## Circular Motion and Introduction to Relativity

### Robert Rosen

#### Description

Students work cooperatively to view, demonstrate, and understand the importance of frame of reference. They present a short skit, based on the information from their research, that describes a trip to a nearby solar system.

#### Objectives

The student knows that all motion is relative to whatever frame of reference is chosen and that there is no absolute frame from which to observe all motion.

The student knows that any change in velocity is an acceleration.

The student knows that investigations are conducted to explore new phenomena, to check on previous results, to test how well a theory predicts, and to compare different theories.

The student knows that scientists assume that the universe is a vast system in which basic rules exist that may range from very simple to extremely complex but scientists operate on the belief that the rules can be discovered by careful, systemic study.

The student uses classical, contemporary, and vocal acting techniques and methods to portray the physical, emotional, and social dimensions of characters from various genres and media.

#### Materials

-Media Center reference materials
-Coil of wire
-Magnet
-Galvanometer or sensitive ammeter
-Ball
-Bucket
-Rope
-Water
-Props for skit (Students or the drama department can supply masks, make-up, and a model of the spaceship.)
-Centifugal Force Rubric (see attached file)
-Skit Rubric (see attached file)

#### Preparations

A discussion of frames of reference, point of view, and relativity may be necessary prior to this activity The following lesson may assist the instructor in preparation.

1. The description of motion depends on the observer’s frame of reference. No reference frame is more valid than another, but the frame of reference in which the description of motion is simplest may be most convenient to use.
Begin by asking students the following question, -Does Earth move around the sun or does the sun move around Earth?- Students have been taught that Earth revolves around the sun. This description of motion is valid for an observer in the sun’s reference frame. A frame of reference can be defined as a set of coordinate axes fixed to some object, like a laboratory or the sun. In the sun’s reference frame, the coordinates of a moving object’s position are determined using coordinate axes fixed to the sun, but it is just as valid to use Earth as a frame of reference. The sun would seem to circle an observer in an Earth-centered coordinate system. (We are ignoring the rotation of Earth for the sake of simplicity.) Both ways of looking at the motion of the Sun-Earth system are equally valid. Ask students why the sun is used as a reference frame in every astronomy text when the solar system is described. Students must remember that the solar system consists of nine planets in order to answer this question. It is possible to describe the motion of the sun and the other eight planets in Earth’s reference frame, but the description is extremely complex. The description of the motion of the nine planets is very simple in the sun’s reference frame, so that is the description that is taught. Both descriptions are correct, however.

2. There are many other examples that illustrate how the description of motion depends on one's frame of reference.
Throw a ball and ask students, -What is the shape of the path followed by a thrown ball?- Students will see the ball follow a curved path that rises and then falls; in fact, the shape of the trajectory is parabolic. If the initial velocity of the ball has a horizontal component in the observer’s frame of reference, then the ball’s trajectory will have a parabolic shape. Next, ask a student to throw a ball of paper up in the air while walking forward. Students at their seats will describe the trajectory of the ball as parabolic. The thrower will see the ball go straight up and then come straight down. The ball and the thrower have the same horizontal velocity, so that in the thrower’s reference frame, the ball has no horizontal component of velocity. Have the thrower describe the trajectory of the ball in his frame of reference. Ask students which description of the ball’s path is correct: a vertical line or a parabola? Of course, both descriptions are equally valid.

3. Questions can sometimes be answered by carefully choosing your frame of reference.
Connect a solenoid (coil of wire) to a galvanometer (or very sensitive ammeter). Move a strong magnet in and out of the coil. The galvanometer will indicate that an alternating current flows in the coil while the magnet is moving. If students have not studied electromagnetic induction, the instructor may wish to point out the practical significance of the phenomenon (generation of electrical power). Next, ask students to predict what will happen if the coil is moved while the magnet is held stationary inside the coil. Use guided questioning to lead students to the conclusion that moving the coil around a stationary magnet is really the same experiment as moving a magnet in and out of a stationary coil. (Suppose we see the magnet moving in and out of the coil in our reference frame. Imagine an ant sitting on the magnet. The ant uses the magnet as his frame of reference. In the ant’s frame, the coil moves back and forth with respect to the magnet. Since the ant’s description of motion and ours are equally valid, we conclude that current will flow in the coil whenever the coil and magnet move relative to each other.)

