# Conservation of Energy on an Inclined Plane

## Inquiry Lab Kit for AP® Physics 1

### Materials Included In Kit

Hexagonal screws, 1" long x ¼" dia, 16
Inclined planes, wood, 18¾" x 4⅝" x ½", 8
Metal sheets, aluminum, 4" x 4", 8
Spheres, steel, ¾" dia., 8
Support rods, 8
Washers, ", 8

(for each lab group)
Clamp holders, 8
Meter sticks, 8
Pencil or chalk
Printer paper, white, 3–4 sheets
Protractors, 8
Scissors, 8
String, nylon, 16 m
Support stands, 8
Textbooks, 3–4 (optional)
Transparent tape

### Safety Precautions

Remind students to quickly retrieve the sphere once it hits the floor. Wear safety glasses. Please follow all laboratory safety guidelines.

### Disposal

All materials may be saved and stored for future use.

### Lab Hints

• This laboratory activity can be completed in two 50-minute class periods. It is important to allow time between the Introductory Activity and the Guided-Inquiry Activity for students to discuss and design the guided-inquiry procedures. Also, all student-designed procedures must be approved for safety before students are allowed to implement them in the lab. Prelab Questions may be completed before lab begins the first day.
• It is important for students to create a smooth curve with the metal sheet so that the sphere does not bounce when it reaches the end of the inclined plane. Place the curved metal sheet approximately 1–2 cm from the edge of the tabletop so the sphere does not roll along the tabletop for an extended period of time.
• If the inclined plane is tilted and the sphere does not roll in a straight line down the inclined plane, use several folded sheets of paper to shim the bottom of the inclined plane until it is level and the sphere rolls in a straight line.

### Teacher Tips

• This investigation should be performed after students have studied such topics as kinematics, projectile motion, and potential and kinetic energy.
• The most significant source of error occurs if the sphere bounces instead of rolling smoothly off the edge of the table. It is important that the metal sheet have a smooth curve between the end of the inclined plane and the tabletop. If the curve is not smooth, or the metal sheet is curved too much, the sphere will bounce off the tabletop, reducing the distance the sphere is expected to travel because the sphere will have both horizontal and vertical speed components. The sphere’s momentum may bend the metal when it rolls over, especially if it is released from a great height. A small amount of crumpled paper can be placed under the curved metal junction in order to give it more support.

### Further Extensions

Opportunities for Inquiry

Using the knowledge gained in the Introductory Activity and Guided-Inquiry experiments, develop a procedure to launch the sphere a certain distance away from the table into a cup.

Alignment to the Curriculum Framework for AP® Physics 1

Enduring Understandings and Essential Knowledge
Interactions with other objects or systems can change the total energy of a system. (4C)
4C1: The energy of a system includes its kinetic energy, potential energy, and microscopic internal energy. Examples should include gravitational potential energy, elastic potential energy, and kinetic energy.

Learning Objectives
4C1.1: The student is able to calculate the total energy of a system and justify the mathematical routines used in the calculation of component types of energy within the system whose sum is the total energy.
4C1.2: The student is able to predict changes in the total energy of a system due to changes in position and speed of objects or frictional interactions within the system.

Science Practices
2.1 The student can justify the selection of a mathematical routine to solve problems.
3.1 The student can pose scientific questions.
3.2 The student can refine scientific questions.
3.3 The student can evaluate scientific questions.
4.1 The student can justify the selection of the kind of data needed to answer a particular scientific question.
4.2 The student can design a plan for collecting data to answer a particular scientific question.
4.3 The student can collect data to answer a particular scientific question.
4.4 The student can evaluate sources of data to answer a particular scientific question.
5.1 The student can analyze data to identify patterns or relationships.
5.2 The student can refine observations and measurements based on data analysis.
5.3 The student can evaluate the evidence provided by data sets in relation to a particular scientific question.
6.1 The student can justify claims with evidence.
6.2 The student can construct explanations of phenomena based on evidence produced through scientific practices.
6.3 The student can articulate the reasons that scientific explanations and theories are refined or replaced.

