Center of Gravity

Introduction

Show students how to locate the center of gravity of irregularly shaped objects and how to position objects so they are as stable and balanced as possible. Then show them how the irregular object spins around its center of gravity.

Concepts

• Center of gravity
• Stability
• Center of mass
• Projectile motion

Background

According to Isaac Newton’s (1642–1727) laws of gravitation, the Earth attracts every tiny particle of mass of every object and pulls them toward the center. For any specific object (composed of many tiny particles), the center of gravity of the object is the location where all the individual gravitational forces acting on the individual particles add up and result in one net downward force. At this point we can assume all of the mass of the object is concentrated, and therefore this point is also referred to as the center of mass. The location of the center of gravity, especially for an irregularly shaped object, is critical for the overall stability and balance of an object on the Earth’s surface. In order for an object to remain stable and balanced, its center of mass must be located directly above or below its supporting base. The Bottle Balance Beam and Finger Balance both demonstration this principle.

In general, when a force acts on an object, it can be assumed that the force acts on the center of mass of the object. If a force is specifically applied to an object at a position other than its center of mass (i.e., to the left, right, up or down) from the center of mass, then this force will cause the object to rotate about its center of mass. If the object is allowed to rotate about a point other than the center of mass, then the object will wobble instead of spin smoothly. This wobble becomes very apparent when riding in a car that has poorly balanced tires.

Materials

Safety Precautions

The materials used in this demonstration set are considered safe. The foam polygon is soft and will not break or damage items in the classroom. However, do not throw the foam object at anyone. Please follow all normal laboratory safety guidelines.

Disposal

The materials should be saved for future demonstrations.

Prelab Preparation

1. Use scissors to cut the string to approximately 40 cm.
2. Tie one end of the string to the fishing sinker.
3. Tie a “looping knot” to the other end of the string (see Figure 1).
   {12511_Preparation_Figure_1}

Procedure

Demonstration 1. Center of Gravity of an Irregular Polygon

1. Show the irregular plastic polygon to the students.
2. Ask students to predict which hole in the polygon is at the center of mass of the polygon. Students can also develop experiments to locate and test the center of mass position, if appropriate.
3. Obtain the handle with hook and the string and sinker.
4. Insert the hook through the hole at one of the corners of the polygon. Allow the polygon to hang freely.
5. Place the “looping knot” onto the hook and allow the sinker to hang freely.
6. Once the polygon and string stop swinging, determine which hole or holes in the center of the polygon is in line with the hanging string. Optional: Hold the string tightly along the polygon and then use a dry erase marker to draw a line on the polygon along the string.
7. Repeat steps 4–6 from a different corner of the polygon. Which hole is in line with the hanging string? Is it the same hole as before?
8. Repeat steps 4–7 one more time. The third trial should verify which hole is located at the center of mass of the polygon.
9. Remove the hook from the polygon and obtain a sharpened pencil.
10. Insert the point of the pencil into the hole determined to be located at the center of mass of the polygon.
11. Spin the polygon on the pencil. Notice that the polygon will rotate smoothly and evenly (little or no wobbling) when it spins at this location.
12. Place the point of the pencil into one of the other holes at the center of the polygon and spin the polygon. Notice how the polygon wobbles when it spins from a location other than the center of mass.

Demonstration 2. Bottle Balance Beam

1. Place the top of a 2-L bottle in the hole of the Balance Beam as shown in Figure 2.
   {12511_Procedure_Figure_2}
2. When the center of gravity of the entire setup is directly above the base of the Balance Beam, the apparatus will balance itself. When the center of gravity is not directly above the foot of the base, the apparatus will not balance and it will fall over.
3. The center of gravity of the apparatus can be changed by adding or removing weight (water) from the bottle. When the right amount of water is in the bottle and distributed properly, the apparatus will balance perfectly in a stable fashion.
4. Start by placing the empty bottle in the Balance Beam hole and trying to balance the apparatus on a flat, stable tabletop. When the bottle is empty, the apparatus will not balance and it will fall over. Where is the center of gravity when the bottle is empty? Why does it fall in the direction it does?
5. Remove the empty bottle and fill it one-half full with water. Replace the cap and place the bottle back in the Balance Beam hole. Try to balance it. Determine the location of the center of gravity relative to the base. Explain why the apparatus still falls over.
6. Continue adding water to the bottle and testing it until the apparatus balances. When it balances, verify that the center of gravity is directly over the foot of the base of the apparatus.
7. What happens if more weight is added to the top of the bottle? Place extra weight on the end of the bottle and the end of the wooden beam. What happens to the center of gravity and which way does the apparatus fall? Get the apparatus to fall in both directions and discuss what happens in terms of the center of gravity shift.

