Materials Included In Kit

Additional Materials Required

Prelab Preparation

Experiment 1. Investigating Pendulums

1. Obtain a support stand and ring and a meter stick.
2. Cut a length of fishing line 75 cm long.
3. Tie the small plumb bob to the end of the string. Keep excess string to a minimum, or cut off the excess string from the knot.
4. Clamp the string to the side of the ring with the clothespin.
5. Position the support stand so that the plumb bob can dangle over the edge of the table and swing freely (see Figure 4 in the Procedure).
6. Repeat steps 1–5 for two more pendulums.

Experiment 2. Archimedes’ Principle

1. Divide the modeling clay into eight (8) 30-g pieces.

Experiment 4. Atwood’s Machine

1. Obtain the Atwood’s machine, support stand, support stand clamp, pulley cord, weight set (or equivalent), soft towel and heavy books.
2. Set up the apparatus on at least a 1.5 m tall platform as shown in Figure 14. Place heavy books on the base of the support stand to prevent it from tipping when the masses hang over the edge of the platform.
   {12633_Preparation_Figure_14}
3. Securely tie one 200-g mass to each end of the pulley cord so the masses are balanced and do not fall. The pulley cord should be long enough to allow one mass to rest on the floor while the other mass hangs just below the Atwood’s machine pulley. Make sure the masses are free to move up and down without interference.
4. Place a soft towel beneath the Atwood’s machine to “catch” the masses and prevent them from being damaged when they hit the floor.

Safety Precautions

The plumb bobs contain lead. Lead is extremely toxic by inhalation (dust form) and ingestion. The clay is considered nontoxic. During the Atwood’s machine experiment, make sure the weights are tightly secured to the pulley cord so they will not come loose during the experiment. If one weight falls off one end of the pulley system, one side will crash down to the ground, while the other side flies up. The quickly falling and/or rising masses could cause injury. The masses may also be damaged if they hit the floor too hard. Students should wear safety glasses during this activity. Remind students to wash their hands with soap and water after completing each laboratory activity. Students should follow all normal laboratory safety guidelines. 

Disposal

Please consult your current Flinn Scientific Catalog/Reference Manual for general guidelines and specific procedures, and review all federal, state and local regulations that may apply, before proceeding. The materials from each lab should be saved and stored in their original containers for future use. Materials from this lab can be reused many times or disposed of following Flinn Suggested Disposal Methods #26a. 

Lab Hints

Investigating Pendulums

• Enough materials are provided in this kit for three student groups to work at the same lab station. This laboratory activity can reasonably be completed in 15 minutes. All materials are reusable.
• To protect against the hazards of lead, the plumb bobs can be dipped into melted wax, or coated with a clear coat of paint or nail polish. The additional mass will not affect the results of the experiment.
• Advise students to release the pendulum so that it swings back and forth in one plane and does not rotate as it oscillates.
• Students should practice releasing the pendulum so that it swings smoothly. Lightly holding the plumb bob from the bottom with only a fingertip as it is pulled to the appropriate release angle and then lowering the finger to release the bob will provide a smooth release and even swing.
• If protractors are not available, simple triangles can be drawn on separate sheets of paper and cut out to create 5° and 15° angles. For the 5° angle, the sides of a right triangle should measure 2.2 cm and 25 cm. The small angle of this triangle (the angle between the long side of the triangle and the hypotenuse) will measure 5°. For the 15° angle, the sides of a right triangle should be 6.7 cm and 25 cm.
• Students should work in pairs with one student measuring the angle of the pendulum with a protractor and releasing the plumb bob, while the other student times and counts the oscillations.
• The Background information and/or extension questions can be given to students based upon their instruction level and your goals for the class.
• For small displacements, the restoring force acting on the plumb bob is directly proportional to the displacement away from the equilibrium position. When the restoring force is directly proportional to the displacement, the oscillations are said to exhibit simple harmonic motion (i.e., the period or time it takes for each complete oscillation will be constant). In simple harmonic motion, the period of oscillation is given by Equation 1 in the Background section. This equation shows that a pendulum’s swing is independent of the mass of the plumb bob and the amplitude of the swing. It depends only on the length of the pendulum and the acceleration of gravity. However, for large amplitudes, the oscillations of the pendulum no longer appear to be simple harmonic. The oscillations will follow a non-linear relationship. Fortunately, the period of a pendulum released at 20 degrees from equilibrium will still vary by less than 1% from the “ideal” simple harmonic motion period.

