Lab 04: Newton's Second Law of Motion

Lab 04: Newton's Second Law of Motion

Learning Objectives

Apply Newton's Second Law to a two-mass dynamic system (Aligned with TLO 4).
Use a simplified Atwood machine to produce measurable acceleration.
Compute experimental acceleration from distance and time data.
Compute theoretical acceleration from the measured system masses.
Compare experimental and theoretical acceleration using percent difference.
Discuss timing, pulley friction, and mass measurement as sources of error.

This experiment uses a simplified Atwood machine. Two mass carriers are connected by a light string over a pulley. When one side has a slightly greater mass than the other, the system accelerates. The measured motion is used to test the relationship between net force, mass, and acceleration.

Target Learning Outcome

TLO 4: Apply Newton's Second Law of Motion to explain a dynamic system and determine its acceleration.

I. Discussion of Theory

Newton's First Law (Law of Inertia)

An object at rest stays at rest, and an object in motion stays in motion with the same speed and in the same direction unless acted upon by an unbalanced force.

Newton's Second Law

Newton's Second Law states that the net external force on a system equals the product of the total system mass and the acceleration.

Newton's Third Law (Action and Reaction)

For every action, there is an equal and opposite reaction. When two objects interact, they apply forces to each other of equal magnitude and opposite direction.

Key Concepts

Net force: The vector sum of all forces acting on a system.
Tension: The pulling force transmitted through a string or cable.
Acceleration: The rate of change of velocity per unit of time.
System mass: The total mass of all moving components in the system.

Newton's Second Law

\sum F = ma

Variables

Symbol	Description	Unit
$\sum F$	net external force	N
$m$	mass	kg
$a$	acceleration	$m/s^2$

Atwood machine model

In an ideal Atwood machine, the difference in the two weights supplies the net driving force. Both sides have the same magnitude of acceleration because they are connected by the same string.

Free-Body Diagrams (FBDs)

Free-Body Diagram for M1 (heavier mass moving downward):
   ^ Tension (T)
   |
  [M1]
   |
   v Weight (M1*g)

Free-Body Diagram for M2 (lighter mass moving upward):
   ^ Tension (T)
   |
  [M2]
   |
   v Weight (M2*g)

Derivation of the Atwood formula

Applying Newton's Second Law to each mass separately: For $M_1$ (accelerating downwards): $M_1g - T = M_1a$ For $M_2$ (accelerating upwards): $T - M_2g = M_2a$

Adding these two equations eliminates tension $T$ : $M_1g - M_2g = M_1a + M_2a$ $(M_1 - M_2)g = (M_1 + M_2)a$

Solving for acceleration without the pulley: $a = \frac{(M_1 - M_2)g}{M_1 + M_2}$

Pulley Inertia Correction

In a real Atwood machine, the pulley is not massless. It has a rotational inertia $I = \frac{1}{2}mr^2$ . The torque required to accelerate the pulley is provided by a difference in tension on the two sides of the string. When we include the pulley's rotational inertia in the equations of motion, the effective mass of the system increases by $\frac{I}{r^2} = \frac{1}{2}m$ . Thus, the pulley mass appears as $\frac{1}{2}m$ in the denominator.

Driving force

F_{\text{net}} = (M_1 - M_2)g

Variables

Symbol	Description	Unit
$M_1$	mass of the heavier carrier and load	g or kg
$M_2$	mass of the lighter carrier and load	g or kg
$g$	acceleration due to gravity	$cm/s^2 or m/s^2$

Theoretical acceleration

a_{\text{theoretical}} = \frac{(M_1 - M_2)g}{M_1 + M_2 + \frac{1}{2}m}

Variables

Symbol	Description	Unit
$m$	mass of the pulley	g or kg
$a_{\text{theoretical}}$	acceleration predicted by the model	$cm/s^2 or m/s^2$

Distance relation

S = \frac{1}{2}at^2

Variables

Symbol	Description	Unit
$S$	measured travel distance	cm or m
$a$	acceleration	$m/s^2$
$t$	time	s

Experimental acceleration

a_{\text{experimental}} = \frac{2S}{t^2}

Variables

Symbol	Description	Unit
$S$	measured travel distance	cm or m
$t$	average travel time	s

Computation Method

Method Explained: We calculate the average time first, and then use that average time to compute one single experimental acceleration for the given configuration. This method reduces the impact of timing outliers before the non-linear computation $a = 2S/t^2$ is applied.

