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Briefing Doc: 🧑‍🔬Modified Atwood Machine Experiment

This briefing doc summarizes key information from the provided source, focusing on understanding the principles and applications of a Modified Atwood Machine experiment for physics education.

Experiment Overview:

The Modified Atwood Machine, invented by George Atwood in 1784, serves as a classic physics experiment to demonstrate and verify:

Newton's Second Law of Motion (F = ma): Investigating the relationship between force, mass, and acceleration.
Gravitational Acceleration (g): Determining the Earth's gravitational acceleration through experimental data analysis.

Experimental Setup:

The basic setup involves:

Two masses (m - hanging mass and M + Madd - mass of cart plus additional mass) connected by a string.
A frictionless pulley over which the string passes.

The experiment can be conducted using real equipment or simulated through a virtual app:

Real: Pulley, string, masses, meter rule, stopwatch, and potentially a photogate and datalogger for more precise measurements.
Virtual: Android/iOS app simulating the Atwood machine.

Key Concepts:

Free Body Diagrams: Analyzing forces acting on individual components and the system as a whole.
Uniform Acceleration: The system experiences uniform acceleration when the masses are unequal (M + Madd ≠ m).
Friction: Understanding the role of friction in the system and how it affects acceleration.
Error Analysis: Identifying potential sources of error and their impact on experimental results.

Procedure:

Setup: Assemble the Atwood machine, ensuring the pulley is frictionless and the string is inextensible.
Measurements: Vary the mass difference (Δm) by transferring small masses between the hanging mass and the cart. Measure the time taken for a known distance for each Δm.

Data Analysis:Plot acceleration versus Δm.
Determine the experimental value for 'g' from the graph's slope.
Analyze the graph's characteristics, including its intercept and the relationship it depicts.

Error Analysis: Calculate the percentage difference between the experimental and accepted values of 'g'. Discuss potential sources of error.

Equations:

The fundamental equation governing the Atwood machine is derived from Newton's Second Law:

Acceleration: ( a = \frac{mg - \mu (M+M_{add})g}{M+M_{add}+m} ) where:
m: hanging mass
g: gravitational acceleration
μ: coefficient of friction
M: mass of the cart
Madd: mass added to the cart

Safety Considerations:

Limit the mass difference to prevent rapid acceleration and potential detachment of masses.
Ensure a clear area beneath the hanging mass to avoid accidents.

Educational Goals:

This experiment offers rich learning opportunities, encouraging students to:

Investigate Relationships: Explore the relationship between mass, acceleration, and force.
Apply Concepts: Use free body diagrams and Newton's laws to analyze the system.
Analyze Data: Interpret graphical data to extract meaningful conclusions.
Develop Scientific Skills: Practice experimental design, data collection, analysis, and error assessment.

Extension Activities:

Modified Atwood Machine: Further exploration of Newton's Second Law by varying the hanging mass.
Friction Analysis: Investigate the significance of friction and determine its value experimentally.

Important Quotes:

"An Atwood’s machine consists of two masses tied together using a massless non-elastic string hung over a massless frictionless pulley."
"Using data acquired from a simple Atwood’s machine, graphically determine the earth’s gravitational acceleration."
"This lab pertains to a system called an Atwood machine that can be modeled with a free body diagram if we treat the entire system as one accelerating object."

Conclusion:

The Modified Atwood Machine experiment provides a valuable hands-on experience for students to delve into the core concepts of dynamics and gravitation. By carefully conducting the experiment, analyzing data, and considering potential errors, students can gain a deeper understanding of Newton's laws and their practical applications.

Modified Atwood Machine Study Guide

Short Answer Questions

Describe the basic components of an Atwood machine and its primary function.
Explain the concept of free body diagrams and their relevance to the Atwood machine experiment.
What is the significance of keeping the total mass of the system constant in an Atwood machine experiment?
Derive the equation for acceleration (a) in an Atwood machine system, considering mass (m), gravitational acceleration (g), and coefficient of friction (μ).
Why is the tension in the string not considered in the equation for acceleration in an ideal Atwood machine?
What are some potential sources of error in an Atwood machine experiment and how might they affect the results?
Explain how the difference in mass (Δm) between the two hanging masses influences the acceleration of the system.
Describe how you would graphically determine the Earth's gravitational acceleration (g) using data from an Atwood machine experiment.
How does the mass and friction of the pulley affect the calculated value of 'g' in a real Atwood machine?
Discuss an extension to the Atwood machine experiment that involves varying the acceleration by changing the hanging mass. What insights can be gained from this modification?

