Orbital Speed

Introduction

How do the radius of an orbit and gravitational forces affect orbital speed? In this demonstration, a simple apparatus will be assembled and used to demonstrate orbital speed and centripetal force.

Concepts

• Orbits
• Orbital speed
• Kepler’s law
• Centripetal force

Materials

Safety Precautions

The very nature of the motion in this activity makes it potentially dangerous. Use caution when twirling the rubber stopper. This demonstration is best conducted in a large open area. Wear safety glasses. Please follow all laboratory safety guidelines.

Procedure

Part I. Orbital Speed and Radius

1. Obtain a two-hole rubber stopper, the handle tube, string and a meter stick.
2. Thread the string through one hole in the rubber stopper and then back through the other hole. Tie the stopper securely to the end of the string. Tie a few knots to make sure the stopper is secure.
3. Thread the free end of the string through the handle tube. Leave about 1 meter of string between the top of the handle and the rubber stopper. See Figure 1 for the basic setup.
4. Hold the bottom of the free end of the string firmly in one hand and the handle tube in the other hand (see Figure 1). Caution: Be sure you are in an open area clear of people and any breakable items
5. Twirl the rubber stopper slowly in a horizontal circle over your head and gradually increase the speed of the rubber stopper until it just stays in a horizontal orbit. Be sure to hold on tight to the bottom of the string.
6. Spin the stopper at a constant rate for 20 seconds. Have students count and record the number of revolutions the stopper makes in a 20-second period when the orbit radius is one meter in the Orbital Speed Worksheet.
7. Shorten the length of the string above the handle to approximately 0.5 meters and repeat steps 4 though 6 for an orbit radius of 0.5 meters.
8. Save the assembled stopper/handle setup for Part II.

Part II. Orbital Speed and Force of Gravity

9. Return the string to its original 1-meter length between the stopper and the tube. Slip a paper clip over the string just below the handle tube (see Figure 3) This is the marker clip.
10. Tie a loop in the free end of the string about 12 inches below the handle tube.
11. Slip the loop through the center of six washers and hold the washers in place by inserting a bent paper clip through the loop as shown in Figures 2 and 3.
12. Holding the tube in one hand, slowly begin to twirl the rubber stopper overhead. Increase the speed of rotation until the marker paper clip is just below the bottom of the handle tube but not touching the handle. This is to ensure that the radius of the stopper’s circular path will be 1 meter.

13. Have the students count and record the number of revolutions the rubber stopper makes in a 20-second period using six washers in the Orbital Speed Worksheet.
14. Increase the number of washers to 18 and repeat steps 12 and 13. Make sure the marker paper clip is still in the proper location. It should be just under the handle tube when the orbit radius is 1 meter in length.
15. Have students answer the Orbital Speed Worksheet Questions.

Student Worksheet PDF

12798_Student1.pdf

Teacher Tips

• This kit contains enough materials to perform the demonstration an unlimited number of times.
• Students will need a calculator to perform the orbital speed calculations in the Orbital Speed Worksheet.
• To get reliable results for Part I, be sure to twirl the stopper just fast enough to maintain a horizontal orbital plane. In extensive trials in our lab, we obtained an average of 17 revolutions per 20 seconds for the 1-meter orbital and 39 revolutions per 20 seconds for the 0.5-meter orbital. Practice spinning the stopper for Part I before performing this demonstration in front of the students.
• For demonstration purposes, the orbit created in this activity is circular in shape. The actual orbits of planets and many satellites are elliptical.
• Discuss the various types of orbital paths (i.e., polar, geosynchronous equatorial, elliptical) that satellites may follow and their significance.
• This is a great demonstration to perform before discussing Kepler’s laws.
• The concepts addressed in this demonstration may be extended to include centripetal acceleration (Equation 1) and centripetal force (Equation 2) if desired. In order to calculate the centripetal force (Equation 2) the mass of the stopper should be measured beforehand.
  