4. Newton’s Laws of Motion are valid in inertial (non-accelerating) reference frames but do not work in non-inertial (accelerating) reference frames.
The description of motion is different in different reference frames. No description is “better” than another; no reference frame is “superior” to another. But this does not mean that all reference frames are equivalent. All inertial (non-accelerating) reference frames are equivalent to each other. However, a non-inertial (accelerating) reference frame is not equivalent to an inertial frame. Newton’s three laws of motion work in all inertial frames but are invalid in non-inertial frames. Suppose a Slurpee is sitting on the dashboard of a car. Ask students to predict what happens if the car makes a sudden right turn. (The Slurpee will fly toward the left wall of the car.) Ask students to explain why this behavior violates Newton’s Second Law. (The Slurpee is observed to accelerate in the driver’s reference frame despite the fact no force acts on the Slurpee.) It seems like a mysterious force is acting on the Slurpee and other objects in the car as it turns. In older physics texts, this mysterious force was called centrifugal force. The word centrifugal means “pointing away from the center of a circle”. You can apply Newton’s Second Law to the Slurpee if you assume a new force, centrifugal force, acts in the driver’s reference frame and causes the Slurpee to accelerate. The centrifugal force pushes objects in the car radially away from the center of the car’s circular path. On the other hand, you can explain the behavior of the Slurpee without using a mysterious new force by viewing the Slurpee from an inertial frame of reference. For example, view the Slurpee from a blimp that is stationary with respect to the earth.
See Figure A in the attached file.

#### Procedures

Knowledge/Skills:
-Students will know that all motion is relative to a frame of reference.
-Students will understand the concept of relativity.
-Students will relate relativity to frame of reference and the laws of motion.
-Students will use reading and writing skills to aid in concept development.

Procedure:
1. Place students in cooperative groups of three to four students.

2. Refer to Figure A in the attached file and use the example of the car, the driver, and the Slurpee described in the Step 4 of the Preparation to have students debate the following:

In a car making a rapid right turn, the driver feels as if he is thrown against the left door of the car as the car turns. The Slurpee is thrown toward the left door during the turn. Centrifugal force seems real enough in the driver’s reference frame. Yet, newly printed physics textbooks classify centrifugal force as a “fictitious force”. Is centrifugal force real or not? Defend your position.

Guide the debate until the following points have been made:

Although the driver is aware he is in an accelerating (turning) reference frame, his brain expects Newton’s laws to be valid because the driver spends most of his life in reference frames where Newton’s laws do hold true. The inside of the car looks like an inertial reference frame to the driver’s brain because objects inside the car are stationary with respect to the driver. The driver may use centrifugal force to “patch” Newton’s Laws so that they work in a non-inertial reference frame that seems like an inertial frame. Centrifugal force is a “real force” if the driver interprets his observations from this point of view.

Viewed from the inertial frame of the blimp, no new force is needed to explain the behavior of the Slurpee inside the car. The observer in the blimp would be correct to say that centrifugal force is a “fictitious force” if he interprets his observations from this point of view.

There is no right answer to the question, “Is centrifugal force real or fictitious?” The answer to the question must be accompanied by a description of the observer’s frame of reference in order for the observer’s answer to have meaning. Scientists consider the description of motion from the blimp observer’s point of view to be simpler because a new natural force does not have to be invented to understand the Slurpee’s motion. Thus, new texts label centrifugal force “fictitious” and no longer use the force in the solution of problems.

3. Demonstration:
In this demonstration, whirl a bucket containing water in a vertical circle. Students will be surprised to see that the water does not spill out of the bucket. Describe the motion of the water using two different reference frames. Consider an ant riding along with the bucket. The ant is in an accelerating reference frame. The ant and water experience centrifugal force. When the bucket is upside down, the centrifugal force on the water balances the weight of the water. Thus, the water does not fall onto the instructor’s head. The instructor is in a non-accelerating reference frame. The instructor sees the water moving on a circular path. The water is therefore accelerating downward (toward the center of its circular path) when the bucket is at its maximum height. According to Newton’s Second Law, there must be a force acting on the water in the direction of the acceleration. The bucket exerts a downward force on the water. In addition, the weight of the water provides part of the necessary downward force. The weight of the water causes it to accelerate downward. The downward acceleration corresponds to circular motion rather than falling motion in this demonstration.