### Science & Engineering Practices

Planning and carrying out investigations
Analyzing and interpreting data
Using mathematics and computational thinking
Constructing explanations and designing solutions

### Disciplinary Core Ideas

HS-PS3.A: Definitions of Energy
HS-PS3.B: Conservation of Energy and Energy Transfer
HS-PS3.C: Relationship between Energy and Forces
HS-ETS1.C: Optimizing the Design Solution

### Crosscutting Concepts

Cause and effect
Scale, proportion, and quantity
Energy and matter

### Performance Expectations

HS-PS3-1. Create a computational model to calculate the change in the energy of one component in a system when the change in energy of the other component(s) and energy flows in and out of the system are known.
HS-PS3-2. Develop and use models to illustrate that energy at the macroscopic scale can be accounted for as a combination of energy associated with the motion of particles (objects) and energy associated with the relative position of particles (objects).
HS-PS3-3. Design, build, and refine a device that works within given constraints to convert one form of energy into another form of energy.
HS-ETS1-2. Design a solution to a complex real-world problem by breaking it down into smaller, more manageable problems that can be solved through engineering.

1. A 195-kg sphere, dropped from 135 m above the ground, requires how much time to hit the ground?

t = (2H/g)½ = [(2 x 135 m)/9.8 m /s2]½ = 5.25 s

2. A sphere rolling down a hill has two types of kinetic energy. Name them and write mathematical equations to quantify them.

Linear kinetic energy: KEl = ½ mv2
Rotational kinetic energy: KEr = ½ Iω2, where I = moment of inertia and ω = angular velocity

3. Use Equations 1, 2, 3, 4 and 5 to solve for the velocity, v, of a sphere rolling down a ramp from a height H and of mass m.

PE = KET
KET = ½mv2 + ½Iω2
KET = ½mv2 + ½(2/5mr2)(v/r)2
KET = ½mv2+ 1/5mv2
KET = 7/10mv2
mgh = 7/10mv2
v2 = 10/7gh

{13786_PreLab_Equation_1}
4. Draw a very general graph that depicts the change in a sphere’s potential energy and kinetic energy as it rolls down a hill. Use two lines, one to represent the sphere’s potential energy and another to represent the sphere’s kinetic energy. The y-axis can be labelled “Energy” and the x-axis “Height.”
{13786_PreLab_Figure_1}
5. Why does a small avalanche of rocks typically travel ⅔ its starting height into an adjacent valley whereas a large avalanche travels significantly farther?

In a large avalanche there is enough material to serve as a frictionless barrier between the top rock layer and the bottom rock layer. The top rock layer can move along the bottom rock layer, which experiences friction.

### Sample Data

Introductory Activity

{13786_Data_Table_1}
Fall Time:
{13786_Data_Equation_1}
{13786_Data_Table_2}
Horizontal Velocity:

vx = dx/t
vx = 0.653 m/0.412 s
vx = 1.58 m/s

Theoretical Horizontal Velocity:
{13786_Data_Equation_3}

vx = 1.64 m/s

Percent Error for Horizontal Velocity:
{13786_Data_Equation_5}

% Error = 3.36%

Data and Analysis
{13786_Data_Table_3}

Theoretical Horizontal Distance:

Vtheoretical x t = Xtheoretical
1.64 m/s x 0.412 s = 0.670 m

Percent Error for Horizontal Distance:

{13786_Data_Equation_7}

Guided-Inquiry Discussion Questions

1. In the Introductory Activity, the sphere was assumed to be rolling. Would the theoretical velocity of the sphere increase or decrease if the sphere slid down the ramp with no friction? Derive a formula solving for velocity to support your claim.

The sphere would be traveling faster if it were sliding down the ramp with no friction.

PE = KE
mgh = ½mv2

When rolling is taken into account, the gh term is multiplied by 10/7, which is less than 2.
2. Examine the formulas derived in Prelab Question 3 and the previous question. For a sphere and box moving down a ramp, what variable(s) can be manipulated to increase the horizontal velocity? Discuss practical methods by which the variable(s) could be changed.

