Lab 05: Torque or Moment of a Force

Lab 05: Torque or Moment of a Force

Learning Objectives

Define torque or moment of a force.
Locate the center of gravity of a meter-stick beam experimentally.
Apply the principle of moments to a balanced beam system.
Determine unknown pan masses using torque equations.
Determine the experimental mass of a meter stick using rotational equilibrium.
Compare experimental and true values using percent difference.

Torque, also called the moment of a force, measures the turning effect of a force about a pivot. This experiment uses a model balance to study rotational equilibrium. By balancing a meter-stick beam with hanging pans and brass weights, students can determine unknown masses and verify the principle of moments.

Target Learning Outcome

TLO 5: Define the moment of a force or torque, and apply the principle of moments.

I. Discussion of Theory

Torque or Moment of a Force

Torque is the turning effect produced by a force acting at a distance from a pivot or axis of rotation. The farther the force is applied from the pivot, the greater its turning effect.

Torque

Torque is the product of the force and the perpendicular moment arm.

\tau = Fd

Variables

Symbol	Description	Unit
$\tau$	torque or moment of a force	N·m, N·cm, g·cm, or kg·m depending on setup
$F$	force applied perpendicular to the beam	N or gram-force
$d$	perpendicular distance from pivot to line of action of force	m or cm

Moment Arm

The moment arm is the perpendicular distance from the pivot to the line of action of the force.

Fulcrum

The fulcrum is the point of support, or pivot point, about which a lever or beam turns.

Center of Gravity

The center of gravity is the point at which the entire weight of an object may be considered to act. For a uniform meter stick, it is expected to be near the 50-cm mark, but the actual value should be measured experimentally.

Clockwise and Counterclockwise Torque

A force that tends to rotate an object in the direction of the hands of a clock produces a clockwise torque. A force tending to rotate the object in the opposite direction produces a counterclockwise torque.

Rotational Equilibrium

A body is in rotational equilibrium when the net torque about any chosen axis is zero. In this condition, clockwise and counterclockwise moments are balanced.

Principle of moments

For a balanced beam, clockwise torque equals counterclockwise torque.

\sum \tau_{\text{clockwise}} = \sum \tau_{\text{counterclockwise}}

\sum \tau = 0

Sign Convention for Torque

When summing torques ( $\sum \tau = 0$ ), it is necessary to assign a positive sign to one direction of rotation and a negative sign to the other. You may choose either clockwise as positive or counterclockwise as positive. The specific choice does not matter, but consistency matters throughout the entire problem.

Why mass can be used in the torque equation

In this lab, balance equations are often written using masses in grams and distances in centimeters. This works because each hanging mass produces a weight $mg$ , and the common factor $g$ appears on both sides of the balanced torque equation. Therefore, for comparison of torques in the same gravitational field, equations may be written in units such as g·cm.

Two-pan balance equation

When two pans balance on opposite sides of the fulcrum, their moments are equal.

m_1 L_1 = m_2 L_2

Variables

Symbol	Description	Unit
$m_1$	mass of pan on the left side of the fulcrum	g
$m_2$	mass of pan on the right side of the fulcrum	g
$L_1$	distance of m1 from the fulcrum	cm
$L_2$	distance of m2 from the fulcrum	cm

Loaded pan balance equation

When a known load P is added to pan m1, the new balance distances are Lx and Ly.

(m_1 + P)L_x = m_2 L_y

Variables

Symbol	Description	Unit
$P$	known added load, usually a 50-g brass weight	g
$L_x$	new distance of the loaded left pan from the fulcrum	cm
$L_y$	new distance of the right pan from the fulcrum	cm

Meter-stick mass from torque

When the meter stick is supported away from its center of gravity, a known hanging mass can balance the stick's weight.

m_3L_3 = ML

M = \frac{m_3L_3}{L}

Variables

Symbol	Description	Unit
$m_3$	total mass of pan m1 plus brass weights used to balance the stick	g
$L_3$	distance of m3 from the pivot support	cm
$M$	computed mass of the meter stick	g
$L$	distance between the meter-stick center of gravity and the pivot support	cm

Percent difference

Use percent difference to compare computed and true values.