Experiment 2. Archimedes’ Principle

• Enough materials are provided in this kit for two groups of students to work at the same lab station. Extra cups, trays and pipets have been provided. This laboratory activity can reasonably be completed in 20 minutes. Prepare enough 30-g clay pieces for each group so that each group can begin with a dry piece of clay.
• All materials are reusable.
• For the floating objects, it is important to fill the cup to the almost overflowing point. The water edge will be nearly to the edge of the cup rim. It may take several practice fills for students to see this level. The initial water level needs to be consistent for each experiment for more accurate results. Be sure to do this experiment on a level surface.
• When placing the clay ball or boat into the water, remind students to let it down gently so there is no splash or waves. Also, remind them to not put their fingers into the water, which will add to the water displacement and skew the results. The fishing line harness prevents this.
• Before removing the cup from the plastic weighing tray, carefully remove some of the bulging water in the cup using a pipet. This will help prevent excess water from spilling out of the cup and into the tray when removing the cup from the tray.

Experiment 3. Center of Gravity

• This kit contains enough materials for three student groups of students to work at the same lab station. Student groups should share the three different polygon shapes and dry erase markers. Each group should locate the center of gravity of all three polygon shapes. Students can reasonably complete this experiment in 15 minutes.
• All materials are reusable.
• Shapes may vary slightly from kit to kit.

Experiment 4. Atwood’s Machine

• Enough materials are provided in this kit for one student group. This laboratory activity can reasonably be completed in 15 minutes as long as the Atwood’s machine is set up prior to the start of the lab. All materials are reusable.
• To save time, provide the friction “mass” to the students instead of requiring them to measure it. For 200-g and 220-g masses, the friction “mass” is about 10 g. The friction “mass” will vary depending on the masses that are used, and a more accurate value can be obtained by measuring it before class.
• In addition, measure the height of the dropping mass before class and provide this information to the students.
• This experiment is very sensitive to timing errors and errors caused by friction. If students measure the friction “mass” with an error of only 2 g, such as, reporting the fraction “mass” to be 12 g when it is really 10 g for example, can lead to a 20% error.
• Tying extra pulley cord to the bottom of both masses so that the two ends of the pulley cord always touch the floor no matter how high the masses are will eliminate any possible errors due to the pulley cord mass moving from one side of the pulley system to the other.
• Tall filing cabinets make great tall platforms for this demonstration.

Teacher Tips

• Set up each lab station as described in the Prelab Setup for each experiment before class. Students should leave the stations as they find them before they move on to the next lab station.
• Before class, prepare copies of the procedures and student worksheets for each student. The Background information for each experiment can also be copied at the instructor’s discretion.

Sample Data

Experiment 1. Investigating Pendulums

Test 1

Test 2 Experiment 2. Archimedes’ Principle Experiment 3. Center of Gravity
*Notice: For the “inverse” triangle polygon, the center of gravity lies outside the physical shape of the object. Experiment 4. Atwood’s Machine
Height of the released mass: ___1.665 m___
Friction “mass”: ___10 g___

Answers to Questions

Experiment 1. Investigating Pendulums

Optional Advanced Questions

1. (Optional) On a separate sheet of paper, make a graph with the swing period squared on the y-axis versus the pendulum length on the x-axis. Plot the data and draw a best-fit line through the data points starting from the origin (0,0). Is the best-fit line a straight line? If so, what does this say about the relationship between the period of the pendulum swing and the length of the pendulum?
   {12633_Answers_Figure_15}
   Yes, the best-fit line is straight. This means that the period of the pendulum is proportional to the square-root of the pendulum length. T ∝ √L.
2. What is the period of oscillation for a 1-m long pendulum, with a 100-g plumb bob, on the surface of the Earth (g = 9.81 m/s²)? What would the period of oscillation be on the moon (g = 1.62 m/s²)?

   Period on Earth’s surface = 2.0 s. Period on the Moon’s surface = 4.9 s.

Post-Lab Questions 

1. Calculate the average number of swings in 30 seconds for each test in Tests 1 and 2. Record the average value in the data tables.

   Sample calculation: (22 + 22 + 22)/3 = 22

2. Calculate the swing period by dividing 30 seconds by the average number of complete swings in 30 seconds for each test in Tests 1 and 2.

   Sample calculation: (30 s)/22 = 1.36 → 1.4 s

   {12633_Answers_Table_1}
3. Compare the swing periods in Test 1. How does the release angle affect the swing period? How does the mass of the plumb bob affect the swing period?