In order to increase the velocity of the box at the bottom of the ramp, the starting height or the acceleration due to gravity can be increased. In both formulas, the velocity is only dependent upon the initial height and gravity. It is easy to change the height of the object—simply move it higher up the ramp. It is not feasible to change gravity by a measurable amount on Earth.

3. Consider two spheres, one of mass m and the other 2m, held 1 meter above the ground.
1. If the spheres are released simultaneously, which sphere will hit the ground first?

There spheres would hit the ground at the same time. Both spheres are accelerating at the same rate, 9.81 m/s2, due to gravity.

2. Which sphere is traveling faster when it hits the ground?

Because the spheres are released from the same height, accelerate equally, and fall for the same amount of time, the change in velocity will be the same for both. The spheres will hit the ground with the same velocity.

4. Two spheres of equal mass are held in place at the tops of two ramps (see Figure 3).
1. Which sphere has greater potential energy at the top of the ramp?

The spheres have equal potential energy. Potential energy is dependent upon the mass of the object, gravity and the height, PE = mgh. The mass of the spheres are the same, as well as gravity. If the heights were different, then the sphere higher up would have greater potential energy.

2. Which sphere is traveling faster at the bottom of the ramp when released?

The spheres will be traveling with the same velocity at the bottom of the ramp. One sphere may get the bottom more quickly due to the angle of the ramp, but the conversion to kinetic energy will be the same for both ramps: PE = KE.

5. Consider the following variables that may affect the horizontal travel distance of the steel sphere roll down an inclined plane: mass, release height and angle of the inclined plane with respect to the table.

Effect of Release Height
To test the effect of release height on horizontal travel distance, the height of the table, the sphere’s mass and the angle of the ramp will be kept constant. The sphere is placed at different starting positions, noting the height. The distance the sphere travels away from the table is measured. As the release height increases, the horizontal travel distance will increase.

Effect of Mass
To test the effect of the sphere’s mass on horizontal travel distance, the height of the table, angle of the ramp, and release height will be kept constant. Spheres are varying mass will be released. The distance the sphere travels away from the table is measured. As the mass changes, there will be no difference in horizontal travel distance.

Effect of Ramp Angle
To test the effect of the angle of the ramp on horizontal travel distance, the height of the table, sphere’s mass, and release height will be kept constant. The support stand will be adjusted so the ramp forms varying angles with respect to the table. The metal foil ramp will also be adjusted for a smooth transition between the ramp and table. The distance the sphere travels away from the table is measured. As the angle changes, there will be no difference in the horizontal travel distance.

Review Questions for AP® Physics 1
1. A 1.15-kg block is released from rest on a frictionless inclined plane. The inclined plane forms an angle of 35° with the ground. The block slides down the inclined plane and compresses a spring at the bottom by 3.5 cm and stops momentarily. If 334 N are required to compress the spring by 2.4 cm, what distance has the block travelled?

Fspring = –kx
334 N = –k(0.024 m)
k = 1.4 x 104 N/m

Wspring = –k(x22– x12)/2 = [–(1.4 x 104 N/m)(0.035 m)2]/2 = –8.6 J

Wspring = Wgravity
–8.6 J = mg x d x sinθ
–8.6 J = 1.15 kg x 9.8 m/s2 x d x sin35°
d = –8.5 J/–6.46 = 1.33 m

2. A student attempts to fire a small steel sphere (mass = 250 g) into a cup from a spring-loaded cannon affixed to a table. The cup is located 74.7 cm from the table’s edge. The height of the table is 85 cm. If compressing the spring by 2.2 cm results in the sphere travelling 82.2 cm, by what distance should the spring be compressed to project the sphere into the cup?

Time to fall:
t = (2H/g)½ = [(2 x 0.85 m)/9.8 m/s2]½ = 0.42 s

vx = 0.822 m /0.42s = 1.96 m/s
KEx = ½mvx2
KEx = ½(0.25 kg)(1.96 m/s)2
KEx = 0.480 J

Espring
= –½kx2
Espring = KEx
–½k(0.022 m)2 = 0.480 J
k = 1.98 x 103 N/m

vx = 0.747 m/0.42 s = 1.78 m/s
KEx = ½mvx2
KEx = ½(0.25 kg)(1.78 m/s)2
KEx = 0.40 J

–½k(x)2 = 0.40 J
x2 = (0.40 J x 2)/k
x2 = 0.80 J/(1.98 x 103 N/m) = 4.04 x 10–4 m2
x = 0.0201 m = 2.01 cm

3. Consider the box and sphere on the ramps below. The box and sphere have the same mass. When released, the box slides down the frictionless ramp and off the table. The sphere rolls down the ramp without slipping and off the table. The table is 1.00-m tall. Which goes farther: the box or the sphere? Explain.