Percent difference

\%\,\text{difference}=\left|\frac{a_{\text{experimental}}-a_{\text{theoretical}}}{a_{\text{theoretical}}}\right|\times100\%

Expected Trends

Increasing Net Force: Increasing the unbalanced mass ( $M_1 - M_2$ ) while keeping total mass constant will result in an increase in acceleration.
Increasing Total Mass: Increasing the total mass of the system while keeping the unbalanced mass ( $M_1 - M_2$ ) constant will result in a decrease in acceleration.

Unit consistency

Use a consistent unit system. For grams and centimeters, use $g=980\,\text{cm/s}^2$ . For kilograms and meters, use $g=9.8\,\text{m/s}^2$ .

II. Equipment / Materials Needed

Equipment or material	Purpose
Atwood machine frame	Holds the pulley and carriers.
Rotating pulley	Guides the string with reduced resistance.
Two equal mass pans or carriers	Hold the slotted weights.
Two-meter string	Connects the two carriers over the pulley.
Set of slotted weights	Creates a controlled mass difference.
Meterstick	Measures travel distance $S$ .
Stopwatch	Measures travel time.
Balance	Measures masses when needed.

Apparatus care and Safety Setup

Check that the pulley, string, and carriers are secure before each trial. Add and remove masses gently, and keep the setup stable during measurements. Ensure weights cannot fall on feet or fragile surfaces.

III. Diagram of Setup

Simplified Atwood machine

                 support frame
             ___________________
                    |   |
                    O   pulley, mass m
                   / \
                  /   \
                 /     \
              M2 side   M1 side

M1 is slightly greater than M2.
Measure the travel distance S and the corresponding time t.

IV. Procedures

A. Apparatus setup

STEP-BY-STEP

Record the mass of the pulley, $m$ , in Table 4.1.
Mount the pulley on the frame.
Adjust the pulley until it rotates freely.
Place the string around the pulley.
Attach one pan or carrier to each end of the string.
Balancing the pans: Add small slotted weights to the lighter side if the carriers are uneven.
Compensating friction: Add a very tiny amount of mass to one side until the system moves at a constant velocity when given a slight push. This compensates for kinetic friction.

B. Measurement procedure

STEP-BY-STEP

Adding unbalanced mass: Add a small extra mass to one carrier so that it becomes $M_1$ .
Label the other carrier as $M_2$ .
Record $M_1$ , $M_2$ , and pulley mass $m$ .
Measuring distance S: Measure the travel distance $S$ that $M_1$ will fall.
Start the system from rest and measure the time required for $M_1$ to move through distance $S$ .
Timing five trials: Repeat the timing for five trials using the same distance $S$ .
Compute the average time.
Computing acceleration: Compute experimental acceleration using $a=2S/t^2$ where $t$ is the average time.
Compute theoretical acceleration using the Atwood machine formula.
Compute percent difference.

Measurement reminder

Start each trial from rest and use the same starting and ending marks for every timing measurement.

V. Student Information

Field	Entry
Name
Schedule
Group No.
Date Performed

VI. Data and Results

Table 4.1. Atwood Machine Data

Quantity	Value	Unit
$M_1$ = mass of heavier carrier and load		g
$M_2$ = mass of lighter carrier and load		g
$m$ = mass of pulley		g
$S$ = travel distance		cm

Trial	Time, $t$ (s)
1
2
3
4
5
Average Time

Summary of Results

Result	Value	Unit
Experimental acceleration		cm/s²
Theoretical acceleration		cm/s²
Percent difference		%

Additional trials

More trials may be obtained by changing $S$ or changing the extra mass added to one carrier.

VII. Computations

Required computations

STEP-BY-STEP

Compute the average time from the five trials.
Compute $t^2$ using the average time.
Compute $a_{\text{experimental}}=2S/t^2$ using the computed $t^2$ .
Compute $a_{\text{theoretical}}=\frac{(M_1-M_2)g}{M_1+M_2+\frac{1}{2}m}$ .
Compute the percent difference.
Show all substitutions with units.