Short Answer Answer Key

An Atwood machine consists of two masses connected by a string that passes over a pulley. Its primary function is to demonstrate and verify the mechanical laws of motion with constant acceleration, specifically Newton's Second Law.
Free body diagrams visually represent all the forces acting on an object or system. In the Atwood machine, they help analyze the forces on each individual mass and the system as a whole, allowing for the derivation of the acceleration equation.
Keeping the total mass constant isolates the effect of the mass difference (Δm) on the acceleration, allowing for a clearer analysis of the relationship between these variables.
The equation for acceleration is a = (mg - μ(M+Madd)g) / (M+Madd+m), where m is the hanging mass, g is gravitational acceleration, μ is the coefficient of friction, M is the mass of the cart, and Madd is the mass added to the cart. This equation is derived by applying Newton's Second Law to both masses and solving for 'a'.
In an ideal Atwood machine, the string is considered massless and inextensible, and the pulley is massless and frictionless. This means the tension in the string is uniform throughout and cancels out when considering the net force on the system.
Potential sources of error include friction in the pulley, the mass of the string, air resistance, and inaccurate measurements. These errors can lead to variations in the calculated acceleration and gravitational acceleration (g).
A larger difference in mass (Δm) between the two hanging masses results in a greater net force acting on the system, leading to higher acceleration.
By plotting the acceleration of the system against Δm and obtaining a linear graph, the slope of the line represents the Earth's gravitational acceleration (g).
The mass and friction of the pulley contribute to the inertia of the system. This can lead to slightly lower measured acceleration values and therefore, an erroneously low value for 'g'.
By varying the hanging mass and measuring the resulting acceleration, you can verify Newton's Second Law, which states that force is proportional to mass and acceleration (F=ma). This modification allows for a more comprehensive understanding of the relationship between force, mass, and acceleration.

Essay Questions

Analyze the forces acting on each mass and the entire system in an Atwood machine using free body diagrams. Explain how Newton's Second Law is applied to derive the equation for acceleration.
Discuss the assumptions made in an ideal Atwood machine model and how these assumptions differ from a real-world scenario. Analyze the implications of these differences on the experimental results.
Explain the concept of conservation of energy in the context of an Atwood machine. Analyze how the potential and kinetic energies of the masses change throughout the motion.
Describe a detailed procedure for conducting an Atwood machine experiment to determine the Earth's gravitational acceleration (g). Discuss the necessary measurements, data analysis techniques, and potential sources of error.
Compare and contrast the Atwood machine with other experimental setups used to demonstrate and verify Newton's Laws of Motion. Discuss the advantages and limitations of each method in illustrating specific concepts.

Glossary of Key Terms

TermDefinitionAtwood MachineA laboratory apparatus consisting of two masses connected by a string that passes over a pulley, used to demonstrate and verify the mechanical laws of motion with constant acceleration.Free Body Diagram (FBD)A visual representation of all the forces acting on an object or system, isolated from its surroundings.Newton's Second LawThe fundamental principle stating that the net force acting on an object is equal to the product of its mass and acceleration (F=ma).Acceleration (a)The rate of change of velocity over time. In the Atwood machine, it refers to the rate at which the masses move due to the net force acting on the system.Gravitational Acceleration (g)The acceleration experienced by an object due to the Earth's gravitational field, approximately 9.8 m/s².Tension (T)The force transmitted through a string, rope, or cable when it is pulled taut by forces acting from opposite ends. In an ideal Atwood machine, the tension is considered uniform throughout the string.Coefficient of Friction (μ)A dimensionless quantity that represents the ratio of the force of friction between two surfaces to the normal force pressing them together. It determines the resistance to motion between the surfaces in contact.Mass Difference (Δm)The difference in mass between the two hanging masses in an Atwood machine. This difference is the driving force behind the acceleration of the system.Error AnalysisThe process of identifying, quantifying, and evaluating potential sources of error in an experiment, and their impact on the results.Conservation of EnergyThe fundamental principle stating that energy cannot be created or destroyed, only transferred or transformed from one form to another. In an Atwood machine, the total mechanical energy of the system remains constant.

Worksheet 📖 ejss_model_AtwoodMachine2wee Newton's second Law by Nawal Nayfeh, Physics instructor at university of sharjah, UAE.docx

Apps

https://play.google.com/store/apps/details?id=com.ionicframework.atwoodmachineapp294376&hl=en

Problem Statement/Description:

An Atwood’s machine consists of two masses tied together using a massless non-elastic string hung over a massless frictionless pulley. Using data acquired from a simple Atwood’s machine, graphically determine the earth’s gravitational acceleration. For this experiment, we will keep the total mass of the system (M + m) constant.