   The swing periods appears to be unaffected by the mass of the plumb bob or the angles at which it is released.

4. Compare the swing periods in Tests 1 and 2. How does the length of the pendulum affect the swing period?

   The swing period is affected by the length of the pendulum. Longer pendulums have longer swing periods, meaning they swing slower than shorter pendulums. The short pendulums swung back-and-forth very quickly.

5. Based upon the data from this experiment, are the following statements true or false? (Circle one.)

   (T) F The period of a pendulum is not affected by the mass of the plumb bob on the end of the pendulum.
   T (F) The period of a pendulum is affected by how high the pendulum is raised before it is released.
   T (F) The period of a pendulum increases as the pendulum length decreases.
   T (F) A grandfather clock will “tick-tock” faster when the pendulum is released with a large swing arc compared to a small swing arc.

Experiment 2. Archimedes’ Principle

1. Calculate the difference between the weight (mass) of the clay in air and when it is submerged.

   Sample calculation: 30 g – 10 g = 20 g

2. Subtract the initial volume from the final volume and record the difference as the volume of the water displaced by the submerged clay in the data table.

   Sample calculation: 80.0 mL – 61.0 mL = 19.0 mL

3. Assume the density of water is 1 g/mL. Explain the similarity between the weight (mass) difference and the volume of water displaced by the clay (Questions 1 and 2). The similarity of the weight difference and the volume of water displaced represents Archimedes’ principle. Write the principle in your own words.

   The mass and volume are similar because the buoyancy of the water is related to how much water is displaced. The clay weighs 20 g less when it is submerged in water compared to in air, so 20 g of water must be displaced by the clay. Since the density of water is 1 g/mL, 20 g is equal to 20 mL of displacement. This is very close to the measured 19.0 mL.

4. Calculate the density of clay. Should the clay sink or float? Explain.

   Sample calculation: 30.0 g/19.0 mL = 1.58 g/mL
   The clay should sink in water because it is more dense than water.

5. Calculate the density of the clay boat. Should the boat sink or float? Explain.

   Sample calculation: 30.0 g/29.5 mL = 1.02 g/mL
   The boat should float because its density is approximately the density of water.

6. The mass of the ball of clay and the mass of the boat-shaped clay are the same. In terms of Archimedes’ principle, why does the boat shape float while the ball shape sink? (Hint: What does the waterline on a floating object indicate?)

   Archimedes’ principle states that an object will displace a fluid equal to the volume of the object. The weight of the volume of fluid displaced will be equal to the buoyant force. The boat shape displaces more water than the ball shape, and once enough water is displaced to hold up the weight of the boat, the boat will float. The waterline represents the equilibrium point in which 30 mL of water has been displaced by the shape of the boat.

7. Predict what will happen to the water level of a lake if a cannonball is fired from a floating ship into the lake. Why?
   1. Water level will go up.
   2. Water level will go down.
   3. Water level will stay the same.

      A cannonball is much denser than water so more water must be displaced when the cannonball is on the ship compared to when the cannonball is in the water.

Experiment 3. Center of Gravity

1. Define the center of gravity of an object.

   See Background information for an appropriate definition.

2. Describe a test that could be used to verify whether a point on an object was indeed the center of gravity.

   If the point is truly the center of gravity, the object can be held at this location and remain balanced. Or, the object could be rotated and the object will rotate around the center of gravity (i.e., the center of gravity (center of mass) will remain “stationary” while the rest of the mass spins around it).

3. Assuming that the objects shown below all have the same uniform density throughout, rank the following objects from most stable to least stable, and explain your reasoning.
   {12633_Answers_Figure_16}
   The first object is stable because the center of gravity is directly above the support base. However, it can easily be tipped to one side or the other. The second object is not balanced over its center of gravity. The third object is balanced over its center of gravity, and the base is wide enough to prevent it from tipping. The third object is the most stable.

