Materials Included In Kit

Additional Materials Required

Safety Precautions

The materials in this lab are considered safe. Projectiles may be inadvertently launched during this activity. Wear safety glasses. 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 Design and Procedure 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.
• Examine the effects of friction on acceleration. Notice that as the hanging mass decreases in value and the system mass increases, the calculated value of g increasingly deviates from the accepted value, 9.8 m/s². This is the case because the cart, when very heavy, presses against the tabletop with more force and friction becomes increasingly significant.
• If motion detectors and air tracks are available they can be used to more accurately measure travel times and minimize the effects of friction, respectively.
• 150 washers are included in this kit. If students find they need additional objects to weigh down the cart, most anything can be used so long as it fits in the cart and can be massed. As an advanced experiment, ask students to determine experimentally the mass of various objects, such as car keys.
• The sample data for the Guided-Inquiry portion of the experiment only provides the average time values for 10 trials. Data analysis of the 10 trials separately yielded values close to those based solely on the average time value. Students may conduct analysis on each trial individually. For formatting and space purposes, we chose to omit the individual trials.

Teacher Tips

• Free-body diagrams are pervasive and important. Students will benefit from extensive practice in this area.
• This investigation provides a means of incorporating linearization techniques into the curriculum.
• Incorporate standard deviation calculations and percent error calculations if time permits.
• The Flinn Scientific Atwood’s Machine Demonstration Kit (Catalog No. AP6495) provides a standard Atwood’s Machine and procedure designed to determine the effects of the pulleys’ mass and friction on acceleration.

Further Extensions

Opportunities for Inquiry

This investigation may be carried out on an inclined plane, as opposed to a horizontal table. A photogate timer or motion detector may be used to more precisely measure the cart’s travel times. In addition, students can be asked to investigate more formally friction’s effects on the system’s acceleration.

Alignment to the Curriculum for AP^® Physics 1 

Enduring Understandings and Essential Knowledge
Objects and systems have properties of inertial mass and gravitational mass that are experimentally verified to be the same and that satisfy conservation principles. (1C)
1C1: Inertial mass is the property of an object or a system that determines how its motion changes when it interacts with other objects or systems.

A gravitational field is caused by an object with mass. (2B)
2B1: A gravitational field g at the location of an object with mass m causes a gravitational force of magnitude mg to be exerted on the object in the direction of the field.

Classically, the acceleration of an object interacting with other objects can be predicted by using a = ΣF/m. (3B)
3B1: If an object of interest interacts with several other objects, the net force is the vector sum of the individual forces.
3B2: Free-body diagrams are useful tools for visualizing forces being exerted on a single object and writing the equations that represent a physical situation.

Learning Objectives
1C1.1 The student is able to design an experiment for collecting data to determine the relationship between the net force exerted on an object, its inertial mass, and its acceleration.
2B1.1 The student is able to apply F = mg to calculate the gravitational force on an object with mass m in a gravitational field of strength g in the context of the effects of a net force on objects and systems.
3B1.1: The students is able to predict the motion of an object subject to forces exerted by several objects using an application of Newton’s second law in a variety of physical situations with acceleration in one dimension.
3B1.2: The student is able to design a plan to collect and analyze data for motion (static, constant, or accelerating) from force measurements and carry out an analysis to determine the relationship between the net force and the vector sumo of the individual forces.
3B1.3: The student is able to reexpress a free-body diagram representation into a mathematical representation and solve the mathematical representation for the acceleration of the object.
3B2.1: The student is able to create and use free-body diagrams to analyze physical situations to solve problems with motion qualitatively and quantitatively.

Science Practices
1.1: The student can create representations and models of natural or man-made phenomena and systems in the domain.
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.
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.

Answers to Prelab Questions

1. Sketch and label the following free-body diagram, filling in all force vector arrows.
   {13783_PreLabAnswers_Figure_1}
2. If a net force of 34.3 N is required to move the system in the figure above 2.3 m in 1.3 seconds, calculate system’s mass and the mass of the cart portion of the system.
   a = 2d/t²
   a = (2 x 2.3 m)/(1.3 s)²
   a = 2.72 m/s²
   F = ma
   34.3 N = m x 2.72 m/s²
   m = 12.6 kg
   mg = 34.3 N
   m_hanging x 9.8 m/s² = 34.3 N
   m_hanging = 3.5 kg
   m_cart = m_system – m_hanging
   m_cart = 12.6 kg – 3.5 kg = 9.1 kg.
   
   
3. According to the equation a = bc, what type of algebraic relationship (direct or inverse) exists between the quantities a and c? Draw a qualitative graph to describe the relationship between a and c. Assume b is a constant.

   A direct relationship exists between a and c. That is, as c increases, a must also increase.

   {13783_PreLabAnswers_Figure_2}
4. According to the graph, what type of algebraic relationship (direct or inverse) exists between b and c?