Equipment depending on resources available and degree of blending real and virtual, best to do Real and then Virtual later.

Option 2: Real: stopwatch, pulley, string, meter- rule, 2 x slotted masses (350g + 2 x 10g + 2 x 5g), retort stand, mat to absorb the impact of falling masses
Option 1: Computer/ Handphone App: https://play.google.com/store/apps/details?id=com.ionicframework.atwoodmachineapp294376&hl=en
Option 3: ICT: stopwatch, pulley, string, meter- rule, 2 x slotted masses (350g + 2 x 10g + 2 x 5g), retort stand, mat to absorb the impact of falling masses and including a photogate (smart pulley) and datalogger

Safety Consideration

If the difference between the masses on each hanger is too great, the masses may fly off the pulley and hanger when released. For this experiment, use values of M - m (Δm) in the range of 10g – 60g.
Make sure that the coast is clear before releasing the mass and do not squat right under the masses as there is a chance that the string holding the masses might break. Place the mat directly below the falling mass m2 to absorb the impact of the fall, while also making sure that mass M does not hit the pulley as m hits the mat.

Related Concepts

Newton’s 2nd law, acceleration of free fall, the 1D motion of uniformly accelerated objects, the principle of conservation of energy

Pre-Lab Discussion

You may be familiar with drawing free body diagrams and forces acting on common objects like blocks sliding down inclines, objects falling through the air, and other projectiles we often treat as point masses. But what about a free body diagram for an awkward shape? This lab pertains to a system called an Atwood machine that can be modeled with a free body diagram if we treat the entire system as one accelerating object.
Show that by applying Newton’s second law for the entire system, we obtain Eq. 1:
\( a = \frac{mg - \mu (M+M_{add})g}{M+M_{add}+m} \)

m is the hanging mass
g is the gravitational acceleration
μ is the coefficient of friction
M is the mass of the cart
M_add is the mass added to the cart

Why is the tension in the string not a factor in the above equation?

Note that the torque required to cause the pulley to begin spinning is assumed to be negligible, as is the mass, friction, and rotational inertia of the pulley.

Design and Conduct the Experiment

What variables from Eq. 1 can be measured directly using equipment from the materials list above? What piece of equipment will you use to measure each variable?
What variables will you change to get at least 5 different data points for the acceleration of the system? What variables must remain constant in order to get a consistent graph of acceleration?
If you move masses from the rising hanger to the falling hanger, will this change the total system mass? Explain why or why not.
What types of acceleration will you observe as the mass difference becomes greater between the two hangers?
Describe the potential sources of errors that could prevent you from gathering accurate consistent data.
Give a short description of the procedure you will use, identifying what measurements you will make, the points at which you will make these measurements, and how each piece of equipment will be used to make these measurements.

Sample Learning Goals

Investigate the relationship between the mass and the acceleration of the system

keep the external force (hanging mass) constant
add mass to the cart

Equations

The Atwood machine (or Atwood's the machine) was invented in 1784 by the English mathematician George Atwood as a laboratory experiment to verify the mechanical laws of motion with constant acceleration. Atwood's machine is a common classroom demonstration used to illustrate principles of classical mechanics.

The idea frictionless Atwood Machine consists of two objects of mass (Mass of cart M plus additional mass added M_add) and m (hanging mass), connected by an inextensible massless string over an ideal massless pulley. [1]

When M+M_add = m, the machine is in neutral equilibrium or net force is equal to zero, regardless of the position of the weights.

When M+Madd ≠ m the system of masses experience uniform acceleration.

We are able to derive an equation for the acceleration by using force analysis or free body diagram. If we consider a massless, inextensible string and an ideal frictionless pulley.

As a sign convention, we assume that a is positive when downward for the mass m and to the right for M+Madd

By considering free body diagram of M+Madd we get

F = ma

T - f = (M+Madd)a

where

T is the tension pointing towards the right on the cart and added mass system

f is the friction force on the cart and added mass system on the wheels (assume to be a single force pointing left, thus the minus sign)

a is the acceleration of the cart and added mass system

Similarly, by considering the free body diagram of the hanging mass

F = ma by continuous motion, downwards as positive,

mg - T = ma

by substitution,

T = (M+Madd)a +f =mg -ma

we get

\( a = \frac{mg - f}{M+M_{add}+m} \)

to get an equation of removing f, consider the forces of the cart and added the mass system in the vertical direction