4. Using Figure B in the attached file, pose the following:

Observer A is stationary with respect to the earth. Bicycle rider B has a speed of 10 m/s with respect to the Earth. The rider throws a ball with a speed of 20 m/s with respect to the moving bicycle. Ask students to predict the speed of the ball as measured by observer A.

Answer: 10 m/s + 20 m/s = 30 m/s. Note that the speed of the ball measured in observer A’s reference frame is different from the speed of the ball measured in observer B’s reference frame.

5. Follow up: Ask students to research the following questions. (Written answers to the questions should be brief.)
a. What are the postulates that Einstein used to develop his Theory of Relativity?
b. Which of Einstein’s postulates were experimentally confirmed in the Michaelson-Morely and Kennedy-Thorndike experiments?

Answers: (a.) All inertial reference frames are equivalent and the speed of light c= 3.00 x 108 m/s is the same for all observers. (b.) In these experiments, the speed of light was measured in reference frames that were moving with different velocities with respect to the light source. The relative motion did not affect the value of the speed of light c that was measured. The experiments confirm that Einstein’s postulates are true.

6. Discussion: Discuss relativity in greater detail.

All inertial reference frames are equivalent means that the laws of physics are the same in every inertial reference frame. This postulate also means there are no special reference frames that are “really at rest”. When a car travels at constant speed, the car isn’t “really moving” with the Earth “really at rest”. Rather, the (approximately) inertial reference frames are moving with respect to each other. Similarly, there is no “at rest reference frame” that could be used to measure an “absolute velocity” of a star, a planet, or light.

7. To illustrate these ideas, have students imagine they are in an airplane whose window shades cover the windows. Challenge students to devise an experiment that must be done completely inside the plane; no data from the outside can be used which would tell them whether the plane was at rest with respect to the ground or flying at constant velocity with respect to the ground.

Einstein’s Theory of Relativity states that it is not possible to distinguish reference frames that are at rest with respect to each other from those that are moving at constant velocity with respect to each other because all experiments done in each inertial reference frame will yield the same results.

The speed of light c= 3.00 x 10 to the 8th power m/s is a constant for all observers. This postulate seems innocent, but it violates the notion of relative velocity. Consider the following example:

See Figure C in the attached file. Ship A is at rest with respect to planet C. Ship B moves toward ship A at half the speed of light (.5 c). An observer on planet C measures the speed of the light beam sent out by ship A. The speed of light that the observer on planet C measures is c= 3.00 x 10 to the 8th power m/s. An observer on ship B also measures the speed of the light sent out by ship A as c= 3.00 x 10 to the 8th power m/s, not 1.5 c (despite the motion of ship B toward ship A). There is only one way to resolve the paradox of a universal speed of light. One must conclude that observers in different reference frames will measure different distances and different elapsed times between events. Length and time adjust themselves so that

c = distance traveled by light ÷ traveling time is a constant in all reference frames.

8. Student research: Ask students to research time dilation and length contraction. These are phenomena predicted by Einstein’s Theory of Relativity. Tell students to use sources such as popular periodicals, encyclopedias, and high school level textbooks. The goal is to understand time dilation and length contraction qualitatively. Material that analyzes the relationship between space and time using invariants or the Lorentz transformation is well beyond the scope of most high school courses and is to be avoided. After students present their qualitative findings, the instructor can introduce formulas for time dilation and length contraction without any explanation or derivation.

Time Dilation Formula
Let -t to the 1st power = time between two events as measured on Earth
-t = time between two events as measured on a spaceship
v = speed of ship relative to Earth
c = speed of light

then

-t to the 1st power = -t ÷ (1 -(v/c)squared)

For example, suppose that astronauts travel at 99% of the speed of light (v= .99 c) for 5.0 years as measured on the ship’s clock. The traveling time for the trip as measured on earth is

-t to the 1st power = 5.0 ÷ ( *(1 - (.99)to the 2nd power ) = 35 years

So an astronaut’s friends on Earth age 35 years while the astronaut is gone. The astronaut returns to Earth only 5.0 years older!