   An inverse relationship exists between b and c. That is, as b increases, c decreases. Or as b increases, 1/c increases.

5. Describe the steps necessary to convert the parabolic b vs. c curve to a linear graph. That is, one in which the relationship between the two variables gives a straight line.

   Linearization is the process by which mathematical operations such as multiplication or division, are applied to a data set so that a graph of said data set vs. another quantity will give a straight line of near constant slope, as opposed to a more parabolic curve. The steps necessary to linearize a graph can be determined by looking at the mathematical equation or law that governs the two quantities. In this case, a = bc denotes an inverse relationship between b and c. Isolating b on one side of the equation gives b = a/c or b = (1/c) x a. In colloquial terms, then, b is equal to 1/c times a proportionality constant, a. So, as b increases, c decreases. Thus, to linearize a graph of b vs. c, each c value must be divided into 1. A plot of b vs. the resulting values, the 1/c values, will give a straight line.

6. Regardless of the distance the system shown in Question 1 traverses, why must its measured acceleration never exceed g, 9.8 m/s²?

   Assuming that random error attributable to manual timing is negligible, frictional forces will prevent the system from reaching free-fall acceleration, which assumes friction is zero.

7. What is the justification for assuming the modified Atwood’s Machine moves as if in free fall?

   This assumption is predicated on the fact that the force attributable to the downward pull of gravity, mg, is the only unbalanced force acting on the system. That is, it is the only force that does not have an equal force in the opposite direction. For example, the tension in the string does not affect the system’s acceleration because it is a balanced, internal force. The direction of the tension acting on the hanging mass is opposite the tension acting on the cart and the tensions are equal in magnitude.

Sample Data

Introductory Activity

For Trial 1:

a = 2d/t² = (2 x 0.6 m)/ (2.4 s)² = 0.208 m/s²
F = m_sa = 0.231 kg x 0.208 m/s² = 0.0481 N

Theoretical Acceleration:

F_net = F_g = m_h x g
m_ta = m_h x g
a = (m_h /m_t ) x g = (0.0083 kg/0.231 kg) x 9.81 m/s²
a = 0.352 m/s²

Precent Error:

% Error = (|Experimental – Theoretical|/Theoretical) x 100
% Error = (|0.198 m/s² – 0.352 m/s²|/0.352 m/s²) x 100
% Error = 43.7%

Theoretical Force:

F_net = F_g = m_h x g
F_g = 0.0083 kg x 9.81 m/s² = 0.081 N
% Error = (|0.0458 N – 0.081 N|/0.081 N ) x 100
% Error = 43.7%

Guided-Inquiry Design and Procedure

Relationship Between Force and Acceleration

1. In the cart--mass system, what force is causing the system to move? Write an equation for this force.

   The force of gravity on the hanging mass is the only unbalanced force on the cart--mass system and causes the system to accelerate.
   F_n = F_g = m_h x g
   

2. Use Newton’s second law to write an equation for the motion of the system.

   F_n = ma = F_g
   m_Ta = m_h x g
   (m_h + m_c) a = m_h x g

3. How could the force identified in Question 1 be increased or decreased?

   The force due to gravity can be increased or decreased by altering how much mass is in the hanging bag. Adding more washers will increase the force on the system; removing washers will decrease the force.

4. If more mass (washers) were added to the bag, would other variables of the system also be changed? Identify all variables.

   If washers were added to the bag, the total mass of the system would also increase. The total mass of the system is the hanging mass and the mass of the cart.

5. Explain how the total mass of the system can remain constant while changing the mass of the hanging bag.

   In order to maintain a constant mass of the system, any mass added to the bag must be removed from the cart. The total mass stays the same because the washers are simply moved from one location to another. But the applied force will change because the hanging mass is changing.

6. Why is it necessary to keep the total mass of the system constant when determining the relationship between the acceleration of the system and the net force applied to the system?

   In any experiment, it is necessary to change only one variable while keeping any others constant. By studying one variable, it is possible to learn the relationship between two variables. If the total mass of the system were changing as well as the applied force, it would be difficult to determine the effect on acceleration because the individual effects are unknown.

7. Write a detailed, step-by-step procedure to determine the relationship between the applied force and acceleration of the cart--mass system. Identify the independent and dependent variables, and any variables that should remain constant.

   The independent variable is the applied force by adding washers to the bag. The dependent variable is the time it takes for the hanging mass to fall. The total mass of the system is to remain constant, as well as the distance the cart travels.
   
   Mass the hanging weight and record this value. Add several washers to the cart and mass the cart. Record this value. Ready the system following the procedure in the Introductory Activity. Record the fall times for 10 trials. Remove washers from the cart and place them in the hanging bag. Record the mass of the hanging bag. Ready the system and record the fall times for 10 trials. Repeat transferring washers from the cart to the bag for a total of four different hanging masses.
   See sample data and graph on next page.