Experiment 4. Atwood’s Machine

1. Review the timing data for all the trials and determine the best average time of descent. (Sample methods for determining the best average time include: calculate the average value for all the time measurements; ignore data points (e.g., the high and low values), and then calculate the average; draw a graph and determine a line of best fit.
   {12633_Answers_Equation_11}
2. Use the following equation to calculate the acceleration of the masses.
   {12633_Answers_Equation_9}

   a = acceleration of the masses
   h = height the released mass
   t = average time of decent

   a = 2(1.665 m)/(3.73 s)² = 0.239 m/s²

3. Use the following equation to calculate the acceleration due to gravity (g).
   {12633_Answers_Equation_10}

   g = acceleration due to gravity
   a = acceleration of masses (calculated in Question 2)
   m₁ = 200 g
   m₂ = 220 g
   ms = pulley sheave mass (5.3 g)
   m_f = friction “mass”

   g = (0.239 m/s²)(200 g + 220 g + 5.3 g)/(220 g – 200 g – 10 g) = 10.2 m/s²

4. Calculate the percent error in this experiment compared to the literature value of g, 9.81 m/s² (at sea level).
   {12633_Answers_Equation_12}
5. List possible sources of error in this experiment and their effects on measuring the acceleration due to gravity. How would friction in the pulley affect the outcome of this experiment—would the calculated value of g be too high or too low as a result of this error?

   Some sources of error include friction in the pulleys, stretching of the pulley cord, and shifting of the pulley cord mass from one side of the Atwood’s machine to the other as the masses moved up and down. Also, since the pulleys are not massless, the rotation of the pulleys will reduce the linear acceleration of the falling (and rising) masses. Timing and friction cause the most error for this experiment. Friction causes the acceleration to be lower than expected. So, if friction were not accounted for, the calculated acceleration due to gravity would be too low.

Introduction

This all-in-one Investigating Gravity Kit is designed to provide students the opportunity to explore the fundamental principles of the force due to gravity. Four hands-on lab stations can be arranged so student groups can experiment with pendulums, center of gravity, acceleration due to gravity and buoyancy (anti-gravity) forces. Students will use a pendulum to learn about simple harmonic motion and the properties that affect a pendulum’s swing period. In the center of gravity experiment, students examine mass distribution and balance. Using Atwood’s machine and a timer, students will indirectly measure the acceleration due to gravity. And finally, students will learn about the anti-gravity buoyant force that allows boats to float.

Concepts

• Gravity
• Period of oscillation
• Pendulums
• Simple harmonic motion
• Archimedes’ principle
• Density
• Buoyancy
• Water displacement
• Balance
• Center of gravity
• Newton’s laws of motion
• Acceleration due to gravity
• Friction
• Atwood’s machine
• Newton’s second law of motion

Background

Experiment 1. Investigating Pendulums

A simple pendulum is composed of a string tied to a rigid object at one end (the anchoring point) and a freely hanging mass, also known as a plumb bob, tied to the other end. When the pendulum is at rest, the plumb bob will hang directly below the anchoring point, and the string will be vertical. When the pendulum is displaced away from the equilibrium (at rest) position and released, the force of gravity will cause the pendulum to swing back and forth along a swing arc at a constant rate. The pendulum will oscillate with simple harmonic motion. That is, the pendulum will oscillate back and forth along an arc, following the same path, reach the same displacement away from the equilibrium position, and each back-and-forth motion (one complete oscillation) will take the same amount of time. The time it takes for one complete oscillation is known as the period of the oscillation.

The external forces acting on the plumb bob are the pull of gravity (mg) and the tension in the string (T) holding the plumb bob up. When the pendulum is vertical, these forces are balanced. When the plumb bob of the pendulum is moved away from its equilibrium (at rest) position along the arc of the pendulum swing and then released, gravity and the tension in the string are still the only forces acting on the plumb bob. However, now these forces are unbalanced. The unbalanced forces result in a restoring force (mg x sinϴ) that moves the plumb bob back toward the equilibrium position along the arc of the swing. Because of the plumb bob’s momentum, however, it will continue to swing past the equilibrium position. Once it passes equilibrium, the plumb bob will swing up along the pendulum’s arc and a restoring force will again act on the plumb bob to slow it down until it momentarily stops, and then falls back down towards its equilibrium position and the cycle will repeat itself. The pendulum will continue to oscillate back and forth this way indefinitely if no other forces (such as friction) act on it. The period of this simple harmonic motion can be calculated theoretically using Equation 1.

See Figure 1 for a diagram of the forces acting on the plumb bob.