Fall Time:

Velocity at Bottom of Ramp:

Box slides with no friction:

PE = KE
mgh = ½mv

Sphere rolls without slipping:

PE = KET
KET = ½mv2 + ½Iω2
KET = ½mv2 + ½ × 2/5 mr2(v/r)2
KET = ½mv2 + 1 5mv2
KET = 7/10mv2
mgh = 7/10mv2
v2 = 10/7 gh

Horizontal Distance:

Dx = v x t
Box: Dx = 4.43 m/s x 0.452 s = 2.00 m
Sphere: Dx = 3.74 m/s x 0.452 s = 1.69 m
The box travels farther because all of its gravitational potential energy is transformed into horizontal kinetic energy. The sphere transforms a smaller amount of its potential energy into horizontal kinetic energy because some energy is transferred into its rotational motion.

4. How long must a 25° ramp be to slow to a momentary stop a runaway truck moving along a truly horizontal highway at 140 miles per hour?

KE = PE
½mv2 = mgh
½v2 = gh
h = [(½v2)/g] = [0.5 x (38.9 m/s)2]/9.8 m/s2 = 77.2 m
sinθ = 77 m/d
d = 77.2 m /sinθ = 77.2 m /sin25° = 182.7 m

5. Why must all of the experimental horizontal travel distances of the steel spheres necessarily be less than the potential, theoretical travel distance?

Because some of the sphere’s energy is converted to heat (thermal energy) owing to friction.

### References

AP* Physics 1: Algebra-Based and Physics 2: Algebra-Based Curriculum Framework; The College Board: New York, NY, 2014.

# Conservation of Energy on an Inclined Plane

## Inquiry Lab Kit for AP® Physics 1

### Introduction

The law of conservation of energy states that energy cannot be created or destroyed, only converted from one form to another. For example, a steel sphere held on an inclined plane (on a table) has potential energy proportional to its height, or distance above the tabletop. When released, the sphere’s potential energy transforms into kinetic energy as it rolls down the inclined plane and launches off the tabletop. In this experiment, a sphere’s horizontal travel distance will be gauged to see how closely it obeys the law of conservation of energy.

### Concepts

• Kinetic energy
• Potential energy
• Conservation of energy
• Projectile motion

### Background

When a sphere is held above a tabletop, it possess potential energy due to gravity. If the sphere were to be released, the potential energy would transform into kinetic energy and could do work on another object. Potential energy is dependent on the mass of the object and its height above a surface as seen in Equation 1.

{13786_Background_Equation_1}
As the sphere falls, the potential energy becomes kinetic energy, which is dependent on the mass and velocity of the object (Equation 2).
{13786_Background_Equation_2}
If the sphere is rolling, such as down a ramp, then some of the potential energy is transformed into rotational motion of the sphere. If the sphere is rotating without slipping, then its rotational kinetic energy is proportional to its translational velocity:
{13786_Background_Equation_3}
where I is the moment of inertia for the sphere and ω is the rotational velocity of the sphere. In this case, the following formulas are used:
{13786_Background_Equation_4}
{13786_Background_Equation_5}
where r is the radius of the sphere, m is the mass of the sphere, and v is the translational velocity.

As the sphere falls toward the ground, the only force acting on it is the downward pull attributable to gravity. Since acceleration due to gravity is constant for all objects, the time it takes for any object, initially at rest, to fall a specific distance will be the same. The distance any falling object travels in a given amount of time can be determined using Equation 6.
{13786_Background_Equation_6}
where

H is the height (sometimes referred to as Δy),
g is the acceleration due to gravity,
t is the time.