Data and Graph

Acceleration and force are directly related, which follows the formula for Newton’s second law: F = ma.

Relationship Between Acceleration and Mass

1. Using Newton’s second law, F = ma, what variable should be held constant in order to determine the relationship between mass and acceleration?

   The applied force on the cart--mass system should remain constant.

2. How could the variable identified in the previous question be held constant?

   The applied force is due to the force of gravity on the hanging mass. In order to keep the applied force constant, the hanging mass cannot be changed.

3. What variable could be changed in order to study the relationship between acceleration and mass? How can that variable be altered?

   The mass of the cart can be changed in order to study the relationship between mass and acceleration. Adding more mass to the cart will increase the total mass of the system while keeping the applied force (hanging mass) constant.

4. Write a detailed, step-by-step procedure to determine the relationship between the mass of the system and acceleration of the cart--mass system. Identify the independent and dependent variables, and any variables that should remain constant.

   The independent variable is the mass of the system by adding washers to the cart. The dependent variable is the time it takes for the hanging mass to fall. The hanging mass is to remain constant, as well as the distance the cart travels. Mass the hanging weight and record this value. Mass the cart and record this value. Ready the system following the procedure in the Introductory Activity. Record the fall times for 10 trials. Mass additional washer(s) and record the total mass of the cart. Place the washer(s) in the cart and ready the system. Record the fall times for 10 trials. Repeat adding additional washers for a total of four different system masses. See sample data and graphs.

Data and Graph

Answers to Questions

Answers to Review Questions for AP^® Physics 1

1. Draw a free-body diagram with all relevant force vector arrows to represent the two masses connected via a string on an inclined plane shown.
   {13783_Answers_Figure_2}
2. Calculate the acceleration of the two-mass system if m₁ = 131.2 g, m₂ = 202.5 g and θ = 32.3°.

   a_m1 = T – m₁ x g x sinθ
   a_m2 = m₂ x g – T

   {13783_Answers_Equation_1}

   a = 3.89 m/s²

3. A pug, a dachshund and a Jack Russell terrier are engaged in a two-dimensional tug-of-war for a three-pronged chew toy. None of the dogs is able to rip the toy away from the others. Essentially, it does not move. According to the diagram shown below, the Jack Russell terrier pulls with a force of 160 N and the dachshund pulls with a force of 205 N. Necessary angles are given in the diagram. Determine the force of the pug’s pull.
   {13783_Answers_Figure_3}

Introduction

The fact that light objects require less force to accelerate than heavy objects is intuitive: pulling a heavy box is more difficult that pulling a light box. Newton’s second law summarizes these observations. What is the relationship between force, acceleration and mass? Test your intuition and discover the relationships behind Newton’s second law.

Concepts

• Newton’s second law
• Acceleration
• F = ma
• Force
• Inertia

Background

Newton’s second law of motion states that an object or system’s acceleration is equal to the net force applied to the object or system divided by the object or system’s mass. Thus, if the same force were applied to two objects of different mass, the less massive object would experience a greater acceleration.

Multiple forces acting in complex systems may be added together to find the resulting acceleration. Acceleration along a given axis can only be the result of all the forces in that axis. Forces in the y-direction will not affect acceleration in the x-direction, and vice versa. For example, consider a game of tug-of-war between two teams pulling on a rope with great force. The whole system will often experience very little acceleration because the two forces are acting in opposition along the same axis.

Free body diagrams are often used to describe the forces which act on objects. For example, the Atwood’s machine in Figure 1 consists of two masses connected with a string suspended on a two-pulley system. The machine was developed in the late 18th century to indirectly measure acceleration due to gravity. If the masses are equal, the net force acting on both masses in the y-direction is 0 N and the system is static. However, if one of the masses is heavier than the other, the lighter mass will accelerate up and the heavier mass down owing to a net nonzero force in the y-direction. In this lab, a modified Atwood’s machine, shown in Figure 2, will be used to quantitatively explore the mathematical relationships associated with Newton’s second law. The acceleration of a hanging weight system will be determined by recording the time it takes the weight to travel a measured distance. The acceleration can be calculated by assuming free-fall conditions and assigning an initial velocity of 0 m/s to the system. As a result, the traditional equation (Equation 2), which describes the distance an object in free fall travels, reduces to Equation 3. where

d is the distance the object moves (in m)
a is the acceleration of the object (in m/s²)
t is the time (in seconds)
v_i is the initial velocity of the object

Experiment Overview

The purpose of this lab is to design experiments to verify that a system’s acceleration increases as the net force applied to the system increases and that a system’s acceleration decreases as its mass increases, provided the net force is constant. The investigation begins with an introductory activity to measure the acceleration of a modified Atwood’s machine of prescribed mass over a prescribed distance. The procedure provides a model for guided-inquiry design of experiments to verify the relationships attributable to Newton’s second law using graphs. Conducting the experiments on an inclined plane, as opposed to a horizontal surface, provides additional opportunities for inquiry.