T = period of oscillation
L = length of pendulum
g = acceleration of gravity constant

Experiment 2. Archimedes’ Principle

Archimedes was born about 287 B.C. in Sicily and was killed by a Roman soldier about 211 B.C. He is generally regarded as the greatest mathematician and scientist of antiquity and one of the three greatest mathematicians of all time, along with Isaac Newton (1643–1727) and Carl Friedrich Gauss (1777–1855). Archimedes was very involved in a wide range of scientific and mathematical studies. The famous “gold crown story” stemmed from the fact that the king was suspicious about the purity of the gold in his crown and asked Archimedes to find a way to determine if it was the real thing. Solving the problem seemed to be nearly impossible because little was known about chemical analysis in Archimedes’ day. The story goes that one day Archimedes was thinking about the problem while he was taking a bath. As he lay floating in a pool, he thought about how his body felt “weightless.” Suddenly he realized that all bodies “lose” a little weight when placed in water, and the bigger their volume, the more weight they lose. He realized that the density of a metal could be found from its weight and its weight loss in water. The weight of the king’s crown and its apparent loss of weight in water could tell him if the crown was made of pure gold. According to the story, when Archimedes realized this experimental design, he ran into the street yelling “Eureka! I have found it.”

Density is a characteristic property of a material and pure elements or compounds may be identified by their density. Density is defined as the mass of a substance per unit of volume (Density = mass/volume). Density is commonly expressed as g/cm³ or g/mL. The density of pure water is 1.00 g/cm³ at 20 °C. Objects with a density greater than 1.00 g/cm³ will sink in pure water. Objects with a density less than 1.00 g/cm³ will float in pure water.

Buoyant forces cause objects to “lose” weight when they are submerged in a liquid or fluid. This “anti-gravity” is generated by any fluid, including air. The buoyant force on a submerged object is equal to the weight of the liquid, or fluid, displaced by the object. The amount of fluid displaced by the object is equal to the volume of the object. Therefore, the density of the object can be determined by dividing the mass of the object by the volume of liquid it displaces. The weight of this displaced fluid is equal to the buoyancy force. The weight of the displaced liquid can be determined using the density of the liquid and the volume displaced.

Experiment 3. Center of Gravity
Gravity is the attractive force between all objects. The most familiar gravitational force is that of the Earth, which pulls all objects toward the ground and is more commonly referred to as an object’s weight. The more massive two objects are, the greater the gravitational force that exists between them. According to Isaac Newton’s (1643–1727) laws of gravitation, the Earth attracts every tiny particle of mass of every object and pulls them toward the center of the Earth. 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. Consequently, the center of gravity is the point where we can assume all of the mass of the object is concentrated, and therefore is also referred to as the center of mass. The location of the center of gravity, especially for irregularly shaped objects, is critical for the overall stability and balance of an object on the Earth’s surface. An object is most stable on the Earth’s surface when the object’s center of gravity is at its lowest point, and is centered about the object’s supporting base. 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 the 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.

When an object hangs from one corner, its center of mass will be located directly below the suspended point. This occurs because the object is most stable when its center of mass is at its lowest possible point. By drawing a vertical line through the hanging location, the line will also go through the center of mass. To pinpoint the center of mass, the object can be hung from another location and another vertical line may be drawn. The intersection of the two lines is the center of mass of the object. Hanging the object from a third location and drawing a third line will help verify the location of the center of mass of the object.

Experiment 4. Atwood’s Machine
The Atwood’s machine was developed by George Atwood (1745–1807) in the late 18th century to indirectly measure the acceleration of gravity. When two unequal masses are tied to the ends of a length of string and hung over a pulley, the larger mass will accelerate down, while the smaller mass accelerates up at the same rate. If the pulleys are assumed to be massless and frictionless, the acceleration depends on the acceleration due to gravity, the total mass of the system and the difference in mass between the two hanging masses. This follows Newton’s second law of motion—force equals mass times acceleration (F = ma).

In the laboratory however, pulleys always have mass and are slowed by friction. Because real pulleys have mass, when they rotate their rotational energy takes away from the total energy that would otherwise be used to move the masses (in the ideal situation). The pulley sheave axles also produce a frictional force that acts against the rotation and decreases the total energy of the system. The frictional force is not constant, but increases as more weight is placed on the axles. In order to obtain real-world results that are closer to the accepted value of gravitational acceleration, the Atwood equation needs to include both the mass of the pulley as well as a term to account for the friction in the pulleys (see Figure 2). Refer to Figure 2 and the following equations for the derivation of the Atwood’s machine equation.