  {12798_Tips_Equation_1}

  a is the centripetal acceleration
  v is the velocity of the stopper (tangent velocity)
  r is the radius of the circular path of the object

  {12798_Tips_Equation_2}

  F is the force (N)
  m is the mass of stopper above handle (kg)
  a is the acceleration (m/s²)

Sample Data

Part I—Orbital Speed and Radius

Use the following equation to calculate orbital speed (velocity, v)

v is the velocity (m/s)
r is the radius of the orbit (m)
T is the period—time for one revolution (s)

Use the orbital speed equation above to calculate the orbital speed of the stopper using:
6 washers 5.98 m/s
18 washers 9.38 m/s

Answers to Questions

1. Using the results from Part I, describe the relationship between orbital radius and orbital speed.
   

   As the orbital radius decreased, the orbital speed of the stopper increased.

2. Using the results from Part II, describe the relationship between gravitational force and orbital speed.
   

   As the gravitational mass increased (addition of washers), the orbital speed of the stopper increased.

3. Predict what would happen to the stopper if the string were suddenly cut during the demonstration.
   

   If the string were cut during this demonstration, the stopper would travel away from the handle tube in a straight line.

4. How is this demonstration similar to the orbits of the planets? How is it different? What does the stopper represent? The tube handle?
   

   This demonstration shows concepts that are similar to the orbits of planets such as the relationship between the distance from a planet to the Sun and the planet’s orbital speed, how gravitational mass effects planet orbital speed, etc. The stopper in this demonstration traveled along a circular orbit. Planets actually travel along an elliptical path around the Sun. The stopper represents a planet and the tube handle represents the Sun (or more massive star).

Discussion

Centripetal force is the “center seeking” force that makes an object move in a circle. According to Newton’s first law, when an object is in motion, it will remain in motion unless acted upon by an unbalanced force. This means an object will travel in a straight line at a constant speed as long as no outside force is acting on it. In order for an object to move in a circle, an inward force is needed. In this demonstration, a rubber stopper is being whirled around on the end of a string. The hand holding the string exerts an inward force (centripetal) on the rubber stopper (see Figure 4). If the string were to break, the stopper would fly outward in a straight line.

The orbital speed of a planet, star or satellite is the speed at which it rotates around a central, more massive body. A specific orbital speed is required to achieve a balance between inertia and the pulling force of gravity of two objects at a given distance. In order for a planet, star or satellite to maintain orbit at a specific distance, the object must travel at a specific speed. Using the moon as an example, if the speed of the moon’s orbit were decreased the moon would fall toward Earth. If the speed of the moon’s orbit were increased, the moon would travel in a straight line away from Earth as described.

In Part I, the stopper moving along the 1-meter radius orbit has a larger orbital path to follow than the stopper following the 0.5-meter radius orbit. Therefore it takes a longer time for the stopper to travel around the 1-meter radius orbit than the 0.5-meter radius orbit. The period of an orbit is defined as the length of time required for a planet to complete one revolution. According to Kepler’s Laws, the orbital speed of a planet (or stopper) decreases as the distance away from the Sun (tube handle) increases. This can be further explored by calculating the orbital speed of the planets Mercury and Jupiter using Equation 3.

v is the velocity (m/s)
r is the radius of the circular path (m)
T is the period – time for one revolution (s)

Mercury is about 36 million miles from the Sun and it takes 88 days for Mercury to complete one revolution around the Sun. Using Equation 3, the velocity of Mercury is roughly 2,570,000 miles/day.

v = 2π (36,000,000 miles)/88 days
v = 2,570,000 miles/day

v = 2π (483,000,000 miles)/4300 days
v = 705,000 miles/day

In Part II, the influence of gravitational force on orbital speed is demonstrated. When additional mass is added to the bottom of the string, the rubber stopper is pulled to the center of its orbit with a greater amount of force. In order to prevent the stopper from being pulled in by this force, a faster velocity is needed to keep the orbital distance at one meter. If the velocity did not increase, the rubber stopper would be pulled toward the tube handle and the radius of its orbit would decrease.