to your data and verify that the slope of your plot is, indeed, the radius of the CD. For this purpose, radians are considered unitless.
θ
R

Orbits Lab Revisited: Basics
40
30
20
10
0
0 1 2 3 4
Angle (radians)
5 6 7

Orbits Lab Revisited: Basics
Acceleration: Last of the basics. Now let ’ s derive the formula for acceleration of an object in circular orbit. Even if the ladybug is going around the circle at a constant speed, her velocity changes because her direction changes. A force is necessary to keep her going in a circle rather than heading off in a straight line.
R
Δθ v
1 v
2
The velocity changes in direction by an amount Δθ in a time Δt .
As we ’ ll see later: Δv = v Δθ .
So: Δv/Δt = v Δθ/Δt
Which means that: a = vω .
Now use v = ωR to show that: a = ω 2 R and/or a = v 2 /R

Orbits Lab Revisited
Three years ago we did the Orbits Lab to see whether a ball on a string is a reasonable model for planetary orbit. We found it is not. Instead of slowing down as the radius increases ( per Kepler
s
Third Law ), the ball actually goes faster. Now it’s time to find out why…
Tenni s Ball
Radius
Tube
Mass

Orbits Lab Revisited
First, let ’ s do some knitting. Here are the pieces to put together:
• The tennis ball has a mass acceleration of a = v 2 /R.
“ m ” , an orbit radius of R, and an
• The hanging mass M applies a force F = Mg to the ball.
• NL2 asserts that a = F/m.
• From above, v = 2πR/T
• Assume that the string has no friction or mass.
Now, knit the pieces together to get a formula for the square of the period T of the orbit as a function of the radius R:

Orbits Lab Revisited
Confirm your formula by experiment. YOU are responsible for the details of the method (including choosing a sufficient number of trials, etc.) and analysis (sufficient to accept or reject your hypothesized formula).
BTW: If your plot just happens to be a straight line, be sure to calculate and INTERPRET the slope!
Write your report using the “ Physics Lab Report ” template in Google
Docs template. Graphs must be completed in a Google Docs spreadsheet.