T₁ = tension in string 1
T₂ = tension in string 2
m₁ = 200 g
m₂ = 220 g
a = acceleration of both masses
g = acceleration due to gravity

A rotating object experiences a torque that is equal to its rotational moment of inertia times its angular acceleration (τ = Iα). A torque is equal to a force times the distance of the lever arm where the force is applied (τ = F x r). In the case of Atwood’s machine, the lever arm is the radius (R) of the pulley sheave (see Figure 2). Therefore, the torque is equal to the net force on the outer rim of the pulley sheave times the radius of the pulley. Rotational moment of inertia is equivalent to the “rotational” mass of the object about an axis of rotation. Assuming the pulley sheave is a solid cylinder with mass m_s and radius, R, its rotational moment of inertia about the center is equal to ½m_sR². Angular acceleration is simply the linear acceleration divided by the radial distance from the center of rotation (α = a/R). The friction along the axle of the pulley sheave produces a torque that acts against the rotation. Since the torque produced by friction acts against the motion, it can be assumed that an additional mass has been added to the smaller mass which causes the acceleration of the masses to be less than expected compared to the ideal situation. The torque produced by this frictional “mass,” m_f, is equal to m_fgR. Since the frictional “mass” is a measure of the total friction produced by the two-pulley Atwood’s machine in this experiment, it is assumed that each pulley produces half the total measured amount of friction. The net force on the outer edge of the pulley sheave is therefore: Equation 5 reduces to Substitute for T₁ and T₂ from Equations 1 and 2: Rearrange to solve for g: The pulley sheave’s mass is a constant. For the Atwood’s machine in this experiment the mass of the pulley sheave is 5.3 ±0.1 grams. The frictional “mass” can be inferred from the static friction of the pulley system and is dependent on the total mass hanging from the pulleys.

Experiment Overview

Experiment 1. Investigating Pendulums 
A swinging pendulum is a very common sight—from children swinging on a swing set to the timekeeping swing of a grandfather clock. What affects the period of a pendulum? Does the release height affect the time it takes for a pendulum to complete one oscillation? What about the weight of the plumb bob? Explore the properties of a pendulum’s swing.

Experiment 2. Archimedes’ Principle
Legend has it that Archimedes ran through the streets of ancient Greece shouting “Eureka!” when he realized how he could utilize his density and water displacement experiments to demonstrate that the king’s crown was made of pure gold and not some imitation. This relationship between density and water displacement is known as Archimedes’ principle. Examine Archimedes’ principle and how it relates to buoyancy (an “anti-gravity” force).

Experiment 3. Center of Gravity
Will a force cause an object to spin, move in a straight line, or fall over? The answers to these questions depend on the location of the center of gravity of the object. Locate the center of gravity of different polygon shapes.

Experiment 4. Atwood’s Machine
Accurately measuring the acceleration of a falling object due to gravity is difficult because a falling object travels short, laboratory-scale distances very quickly. However, by slowing down an object’s fall according to the rules of Newton’s second law of motion, the acceleration due to gravity can be accurately determined by an indirect method. A device that is used to indirectly measure the acceleration due to gravity is called an Atwood’s machine.

Materials

Safety Precautions

The plumb bobs contain lead. Make sure the weights are tightly secured to the pulley cord so they will not come loose during the experiment. If one weight falls off one end of the pulley system, one side will crash down to the ground, while the other side flies up. The quickly falling and/or rising masses could cause injury. The masses may also be damaged if they hit the floor too hard. Wear safety glasses. Wash your hands with soap and water after completing this laboratory activity. Please follow all normal laboratory safety guidelines.

Procedure

Experiment 1. Investigating Pendulums

Test 1. Measure the period of the pendulum’s swing

1. Use a meter stick and adjust the length of the pendulum so that it is 50 cm (0.50 m) from the bottom of the plumb bob to the bottom of the support stand ring (see Figure 3).
   {12633_Procedure_Figure_3_Pendulum setup}
   Use your finger to hold the fishing line in place on the ring, and wrap the excess line around the ring two or three times so it does not slip. Then, secure the fishing line by clamping it with a clothespin clamp (see Figure 4).
   {12633_Procedure_Figure_4_Clamp setup}
2. Pull the (small) plumb bob along its swing arc to an angle of 5 degrees from its vertical equilibrium. Use a protractor to measure the angle of the string.
3. With a stopwatch or watch with second hand ready, release the plumb bob and begin timing. Count the number of times the plumb bob swings through one complete cycle for a total time of 30 seconds. (The easiest way to count is to count how many times the plumb bob returns to the initial release point.)
4. Record the number of complete swings under Trial 1 in the data table for Test 1.
5. Repeat steps 2–4 two more times (Trials 2 and 3).
6. Next, pull the plumb bob along its swing arc to an angle 15 degrees from equilibrium.
7. Release the plumb bob and begin timing. Count the number of times it swings through one complete cycle for 30 seconds. Record the number of complete swings in the appropriate space in the data table.
8. Repeat steps 6 and 7 two more times (Trials 2 and 3).
9. Untie or cut the fishing line to remove the small plumb bob and then tie the large plumb bob to the end of the string.
10. Use the meter stick and adjust the length of the pendulum so that it is 50 cm from the bottom of the plumb bob to the bottom of the support stand ring. Wrap the excess string around the ring and clamp it with the clothespin clamp.
11. Repeat Procedure steps 6–8. Record your data in the appropriate spaces in the data table.