Sample computation: Experimental, Theoretical, and Percent Difference

Given $M_1=54\,\text{g}$ , $M_2=50\,\text{g}$ , $m=20\,\text{g}$ , $S=80\,\text{cm}$ , and average $t=2.10\,\text{s}$ :

Experimental acceleration:

a_{\text{experimental}}=\frac{2(80)}{(2.10)^2}=36.3\,\text{cm/s}^2

Theoretical acceleration:

a_{\text{theoretical}}=\frac{(54-50)(980)}{54+50+\frac{1}{2}(20)}=34.4\,\text{cm/s}^2

Percent difference:

\%\,\text{difference}=\left|\frac{36.3-34.4}{34.4}\right|\times100\%=5.5\%

Sample computation: Simple Atwood Tension and Acceleration

Consider an ideal Atwood machine with $M_1=5\,\text{kg}$ and $M_2=3\,\text{kg}$ . Ignore the pulley mass ( $m=0$ ) and assume $g=9.8\,\text{m/s}^2$ . Find the acceleration and tension.

Acceleration:

a = \frac{(M_1 - M_2)g}{M_1 + M_2} = \frac{(5 - 3)(9.8)}{5 + 3} = \frac{2(9.8)}{8} = 2.45\,\text{m/s}^2

Tension: Using the FBD of $M_1$ : $M_1g - T = M_1a$

T = M_1g - M_1a = 5(9.8) - 5(2.45) = 49 - 12.25 = 36.75\,\text{N}

VIII. Error Analysis

Common sources of error

Stopwatch reaction time.
Pulley friction not perfectly compensated.
String slipping on the pulley.
Pulley not rotating smoothly.
Swinging of the carriers.
Inaccurate measurement of $S$ .
Inconsistent starting method.
Masses not centered in the carriers.
Mass of the string neglected.

Ways to improve accuracy

Use the same starting method for every trial.
Use five or more trials and average the time.
Keep the string seated properly on the pulley.
Check the pulley motion before each run.
Use photogates or video timing if available.

IX. Observations and Conclusions

Conclusion guide

State the experimental acceleration, theoretical acceleration, percent difference, and whether the results support Newton's Second Law. Explain how the unbalanced force $(M_1-M_2)g$ affected the acceleration. Discuss the largest sources of error.

X. Lab Report Format

Lab Report Structure

Title Page: Experiment title, student names, and date.
Objectives: Restate the purpose of the lab.
Theory: Brief summary of the Atwood machine equations.
Data: Completed tables with proper units.
Calculations: Show step-by-step math for one complete trial.
Error Analysis: Discuss specifically what affected your results.
Conclusion: Summarize findings based on the guide.
Questions: Answers to the post-lab problems.

XI. Questions and Problems

State Newton's three laws of motion.
Two objects with masses of $3.00\,\text{kg}$ and $5.00\,\text{kg}$ are connected by a light string over a frictionless pulley. Determine the tension, acceleration, and distance traveled in the first second if both objects start from rest.
Why is the pulley mass included in the denominator of the theoretical acceleration formula?
What happens to acceleration when $M_1-M_2$ is increased while the total mass is nearly constant?
What happens to acceleration when the total mass is increased while the unbalanced mass difference is the same?

Selected Answer Key

Answer Key for Problem 2

Acceleration: $a = 2.45\,\text{m/s}^2$
Tension: $T = 36.75\,\text{N}$
Distance in 1s: $S = \frac{1}{2}at^2 = \frac{1}{2}(2.45)(1)^2 = 1.225\,\text{m}$

XII. References

Bueche, F. J., & Hecht, E. (1997). Schaum's Outline of Theory and Problems of College Physics (9th ed.). New York: McGraw-Hill.

Instructor note

The original HTML worksheet was converted into MDX and expanded with theory, variable definitions, setup diagrams, improved data tables, sample calculations, error analysis, and interpretation questions. HTML-only controls were converted into printable MDX content for the existing CE renderer.

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