F = ma

R - (M+Madd) g = 0

in addition, the frictional model is related by this equation

f ≤ μR

in motion, assume f = μR

therefore, the equation of acceleration can be simplified to

\( a = \frac{mg - \mu R}{M+M_{add}+m} \)

\( a = \frac{mg - \mu (M+M_{add})g}{M+M_{add}+m} \)

m is the hanging mass
g is the gravitational acceleration
μ is the coefficient of friction
M is the mass of the cart

For Teachers

This simulation is to recreate Walter Fendt's Newton's Second Law applet (http://www.walter-fendt.de/ph14e/n2law.htm). The purpose is to align with Modelling Instructions approach to formulate F = ma with the experiment conducted with motion sensor rather than the light gate.

example data set collected across varying m from 0.30 to 0.50 kg which results in increasing amount of curve in the parabolic of the model form, x = x0 + vo*t +0.5*a*t^2Modified Atwood Machine JavaScript HTML5 Applet Simulation Model by Tat Leong Lee and Loo Kang Wee

example data set collected across varying addM from 0 to 0.30 kg which results in decreasing amount of curve in the parabolic of the model form, x = x0 + vo*t +0.5*a*t^2
Modified Atwood Machine JavaScript HTML5 Applet Simulation Model by Tat Leong Lee and Loo Kang Wee

6.4 Dynamics: Map of key concepts and ideas covered at upper secondary level Motion of objects Key inquiry question:

Why do objects on Earth and in the Universe move the way they do?

1. Mass, weight, and density

• Mass is a measure of the amount of substance in a body. A body’s mass resists a change in the state of rest or motion of the body (inertia).

• The density of a substance is the mass per unit volume of the substance i.e. density = mass/volume (ρ = m/V). Density can be measured as the mass of 1 cm3 of any substance.

• Mass, a measure of the amount of substance in an object has magnitude only (scalar), whereas weight, which is the force acting on a body in a gravitational field, has both magnitude and direction (vector).

• Using Newton’s 2nd law, Force = mass x acceleration; Weight = mass x acceleration due to gravity (or gravitational field strength) (W = mg).

2. Newton’s laws of motion

• Newton’s Laws of motion may be applied to: describe the effect of balanced and unbalanced forces on a body; describe the ways in which a force may change the motion of a body, and identify action-reaction pairs acting on two interacting bodies.

• Newton’s 1st law states that an object remains at rest or continues with constant speed in a straight line when no resultant force acts on it (no resultant force means that all the forces acting on the object are balanced). • Newton’s 2nd law states that the resultant force on a body = the mass of the body x acceleration of the body (F = ma). The direction of the acceleration is the same as the direction of the resultant force acting on the body.

• Newton’s 3rd law states that the forces of two bodies on each other (action-reaction pair) are always equal and act along the same line in opposite directions. The two forces (action-reaction pair) are of the same type. Force always appears in a pair. The existence of a single force is impossible.

3. Free-body diagrams

• Free-body diagrams and vector graphical diagrams may be used to represent and analyze the forces acting on a body.

• A free-body diagram shows the forces acting on a body only, not the forces the body exerts on other bodies.

• The resultant force acting on the body can be found using graphical methods (parallelogram method or ‘head-to-tail’ method).

Students’ prior knowledge of Dynamics and Gravitational field

Primary level:

Students learn about mass (a measure of the amount of matter in a body) and volume (the amount of space that a body occupies) and the use of appropriate apparatus to measure these quantities (e.g. use of a lever balance, an electronic balance, a measuring cylinder, a syringe, and a measuring jug). However, the concept of density is not introduced although students do simple experiments to investigate the ability of objects of different materials (plastics, wood, rubber, and metals) to float/sink in water.

Students recognize that objects have weight because of the gravitational force between them and the Earth and that weight is different at different places and can be measured using a spring balance or a weighing scale.

Lower secondary level:

Students learn that:

• the density of a substance is the mass of the substance per unit volume and can be used to predict whether objects sink or float.

• gravity exists between any two objects (e.g. ball and Earth). The weight of an object depends on the force of gravity pulling on that object. Students’ common misconceptions and learning difficulties in Dynamics and Gravitational field

Newton’s first law:

Students often think that:

• a force is required to maintain an object in its motion;

• if there is no motion, there is no force acting;

• constant speed results from a constant force;

• friction (instead of inertia) causes objects to resist a change in its state of rest or motion.

Newton’s second law:

Students often think that:

• a larger velocity means a larger resultant force;

• acceleration implies increasing force;

• greater mass implies greater force;

• a force cannot move an object unless it is greater than the object’s weight;

• heavier objects fall faster than light objects.