Test 2. The period of the pendulum versus the length of the pendulum

12. With the large plumb bob tied to the fishing line, adjust the pendulum length to 20 cm (0.20 m).
13. Pull the plumb bob along its swing arc to an angle of 15 degrees from equilibrium.
14. With a stopwatch or watch with second hand ready, release the plumb bob and begin timing. Count the number of times it swings through one complete cycle for a total time of 30 seconds.
15. Record the number of complete swings in the appropriate space in the data table for Experiment 2.
16. Repeat steps 13–15 two more times (Trials 2 and 3).
17. Adjust the length of the pendulum to 70 cm (0.70 m).
18. Repeat Procedure steps 13–16. [Use the data from the Test 1 for the 50-cm pendulum length (large plumb bob from a release angle of 15 degrees from equilibrium).]
19. Untie or cut the fishing line to remove the large plumb bob and then tie the small plumb bob to the end of the string for the next group of students.

Experiment 2. Archimedes’ Principle

1. Obtain a piece of modeling clay that is approximately 30 g and a paper clip.
2. Bend the paper clip as shown in Figure 5.
   {12633_Procedure_Figure_5}
3. Cut 30 cm of fishing line and tie one end to the bent paper clip as shown in Figure 5. Tie a looping knot at the other end of the fishing line (see Figure 6).
   {12633_Procedure_Figure_6_Looping knot}
4. Roll the piece of clay into a ball.
5. Push the “hook” end of the paper clip into the clay so that it remains secure. Hang the line on the hook of a spring scale and measure the weight (mass) of the clay piece. Record the mass in the data table as the weight of clay in air on the Archimedes’ Principle Worksheet.
6. Fill a 9-oz plastic cup about ¾-full with water. With the clay piece still hanging from the spring balance, lower the clay into the water until it is fully submerged. Do not let the clay touch the sides or bottom of the cup. Record the weight (mass) of the clay while it is submerged in the water. Hold the apparatus steady while taking the reading. Record the mass in the data table as the weight of clay submerged in water on the Archimedes’ Principle Worksheet.
7. Remove the clay from the cup and then remove the paper clip from the clay piece. Dry the clay and roll it into a thin cylinder shape so that it can be lowered into a 100-mL graduated cylinder.
8. Push the “hook” end of the paper clip into the end of the cylinder-shaped clay piece (see Figure 7).
   {12633_Procedure_Figure_7}
9. Fill a 100-mL graduated cylinder with about 65 mL of water. Measure the precise volume of the water in the cylinder and record this as the initial volume on the Archimedes’ Principle Worksheet.
10. Lower the clay cylinder by the fishing line into the graduated cylinder until the clay is completely submerged, and is not touching the sides or bottom of the graduated cylinder.
11. Carefully measure the new (combined) volume of water and the clay piece in the graduated cylinder and record it as the final volume on the Archimedes’ Principle Worksheet.
12. Remove the clay from the graduated cylinder, remove the paper clip, and completely dry the clay.
13. Flatten the clay into a thin piece. Use your hands to shape the clay into a clay boat. Keep shaping the boat and testing it in a cup of water to be sure it floats. The clay boat needs to fit inside the top of the plastic cup without touching the sides (see Figure 8).
    {12633_Procedure_Figure_8_Boat in cup}
14. Completely dry the finished boat with paper towels.
15. Use a large weighing tray as a catch basin and place the 9-oz plastic cup in the middle of the catch basin. Use a beaker to slowly fill the cup with water. When the water gets to the very brim, add water very slowly with a Beral-type pipet. Fill the cup so that the water actually appears to be “bulging” up over the top plane of the cup (see Figure 9). Fill the cup with as much water as possible without the water spilling out of the cup. If water spills out of the cup, remove the cup from the tray, dry the tray with paper towels, place the cup back on the tray and start filling the cup again using the pipet.
    {12633_Procedure_Figure_9_Cup nearly overflowing}
16. Carefully press the “hook” of the paper clip into the bottom center of the boat. Press the clay around the paper clip hook on the bottom of the boat to seal any holes in the boat and to secure the paper clip to the boat (see Figure 10).
    {12633_Procedure_Figure_10}
17. Slowly, without splashing, lower the boat into the bulging cup of water until the boat is floating completely. (Water will spill out of the cup and into the catch basin.)
18. Note the spot on the side of the boat where the water level is when the boat is floating. Use a pencil to lightly mark the level (see Figure 8).
19. Using the harness, slowly and carefully lift the boat out of the water being careful to not splash any extra water out of the cup.
20. Set the boat aside and carefully lift the cup out of the catch basin. It will take a steady hand not to spill any water from the cup into the catch basin!
21. Set the cup aside. Pour the water from the catch basin into an empty graduated cylinder. Measure the volume of water carefully and record it in the data table as the volume of water displaced by the floating boat on the Archimedes’ Principle Worksheet.
22. If time permits, perform a second trial.