Demonstration 3. Finger Balance

1. Obtain the acrylic balance apparatus and plastic tubing.
2. Show the balancing apparatus to your students and attempt to balance it on the tip of your finger. (It will not balance.)
3. Explain to your students that by adding a heavy object to the end of the balancing apparatus, the apparatus will now balance on your fingertip.
   {12511_Procedure_Figure_3}
4. Insert the midpoint of the plastic tubing into the slot of the balancing apparatus. Bend or flex the tubing if necessary so that the ends of the tubing hang at the same level, as shown in Figure 3.
5. Twist the plastic tubing about a quarter turn towards the tip of the balancing apparatus. This should force the end of the tubing to extend back towards the tip of the balancing apparatus with the tubing arms curving down (see Figure 3). The friction between the tubing and the acrylic slot will keep the tubing in this twisted position.
6. Place the tip of the balancing apparatus on a fingertip or the corner of a tabletop.
7. Release the balancing apparatus and allow it to hang as if it is suspended in midair.
8. Discuss the results with students. Note: If the tubing is not angled toward the tip of the balancing apparatus, the balancing apparatus will not balance because the center of gravity is not located below your finger. When discussing center of gravity, show the case in which the tubing is not twisted towards the tip (so that the apparatus does not balance with the hanging tube). This may help to reinforce the concept of center of gravity and why the system was balanced in the first case.

Demonstration 4. Center of Gravity Toss

1. Show the foam polygon to the students.
2. Toss the foam polygon forward with a backspin so the students can see the irregularly shaped object wobble as it travels along its trajectory (see Figure 4).
   {12511_Procedure_Figure_4}
3. Ask students if there was a point on the object that appeared to follow a stable parabolic path during its motion. Toss the object again, if necessary.
4. Attach the center of the binder clip to the very edge of one corner of the foam object (see Figure 5).
   {12511_Procedure_Figure_5}
5. Obtain the handle with hook and the string and sinker.
6. Insert the hook through the wire loops of the binder clip. Allow the polygon to hang freely.
7. Place the “looping knot” onto the hook and allow the sinker to hang freely.
8. Hold the hook and handle parallel to the ground and high enough to allow the foam object, and string and sinker to hang freely (see Figure 5). Make sure the string does not get “hung up” on the foam.
9. Once the foam object and string stop swinging and come to rest, have a student place two pushpins into the foam object along the path of the vertically hanging string.
10. With the pushpins in place, repeat steps 4–6 at a different corner on the foam object. The student should place a third pushpin at the crossing point between the path of the vertically hanging string and the imaginary line between the two pushpins that are stuck in the foam. (This third pushpin should be the center of gravity of the foam object.)
11. As a check, repeat steps 4–6 at a third corner. The path of the string should cross the location of the third pushpin (the center of mass).
12. Remove the “center of mass” pushpin and replace it with the Velcro dot. Note: Position the Velcro dot as close as possible to the location of the (removed) “center of mass” pushpin.
13. Remove the two remaining pushpins and the binder clip.
14. Repeat step 2 so students can observe the spinning, thrown irregularly shaped object in motion. (Is there a point that appears to follow a stable parabolic path?)