Rearranging Equation 6 to solve for t produces Equation 7, which can be used to calculate the time it will take for any object to fall a certain distance.
{13786_Background_Equation_7}
The time calculated in Equation 7 is the total time it takes for the sphere to fall from the edge of the tabletop to the floor. Since no horizontal force acts on the sphere, the horizontal speed is constant and the distance the sphere travels horizontally can be determined by multiplying the horizontal speed by the total flight time of the sphere (Equations 8 and 9).
{13786_Background_Equation_8}
where

D is the horizontal distance (sometimes referred to as Δx),
vx is the horizontal speed,
t is the time.

Substituting Equation 7 into Equation 8 yields:
{13786_Background_Equation_9}
The initial horizontal speed can be evaluated by rearranging Equation 9 to solve for vx.
{13786_Background_Equation_10}
Equation 10 may be used to calculate the experimental speed of the sphere as it leaves the tabletop. Determination of the sphere’s theoretical, horizontal travel distance and velocity requires manipulation of the mathematical equations associated with the conservation of energy principle.

### Experiment Overview

In this advanced guided-inquiry experiment, kinematics will be combined with the law of conservation of mass to predict the horizontal motion of a steel sphere. In the Introductory Activity, a steel sphere will be released from a predetermined height on a ramp and allowed to roll off the table. The distance the sphere travels will be measured. The Guided-Inquiry Design and Procedure section provides leading questions to consider factors that may affect how far the sphere can travel. As an optional extension challenge, you may calculate the height that the sphere would need to be released from in order to land in a cup that is a certain distance away from the table.

### Materials

Clamp holder
Inclined plane, with hexagonal screws
Metal sheet, 4" x 4"
Meter stick
Pencil or chalk
Printer paper, white, 3–4 sheets
Protractor
Scissors
Sphere, steel, ¾" dia.
Support rod
Support stand
Textbooks, 3–4 (optional)
Transparent tape
Washer

### Prelab Questions

1. A 195-kg sphere dropped from 135 m above the ground requires how much time to hit the ground?
2. A sphere rolling down a hill has two types of kinetic energy. Name them and write mathematical equations to quantify them.
3. Use Equations 1, 2, 3, 4 and 5 to solve for the velocity, v, of a sphere rolling down a ramp from a height H and of mass m.
4. Draw a very general graph that depicts the change in a sphere’s potential energy and kinetic energy as it rolls down a hill. Use two lines, one to represent the sphere’s potential energy and another to represent the sphere’s kinetic energy. The y-axis can be labelled “Energy” and the x-axis “Height.”
5. Why does a small avalanche of rocks typically travel  its starting height into an adjacent valley whereas a large avalanche travels significantly farther?

### Safety Precautions

Be sure to quickly retrieve the sphere once it hits the floor. Wear safety glasses. Please follow all laboratory safety guidelines.