Experiment 3. Center of Gravity

1. Cut a piece of fishing line approximately 30 cm long.
2. Tie a “looping knot” at one end of the string (see Figure 11).
   {12633_Procedure_Figure_11}
3. Tie the other end of the fishing line around the washer.
4. Obtain one of the three polygon shapes at the lab station. Notice that there are holes in the corners of the polygon.
5. Set up a support stand with a clamp and two S-hooks. Hang the shape from one of its holes on the S-hook as shown in Figure 12.
   {12633_Procedure_Figure_12}
6. Hang the string from the S-hook. Make sure the support stand clamp is high enough to allow the string and polygon shape to hang freely.
7. Once the string and shape have stopped swinging, hold the string at the bottom of the polygon with one hand, and with your other hand use a dry erase marker to draw a straight line on the hanging polygon shape to mark the path of the vertically hanging string.
8. Repeat steps 5–7 two more times, hanging the shape from two different corners (holes) and drawing the new line on the shape each time.
9. Sketch the lines drawn on the polygon shape onto the corresponding shape in the data table.
10. Erase the marker lines on the polygon shape with a paper towel and return the shape to the distribution area at the lab station.
11. Repeat steps 5–10 using the other two polygon shapes.

Experiment 4. Atwood’s Machine

A. Measure the height of the released mass

1. Raise one of the 200-g masses all the way to the top of the pulley system until the lower mass touches the floor. If the lower mass does not reach the floor, adjust the height of the clamp holding the Atwood’s machine until the lower mass touches the floor (see Figure 13).
   {12633_Procedure_Figure_13}
2. Use two meter sticks, or a tape measure, to measure the height above the floor from which the mass will be released (measure from the bottom of the mass). The lower mass should be touching the floor. Record the height in the data table on the Atwood’s Machine Worksheet.

B. Measure the friction “mass” of the pulley system

3. Position the two 200-g masses along the pulley system so they are at the same height.
4. Add one 1- to 2-g mass onto one of the 200-g masses.
5. Lightly push down on this mass.
6. If the masses continue to move, then the static friction has been overcome. Record the additional mass added as the friction “mass” on the Atwood’s Machine Worksheet.
7. If the masses do not continue to move, reposition the two 200-g masses at equal heights and add another 1- or 2-g mass to the same 200-g mass as before.
8. Repeat steps 5–7 until the masses continue to move after the light downward push.
9. Record the additional mass as the friction “mass” on the Atwood’s Machine Worksheet.

C. Measuring acceleration

10. Add a 20-gram slotted mass to one of the 200-g masses.
11. Raise the 220-g mass until the 200-g mass touches the floor.
12. Obtain a stopwatch.
13. Position the 220-g mass at the appropriate release height.
14. When the 220-g mass is at the correct height, and the 200-g mass is stationary on the floor, simultaneously release the 220-g mass and begin timing with the stopwatch.
15. Stop timing when the 220-g mass just touches the floor. Record the time in the data table on the Atwood’s Machine Worksheet.
16. Repeat steps 6–8 nine more times for a total of 10 trials. Record the times in the data table.
17. Remove the additional 20-g mass and leave the setup for the next group.

Student Worksheet PDF

12633_Student1.pdf