\%\,\text{difference} = \left|\frac{\text{computed value} - \text{true value}}{\text{true value}}\right| \times 100\%

II. Equipment / Materials Needed

Equipment or material	Purpose
Student's model balance and accessories	Supports the beam and knife-edge/pivot support system.
Meter stick or balance beam	Acts as the beam for torque measurements.
Two wide pans with hangers	Unknown masses to be determined by torque.
Brass weights	Known loads for balancing and calculation.
Platform balance	Measures true values of pan and meter-stick masses.
Iron stand and metal frame	Holds the knife-edge/pivot support and beam.

Apparatus care

Keep the beam, knife-edge/pivot support, and pans stable while changing loads. Add weights gently and avoid sudden movement that may displace the beam from the support. Avoid applying excessive force that could damage the knife-edge/pivot support.

III. Diagram of Setup

Center of gravity of the beam

       meter stick / beam
  ---------------------------------
                 ^
                 |
             pivot support

Adjust the pivot support until the beam balances horizontally.
The balance point is the center of gravity of the beam.

Two-pan torque balance

       m1                         m2
       |                          |
       v                          v
  -----+-------------^------------+-----
       <--- L1 ----->|<--- L2 --->
                   fulcrum

For equilibrium:  m1 L1 = m2 L2

Loaded-pan torque balance

    m1 + P                       m2
       |                          |
       v                          v
  -----+-------------^------------+-----
       <--- Lx ---->|<--- Ly ---->
                   fulcrum

For equilibrium:  (m1 + P)Lx = m2 Ly

Meter-stick mass determination

  center of gravity        pivot support        m3
          |                    ^                |
          v                    |                v
  --------+--------------------^----------------+---
          <--------- L --------> <---- L3 ----->

For equilibrium:  M L = m3 L3

IV. Procedure

A. Locate the center of gravity of the beam

STEP-BY-STEP

Place the beam or meter stick on the pivot support.
Adjust the position of the support until the beam balances horizontally by itself.
Fix the support at this position.
Record the balance position as the center of gravity, c.g., of the beam.

B. Determine the pan masses by torque

STEP-BY-STEP

Hang pan $m_1$ on the left side of the fulcrum.
Hang pan $m_2$ on the right side of the fulcrum.
Adjust the positions of the pans until the beam balances horizontally.
Measure the distance of $m_1$ from the fulcrum and record it as $L_1$ .
Measure the distance of $m_2$ from the fulcrum and record it as $L_2$ .
Write the first torque equation: $m_1L_1 = m_2L_2$ .

C. Add a known load and form a second torque equation

STEP-BY-STEP

Add a known load $P$ to pan $m_1$ . A $50\,\text{g}$ brass weight is commonly used.
Adjust the positions of the two pans until the beam balances horizontally again.
Measure the new distance of the loaded pan from the fulcrum and record it as $L_x$ .
Measure the new distance of $m_2$ from the fulcrum and record it as $L_y$ .
Write the second torque equation: $(m_1+P)L_x = m_2L_y$ .
Use the two torque equations to solve for $m_1$ and $m_2$ .
Measure the true masses of the pans using the platform balance.
Compute percent difference for $m_1$ and $m_2$ .

D. Determine the mass of the meter stick

STEP-BY-STEP

Support the meter stick at the 75-cm mark using the pivot support.
Hang pan $m_1$ on the shorter arm of the beam.
Add brass weights to the pan and adjust its position until the beam balances horizontally.
Measure the distance of the loaded pan from the pivot support and record it as $L_3$ .
Determine $m_3$ , the total mass of pan $m_1$ and the brass weights placed on it.
Measure $L$ , the distance between the center of gravity of the meter stick and the pivot support.
Write the third torque equation: $m_3L_3 = ML$ .
Compute the experimental mass $M$ of the meter stick.
Measure the true mass of the meter stick using the platform balance.
Compute the percent difference between the computed mass and the true mass.