Materials

Prelab Questions

1. Sketch and label the following free-body diagram, filling in all force vector arrows.
   {13783_PreLab_Figure_1}
2. If a net force of 34.3 N is required to move the system in the figure to the right 2.3 m in 1.3 seconds, calculate the system’s mass and the mass of the cart portion of the system.
3. According to the equation a = bc, what type of algebraic relationship (direct or inverse) exists between the quantities a and c? Draw a qualitative graph to describe the relationship between a and c. Assume b is a constant.
4. According to the following graph, what type of algebraic relationship (direct or inverse) exists between b and c?
   {13783_PreLab_Figure_2}
5. Describe the steps necessary to convert the parabolic b vs. c curve to a linear graph. That is, one in which the relationship between the two variables gives a straight line.
6. Regardless of the distance the system shown in Question 1 traverses, why must its measured acceleration never exceed g, 9.8 m/s²?
7. What is the justification for assuming the modified Atwood’s machine used in this investigation engages in free fall?

Safety Precautions

Projectiles may be inadvertently launched during this activity. Wear safety glasses. Please follow all normal laboratory safety guidelines.

Procedure

Introductory Activity

1. Measure and mark distances of 0.1 m and 1.1 m from the edge of a tabletop with a meter stick or ruler and masking tape. The cart will traverse this distance. The distance may be varied to accommodate the height of available tables. Low tables will require shorter distances.
2. Secure a table pulley to the table’s edge.
3. Attach a plastic bag to one end of a string, approximately 130-cm long, using a looping knot (see Figure 3). Attach the other end of the string to the cart.
   {13783_Procedure_Figure_3}
4. Weigh one washer and place it inside of the plastic bag.
5. Add four washers to the cart and weigh.
6. Place the cart at the 1.1 m tape mark, farthest from the table pulley. Hold the cart in place and lay the string over the top of the pulley.
7. Release the cart and use a stopwatch to time its travel between the tape marks.
8. Perform steps 6–7 ten times and calculate the cart’s average acceleration.

Analyze the Results

Display the collected mass and time data in an appropriate data table. Calculate the following values for the cart:

• Acceleration for each trial
• Average acceleration
• Force from the hanging mass for each trial
• Average force from the hanging mass

Calculate the theoretical acceleration of the cart and the theoretical force of the hanging mass. Determine the percent error between the experimental average values of acceleration and force and the theoretical values. What might be done to minimize the sources of error in this experiment?

Guided-Inquiry Design and Procedure

Relationship between Force and Acceleration

1. In the cart--mass system, what force is causing the system to move? Write an equation for this force.
2. Use Newton’s second law to write an equation for the motion of the system.
3. How could the force identified in Question 1 be increased or decreased?
4. If more mass (washers) were added to the bag, would other variables of the system also be changed? Identify all variables.
5. Explain how the total mass of the system can remain constant while changing the mass of the hanging bag.
6. Why is it necessary to keep the total mass of the system constant when determining the relationship between the acceleration of the system and the net force applied to the system?
7. Write a detailed, step-by-step procedure to determine the relationship between the applied force and acceleration of the cart--mass system. Identify the independent and dependent variables, and any variables that should remain constant.

Analyze the Results

• Display the collected mass and time data in an appropriate data table. Calculate the following values for the cart: average acceleration and average force from the hanging mass.
• Construct a graph of “Average Acceleration vs. Average Force” and identify the relationship between the variables.
• Calculate the theoretical acceleration of the cart and the theoretical force of the hanging mass. Determine the percent error between the experimental average values of acceleration and force and the theoretical values. What might be done to minimize the sources of error in this experiment?

Relationship between Acceleration and Mass

1. Using Newton’s second law, F = ma, what variable should be held constant in order to determine the relationship between mass and acceleration?
2. How could the variable identified in the previous question be held constant?
3. What variable could be changed in order to study the relationship between acceleration and mass? How can that variable be altered?
4. Write a detailed, step-by-step procedure to determine the relationship between the mass of the system and acceleration of the cart--mass system. Identify the independent and dependent variables, and any variables that should remain constant.

Analyze the Results

• Display the collected mass and time data in an appropriate data table. Calculate the following values for the cart: average acceleration and average force from the hanging mass.
• Construct a graph of “Acceleration vs. Mass of System” and perform appropriate mathematical operations to achieve a linear relationship between the variables.
• Calculate the theoretical acceleration of the cart and the theoretical force of the hanging mass. Determine the percent error between the experimental average values of acceleration and force and the theoretical values. What might be done to minimize the sources of error in this experiment?

Student Worksheet PDF

13783_Student1.pdf