### Procedure

Introductory Activity

1. Obtain a 4" x 4" metal sheet.
2. Set up the inclined plane as shown in Figure 1. The angle of the incline should be between 15° and 45°. Position the bottom of the inclined plane approximately 10 cm from the edge of the table. Make sure there is at least 2–3 meters of open space on the floor around the edge of the tabletop so the steel sphere will have enough space to launch off the end of the table.
{13786_Procedure_Figure_1}
3. Use transparent tape to tape the edge of the 4" x 4" metal sheet to the bottom end of the inclined plane.
4. Bend the metal sheet slightly to form a smooth transition curve between the end of the inclined plane and the tabletop surface. The free end of this sheet should be within 1–2 cm of the tabletop’s edge. When the metal sheet forms a smooth curve, tape the other end of the sheet to the tabletop using transparent tape. Make sure the tape is flat and smooth at both ends of the metal sheet (see Figure 2).
{13786_Procedure_Figure_2}
5. Practice rolling the sphere down the inclined plane and off the edge of the tabletop. Make sure the sphere does not hit the tabletop hard as this may cause the sphere to bounce upward. Adjust the curve in the sheet until the sphere makes a smooth transition from the inclined plane to the tabletop. If the metal appears to bend down as the sphere rolls over it, small amounts of crumpled paper can be used under the sheet to add stability.
6. Cut 1.5–2 m of fishing line (depending on table height). Tie one end of the fishing line to a washer to create a plumb bob.
7. Hang the plumb bob from the end of the tabletop where the inclined plane will launch the steel sphere.
8. Place a strip of transparent tape, or use erasable chalk or pencil, to mark the spot on the floor directly below the hanging plumb bob.
9. Use a meter stick to measure the height of the edge of the tabletop from the floor. Make sure to measure from the mark made below the hanging plumb bob.
10. Use a pencil to lightly mark on the inclined plane approximately 10 cm from the high end of the inclined plane. This will be the release position of the sphere.
11. Use a ruler or meter stick to measure the height of this release point above the tabletop.
12. Use a protractor to measure the angle the inclined plane makes with respect to the tabletop.
13. When performing the experiments, one lab partner will place the steel sphere on the release line marked on the inclined plane and then release it. The other lab partner needs to be in a position to stop the sphere after it launches off the end of the tabletop and hits the floor.
14. Place the center of the steel sphere at the release line. (One person should be ready to stop the sphere after the first bounce.) Gently release the sphere, making sure not to give it any additional push. The sphere should roll straight down the inclined plane. Take note of the general area on the floor where the sphere lands.
15. Securely tape 3 or 4 sheets of white paper along the horizontal path that the launched sphere followed during the test run in step 14. Place a sheet or two beyond the point where the sphere hit during the practice run.
16. Repeat step 14 six or more times. Release the sphere from the same release height for each trial. The sphere should leave a dark mark on the white paper where it lands. Between each trial, use a pencil to circle the mark on the sheet of paper and label it with the trial number.
17. Use a meter stick to measure the horizontal distance between the plumb bob mark and the initial sphere marks on the paper for each trial. Record these distances in the data table. Note: For each trial, measure the first sphere mark only—the mark closest to the inclined plane. Do not measure additional marks left by the bouncing sphere.
Analyze the Results
Organize a data table for the values of the ramp angle, release height and distance the sphere traveled. Calculate the horizontal velocity of the sphere when it contacted the ground. Use the formula for calculating velocity derived in Prelab Question 3 to calculate the theoretical horizontal velocity. Determine the percent error of the experimental horizontal velocity for each trial.

Guided-Inquiry Design and Procedure
1. In the Introductory Activity, the sphere was assumed to be rolling. Would the theoretical velocity of the sphere increase or decrease if the sphere slid down the ramp with no friction? Derive a formula solving for velocity to support your claim.
2. Examine the formulas derived in Prelab Question 3 and the previous question. For a sphere and box moving down a ramp, what variable(s) can be manipulated to increase the horizontal velocity? Discuss practical methods by which the variable(s) could be changed.
3. Consider two spheres, one of mass m and the other 2m, held 1 meter above the ground.
1. If the spheres are released simultaneously, which sphere will hit the ground first?
2. Which sphere is traveling faster when it hits the ground?
4. Two spheres are held in place at the tops of two ramps (see Figure 3).
{13786_Procedure_Figure_3}
1. Which sphere has greater potential energy at the top of the ramp?
2. Which sphere is traveling faster at the bottom of the ramp when released?
5. Consider the following variables that may affect the horizontal travel distance of the steel sphere roll down an inclined plane: mass, release height and angle of the inclined plane with respect to the table.
1. Predict the effects the three variables will have on the sphere’s horizontal travel distance.
2. Design an experiment to test your predictions.
3. Determine the independent and dependent variable for each experiment, as well as any variables that should remain constant.
Analyze the Results
Organize a data table for the values of the ramp angle, release height and distance the sphere traveled. Calculate the horizontal velocity of the sphere when it contacted the ground. Use the formula for calculating velocity derived in Prelab Question 3 to calculate the theoretical horizontal velocity. Determine the percent error of the experimental horizontal velocity for each trial. Use the theoretical velocity value to determine the theoretical horizontal travel distance. Determine the percent error of the horizontal travel distance for each trial.

### Student Worksheet PDF

13786_Student1.pdf

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