Student Worksheet PDF

12511_Student.pdf

Teacher Tips

• All the materials in this demonstration kit can be used indefinitely. Extra pushpins, Velcro dots, pinch clips, fishing sinkers and string have been provided.
• The Bottle Balance Beam can be set out in the classroom or in a display and will attract students’ attention. Additional elements can be added to the bottle to teach other concepts. For example, fill the bottle half full with clear lamp oil and the rest with water tinted blue with food coloring. The clear oil will float on top of the blue water. After several days, empty the bottle and fill it half-full with blue lamp oil and clear water. The blue goes to the top! See who notices. Tell them that it is blue water from Australia (down under) where gravity is reversed!
• A rigid leather belt also works well in place of the plastic tubing for the Finger Balance. The belt needs to fit snuggly in the balance apparatus slot so that the leather strap angles back towards the tip. If the belt is too loose in the slot, it may only hang straight down and the apparatus will not balance because the center of gravity will not be under your finger. Adjust the position of the belt buckle so the belt’s mass is evenly distributed and the apparatus balances and remains upright.
• Remove the protective paper covering from both sides of the acrylic Finger Balance before using.
• When determining the center of gravity of the foam polygon, make sure the pinch clip is attached to as little foam as possible, the clip is centered on the foam edge, and the clip remains parallel to the floor. This will help to eliminate any “extra mass” from affecting the center of mass of the foam object.
• Not all foam shapes will look exactly the same from kit to kit.
• Use a paper cutter to change the shape of the object, if desired. Draw the desired shape before cutting—measure twice, cut once.
• The plastic polygon may also be tossed vertically into the air with a small rotation and one of the holes at the center of the polygon will appear to remain “still” while all the other holes rotate around it (i.e., rotate around the center of mass).
• Additional weights, such as clay or paper clips, may be added to the edge of the plastic polygon to attempt to “balance” the polygon when it spins from a hole other than hole at the center of mass. This demonstrates the importance of having balanced car tires, and how the balance is achieved by adding small weights to the rims of the wheels.

Answers to Questions

Demonstration 1. Center of Gravity of an Irregular Polygon

1. Define center of gravity.

   The center of gravity of the object is the location where all the individual gravitational forces acting on the individual particles add up and result in one net downward force. At this point we can assume all of the mass of the object is concentrated, and therefore this point is also referred to as the center of mass.

2. Explain how the center of gravity of the polygon was determined.

   The center of gravity was located by hanging the object from a corner and hanging string from the same corner. The hanging string lines up with the center of gravity of the object. Hanging the object and string from a second corner allows the center of gravity to be pinpointed—it is at the intersection between the imaginary lines created by the hanging string. Hanging the object and string from a third corner helps to verify the location of the center of gravity.

3. Describe how the polygon spins when it rotates about its center of gravity

   The polygon spins smoothly about its center of mass, and it rotates for a long time.

4. Describe how the polygon spins when it rotates at a point other than the center of gravity.

   The polygon wobbles when it rotates from any of the other holes that are not located at the center of gravity of the polygon. The polygon does not spin smoothly and it slows down quickly.

Demonstration 2. Bottle Balance Beam

5. When a person is standing on two feet, where is his/her center of gravity?

   The center of gravity is located in the center of the body near the waist.

6. Where is his/her center of gravity when standing on one foot?

   The center of gravity is still located in the center of the body near the waist. This is why the body shifts to the left or right to remain balanced over the one foot. The body shifts to move the center of gravity over the support.

7. Why do football players on the line try to “stay low to the ground”?

   Football players try to stay low in order to keep their center of gravity low to the ground. With a low center of gravity, it is more difficult to be knocked over, and it is easier to move heavier objects, such as nose tackles.

Demonstration 3. Finger Balance

8. How did adding a heavy object to the end of the apparatus make it balance on a finger?

   The heavy tube shifted the center of gravity of the object so that it was below the finger supporting the edge of the teardrop shaped object.

9. Draw the position of the balanced Finger Balance and tube and indicate on the drawing the location of the center of gravity (see Figure 6).
   {12511_Answers_Figure_6}

Demonstration 4. Center of Gravity Toss

10. Describe the motion and “travel” of the irregularly shaped object after it was tossed into the air.

    The foam object appeared to wobble as it flew through the air. However, all the material appeared to be spinning around a central location.

11. When the irregularly shaped foam sheet was spun and tossed into the air, what was the shape of the path that the center of mass followed? Draw the shape the center of mass follows.

    The center of mass followed a stable parabolic trajectory.