Newton’s third law:

Students often think that force is a single physical quantity associated with a single object rather than as an interaction between two objects which must, therefore, exist as an action-reaction pair. They have difficulty in understanding that:

• two objects of greatly differing masses (e.g. Earth and us) exert forces of equal magnitude on each other;

• the normal force on an object and the weight of the object do not always have equal magnitudes;

• gravity acts on an object all the time (not just when it is falling).

Analysis

Present your data in the form of a labeled graph with the corresponding data table, or any other way you find useful to graphically determine the earth's gravitational acceleration g.

Δm represents the difference in mass that led to the acceleration of the system. Draw a free body diagram with the forces acting on the system.
Use your graph of acceleration versus Δm to determine an experimental value for g. Explain the procedure you used to determine this value.
Should your graph pass through the origin (0, 0)? Would you consider (0,0) as an experimental point?
Calculate the % difference in your experimental value for g.
What graph would you plot to verify Newton’s second law? How can you tell from your graph that your data supports Newton’s second law?
How would the accelerations you measured in this lab change if you were still moving the same small masses from one hanger to the other for each trial, but the total mass of the system was much larger?
For low acceleration values, how would the mass and friction of the pulley affect the values for g? Would they be erroneously high or low? Explain your answer.
For high values of acceleration, the string may actually slip on the pulley wheel. How would this error affect the observed values for system acceleration?

Extension - Modified Atwood Machine

Vary the acceleration of the block by changing the mass on the hanger – verify Newton’s second law
How can we tell if friction between the block and the table is significant? If so, how can we experimentally determine its value, without the help of the smart pulley system and datalogger? (Hint: You will not need to use a stopwatch; just use energy considerations.)

Video

https://youtu.be/lagGL0catfI by Nawal Nayfeh

Worksheets

ejss_model_AtwoodMachine2wee Newton's second Law by Nawal Nayfeh, Physics instructor at university of sharjah, UAE.docx

Version:

Other Resources

http://www.walter-fendt.de/html5/phen/newtonlaw2_en.htm JavaScript version by Walter Fendt
http://www.walter-fendt.de/ph14e/n2law.htm
http://lectureonline.cl.msu.edu/~mmp/kap4/cd097a.htm
https://www.geogebra.org/m/hh9CGse3#material/K6afWTwQ
https://www.geogebra.org/m/ztUKm6QV Moment of Inertia: Rolling and Sliding Down an Incline by ukukuku
The Atwood machine revisited using smartphones

Modified Atwood Machine FAQ

What is an Atwood Machine?

An Atwood machine is a simple physics apparatus consisting of two masses connected by a string that passes over a pulley. It is used to demonstrate the principles of classical mechanics, particularly Newton's Second Law of Motion.

What is the purpose of the Modified Atwood Machine experiment?

This experiment uses a modified Atwood machine to graphically determine the Earth's gravitational acceleration (g). It involves manipulating the masses and measuring the system's acceleration to establish a relationship between these variables.

How does the Modified Atwood Machine differ from a traditional Atwood Machine?

The modified Atwood machine often includes additional features like a cart with added mass or a track with friction. These modifications allow for more complex investigations into forces and motion, exploring concepts beyond just gravitational acceleration.

What safety precautions should be taken when conducting this experiment?

Ensure the mass difference on either side of the pulley is within a safe range (10g-60g) to prevent masses from flying off.
Keep the area clear before releasing the masses.
Place a mat beneath the falling mass to absorb impact and prevent damage.
Avoid squatting directly under the falling mass due to the risk of string breakage.

What are the key concepts related to this experiment?

Newton's Second Law of Motion (F=ma)
Acceleration due to gravity (g)
One-dimensional motion of uniformly accelerated objects
Principle of conservation of energy
Free body diagrams and force analysis
Friction and its influence on motion

How can I analyze the data collected from the experiment?

Plot a graph of acceleration versus the difference in mass (Δm).
Use the slope of the graph to determine the experimental value for g.
Consider the potential sources of error and their impact on the results.
Analyze the graph to verify Newton's Second Law and observe the relationship between mass and acceleration.

What factors could affect the accuracy of the experimental results?

Friction in the pulley
Mass of the pulley
Slipping of the string on the pulley
Measurement errors in mass and acceleration
Air resistance

What are some extensions or modifications to the experiment?

Investigating the significance of friction and its experimental determination.
Varying the acceleration by changing the hanging mass to further verify Newton's Second Law.
Exploring the impact of system mass on acceleration while maintaining the same mass difference.