Balance condition

Read distances only when the beam is as close to horizontal as possible. A tilted beam changes the perpendicular moment arm and introduces error.

V. Student Information

Field	Entry
Name
Schedule
Group No.
Date Performed

VI. Expected Trend

In rotational equilibrium, a smaller mass placed farther from the fulcrum can balance a larger mass placed closer to the fulcrum. This is because torque is the product of force and distance; increasing the moment arm compensates for a smaller force. Achieving balance ultimately requires equal clockwise and counterclockwise moments around the pivot point.

VII. Data and Results

Table 5.1. Torque and Mass Data

Parameter	Description	Value	Unit
c.g.	center of gravity of beam		cm
$P$	known added load	50	g
$L_1$	distance of $m_1$ from fulcrum		cm
$L_2$	distance of $m_2$ from fulcrum		cm
$L_x$	distance of loaded $m_1+P$ from fulcrum		cm
$L_y$	distance of $m_2$ from fulcrum after adding $P$		cm
$L$	distance between c.g. and pivot support		cm
$L_3$	distance of $m_3$ from pivot support		cm
$m_1$ computed	pan mass from torque equations		g
$m_2$ computed	pan mass from torque equations		g
$m_1$ true value	measured using platform balance		g
$m_2$ true value	measured using platform balance		g
percent difference for $m_1$	comparison of computed and true value		%
percent difference for $m_2$	comparison of computed and true value		%
$m_3$	mass of pan plus brass weights		g
$M$ computed	meter-stick mass from torque		g
$M$ true value	measured using platform balance		g
percent difference for $M$	comparison of computed and true value		%

Table 5.2. Torque Equation Record

Equation	Balanced condition	Torque equation	Unknown solved
1	two empty pans balanced	$m_1L_1 = m_2L_2$	relation between $m_1$ and $m_2$
2	known load $P$ added to $m_1$	$(m_1+P)L_x = m_2L_y$	$m_1$ , $m_2$
3	meter stick balanced about 75-cm mark	$m_3L_3 = ML$	$M$

VIII. Computations

Required computations

STEP-BY-STEP

Write equation 1 from the first balanced pan setup.
Write equation 2 from the loaded pan setup.
Solve the two equations simultaneously for $m_1$ and $m_2$ .
Compare the computed pan masses with the true pan masses from the platform balance.
Compute percent difference for each pan.
Write equation 3 for the meter-stick mass setup.
Solve for $M$ using $M=m_3L_3/L$ .
Compare computed $M$ with the true meter-stick mass from the platform balance.
Compute the percent difference for $M$ .

Sample pan-mass computation method

Suppose the first balance equation is:

m_1 (40\,\text{cm}) = m_2 (30\,\text{cm})

We can express $m_2$ in terms of $m_1$ :

m_2 = \frac{40}{30} m_1 = \frac{4}{3} m_1

After adding a $50\,\text{g}$ load to $m_1$ , the new balance equation is:

(m_1 + 50) (25\,\text{cm}) = m_2 (35\,\text{cm})

Substitute $m_2$ from the first equation into the second:

(m_1 + 50) (25) = \left( \frac{4}{3} m_1 \right) (35)

Solve for $m_1$ :

25 m_1 + 1250 = \frac{140}{3} m_1

Multiply everything by 3:

75 m_1 + 3750 = 140 m_1

3750 = 65 m_1

m_1 = \frac{3750}{65} \approx 57.69\,\text{g}

Now find $m_2$ :

m_2 = \frac{4}{3} (57.69) \approx 76.92\,\text{g}

Sample meter-stick mass computation

Suppose $m_3 = 80\,\text{g}$ (pan + weights), located at a distance $L_3 = 15\,\text{cm}$ from the pivot support. The distance between the center of gravity and the pivot support is $L = 20\,\text{cm}$ .