Length Contraction Formula
Let
L subscript 1= length of spaceship measured in a reference frame in which the ship is
moving with respect to the observer
L = length of ship measured in a reference frame in which the ship is at rest
with respect to the observer; for example, the astronauts are in such a
reference frame
v = speed of each reference frame with respect to the other
c = speed of light

then

L subscript 1 = L(*(1-(v/c)squared)

This formula means that objects moving with respect to an observer appear to be contracted (in the direction parallel to the relative motion). For example, suppose a spaceship's measured length is 1000 m while at rest on Earth. This is also the length astronauts on the ship measure, since they are at rest with respect to the ship. Suppose the ship passes Earth at 99% of the speed of light. The length of the ship as measured on Earth would be

L subscript 1 = 1000(*(1-(.99)squared) = 140 meters

If astronauts observe a copy of their ship on Earth while they pass at 99% of the speed of light, that ship would appear 140 meters long to the astronauts since the ship on Earth is moving at 99% of the speed of light with respect to their reference frame.

The instructor should give students several problems based on these formulas. Students will enjoy these calculations.

9. Student presentations: On the basis of their research and the lessons presented by the instructor, students are to put on a skit that describes a trip to a nearby solar system in a spaceship that travels at more than 90% of the speed of light. The instructor should explain that some students should describe the trip from the astronauts’ reference frame and others should describe what observers on Earth see. The instructor should make it clear what features must be included in the skit. Suggested features include a dramatization of the effects of space travel on the passage of time. Students might compare the age of a returning astronaut to that of a twin brother who watches from Earth. The instructor should suggest that students pay attention to details such as the length of the ship as measured from Earth. There are aspects of space travel that were not covered in the lessons. For example, students might want to illustrate communication problems that arise when the ship is very far from earth. For extra credit, students might demonstrate the effect that relativistic speeds have on mass. Students should be instructed to be creative, use make-up, simple props, music, and other devices that make their presentation entertaining, clear, and informative.

10. Assess student understanding of concept material.

Note: This online site does not allow certain symbols, including superscripts and subscript. You will want to use these where indicated, in place of the words used in the formulas and examples above.

#### Assessments

Material on circular motion can be evaluated by using a quiz with questions covering traditional topics such as period, frequency and calculation of centripetal accelerations. The centrifugal force activity described in the Procedure can be evaluated using the Centrifugal Force Rubric provided in the attached file.

The student skit may be assessed using the Skit Rubric provided in the attached file.

The following questions may also be used to assess student understanding:

1. Consider a red car traveling 40 mph west, and a blue car traveling 50 mph east. The two cars pass each other traveling in opposite directions. Which of the following is true for the red car?
a. The red car is traveling 40 mph relative to Earth.
b. The red car is traveling 0 mph relative to the driver.
c. The red car is traveling 90 mph relative to the blue car.
d. All of these are true.

(Answer d: Relative to each frame of reference, these are all true.)

2. Consider a red car traveling 40 mph and a blue car traveling 50 mph in the same direction. The blue car passes the red car. Which of the following statements are true?
a. The red car is traveling 40 mph relative to the blue car.
b. The blue car is traveling 40 mph relative to Earth.
c. The blue car is traveling 10 mph relative to the red car.
d. All of these are true.

(Answer c: Relative to the red car, the blue car is traveling 10 mph.)

3. Which of the following are constant based on relativity?
a. Mass of an object
b. Length of an object
c. Speed of light
d. All of the above

(Answer c: Only the speed of light is constant in relativity.)

Self Reflection:
Write a brief essay on one of the following:
•What aspects of Einstein’s Theory of Relativity do you find most fascinating and why?
•Can the idea of frames of reference be applied to more than just physics? How can we use this idea to learn to be more understanding of others and become better citizens?

#### Extensions

Enhancement:
Centrifugal force comes into play when one applies Newton’s laws in a reference frame that seems inertial but is really accelerating. Earth is not strictly an inertial reference frame because it slowly rotates. Yet, Newton’s laws are used with very little error in labs on Earth. Earth's rotation can not be ignored when the motion in the oceans or atmosphere is studied because the motion occurs over a long period of time. What force is used to “patch” Newton’s laws in order to correct for the rotation of Earth?

#### Attached Files

The illustrations and the rubrics mentioned in the lesson.     File Extension: pdf