M L = m_3 L_3

M = \frac{m_3L_3}{L} = \frac{80(15)}{20} = 60\,\text{g}

Sample center-of-gravity problem with two scale readings

A light plank rests on two scales separated by $2.00\,\text{m}$ . An object is placed on the plank. The left scale reads $380\,\text{N}$ and the right scale reads $320\,\text{N}$ . Find the location of the system's center of gravity measured from the left scale.

Let the left scale be the pivot point (fulcrum). The sum of torques about this pivot must be zero. Let $W = 380\,\text{N} + 320\,\text{N} = 700\,\text{N}$ be the total weight acting at the center of gravity, at distance $x$ from the left scale.

Clockwise torque from the total weight: $700x$ Counterclockwise torque from the right scale: $320 \times 2.00 = 640\,\text{N\cdot m}$

Equating the torques:

700 x = 640

x = \frac{640}{700} \approx 0.914\,\text{m}

The center of gravity is $0.914\,\text{m}$ from the left scale.

IX. Error Analysis

Common sources of error

Beam not exactly horizontal when distances are read.
Pivot support not positioned exactly at the center of gravity.
Friction at the support point.
Pan hangers not vertical.
Distances read from the wrong reference point.
Brass weights or pans swinging during balancing.
Parallax error when reading the meter-stick scale.
True masses measured with an unzeroed platform balance.

Ways to improve accuracy

Wait for the beam to settle before recording distances.
Read the meter-stick scale at eye level.
Measure all distances from the fulcrum to the line of action of the hanging mass.
Repeat the balancing process and average the distances if possible.
Use small adjustments in pan position when approaching balance.
Verify true masses using a calibrated platform balance.

X. Lab Report Format

A complete lab report for this experiment should include:

Title Page: Experiment title, student names, and dates.
Objectives: Restatement of the goals.
Data Tables: Completed Tables 5.1 and 5.2.
Computations: Clear derivation and substitution of values for $m_1$ , $m_2$ , and $M$ .
Error Analysis: Calculations of percent difference.
Observations and Conclusions: Summary of results matching the conclusion guide.
Post-Lab Questions: Detailed answers to the questions provided.

XI. Observations and Conclusions

Conclusion guide

A strong conclusion should state whether the beam satisfied the principle of moments, compare computed and true masses, report percent differences, and identify the largest likely source of error. Mention how moving a small mass farther from the fulcrum can balance a larger mass closer to the fulcrum.

XII. Questions and Problems

State in words the condition for rotational equilibrium.
A light plank rests on two scales separated by $2.00\,\text{m}$ . The left scale reads $380\,\text{N}$ and the right scale reads $320\,\text{N}$ . Find the location of the system's center of gravity measured from the left scale.
Why can masses in grams be used directly in some torque equations instead of first converting them to weights?
Why should the lever arm be measured from the fulcrum to the line of action of the force, not merely to the edge of the pan?
If the beam is balanced and one mass is moved closer to the fulcrum, what must be done to restore balance?

XIII. Selected Answer Key

Question 1: The net torque about any chosen axis must be zero, meaning the sum of clockwise torques equals the sum of counterclockwise torques. Question 2: As solved in the example, setting the left scale as the pivot gives $700x = 320(2.00)$ , yielding $x \approx 0.914\,\text{m}$ . Question 3: Since torque is defined by weight (which is mass $\times$ gravity), the constant $g$ cancels out on both sides of a balanced torque equation. Question 4: Torque depends on the perpendicular distance to the actual line of action of the weight force, which passes through the center of the pan where it hangs, not the edge. Question 5: To restore balance, the other mass must also be moved closer to the fulcrum, or its mass must be proportionally decreased, to equalize the moments again.

XIV. 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, torque equations, printable diagrams, improved data tables, computation guidance, error analysis, and post-lab questions. HTML-only input fields and theme controls were converted into MDX tables and notes for compatibility with the CE content renderer.

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