1: Vector Addition of Forces Lab

Last updated

Jul 8, 2025
Save as PDF
- Licensing
- 2: Alternative Vector Addition of Forces Lab

$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$

$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$

$\newcommand{\id}{\mathrm{id}}$ $\newcommand{\Span}{\mathrm{span}}$

( \newcommand{\kernel}{\mathrm{null}\,}\) $\newcommand{\range}{\mathrm{range}\,}$

$\newcommand{\RealPart}{\mathrm{Re}}$ $\newcommand{\ImaginaryPart}{\mathrm{Im}}$

$\newcommand{\Argument}{\mathrm{Arg}}$ $\newcommand{\norm}[1]{\| #1 \|}$

$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$

$\newcommand{\Span}{\mathrm{span}}$

$\newcommand{\id}{\mathrm{id}}$

$\newcommand{\Span}{\mathrm{span}}$

$\newcommand{\kernel}{\mathrm{null}\,}$

$\newcommand{\range}{\mathrm{range}\,}$

$\newcommand{\RealPart}{\mathrm{Re}}$

$\newcommand{\ImaginaryPart}{\mathrm{Im}}$

$\newcommand{\Argument}{\mathrm{Arg}}$

$\newcommand{\norm}[1]{\| #1 \|}$

$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$

$\newcommand{\Span}{\mathrm{span}}$ $\newcommand{\AA}{\unicode[.8,0]{x212B}}$

$\newcommand{\vectorA}[1]{\vec{#1}} % arrow$

$\newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow$

$\newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$

$\newcommand{\vectorC}[1]{\textbf{#1}}$

$\newcommand{\vectorD}[1]{\overrightarrow{#1}}$

$\newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}}$

$\newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}}$

$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$

$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$

$\newcommand{\avec}{\mathbf a}$

$\newcommand{\bvec}{\mathbf b}$

$\newcommand{\cvec}{\mathbf c}$

$\newcommand{\dvec}{\mathbf d}$

$\newcommand{\dtil}{\widetilde{\mathbf d}}$

$\newcommand{\evec}{\mathbf e}$

$\newcommand{\fvec}{\mathbf f}$

$\newcommand{\nvec}{\mathbf n}$

$\newcommand{\pvec}{\mathbf p}$

$\newcommand{\qvec}{\mathbf q}$

$\newcommand{\svec}{\mathbf s}$

$\newcommand{\tvec}{\mathbf t}$

$\newcommand{\uvec}{\mathbf u}$

$\newcommand{\vvec}{\mathbf v}$

$\newcommand{\wvec}{\mathbf w}$

$\newcommand{\xvec}{\mathbf x}$

$\newcommand{\yvec}{\mathbf y}$

$\newcommand{\zvec}{\mathbf z}$

$\newcommand{\rvec}{\mathbf r}$

$\newcommand{\mvec}{\mathbf m}$

$\newcommand{\zerovec}{\mathbf 0}$

$\newcommand{\onevec}{\mathbf 1}$

$\newcommand{\real}{\mathbb R}$

$\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}$

$\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}$

$\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}$

$\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}$

$\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$

$\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$

$\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$

$\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$

$\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}$

$\newcommand{\laspan}[1]{\text{Span}\{#1\}}$

$\newcommand{\bcal}{\cal B}$

$\newcommand{\ccal}{\cal C}$

$\newcommand{\scal}{\cal S}$

$\newcommand{\wcal}{\cal W}$

$\newcommand{\ecal}{\cal E}$

$\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}$

$\newcommand{\gray}[1]{\color{gray}{#1}}$

$\newcommand{\lgray}[1]{\color{lightgray}{#1}}$

$\newcommand{\rank}{\operatorname{rank}}$

$\newcommand{\row}{\text{Row}}$

$\newcommand{\col}{\text{Col}}$

$\renewcommand{\row}{\text{Row}}$

$\newcommand{\nul}{\text{Nul}}$

$\newcommand{\var}{\text{Var}}$

$\newcommand{\corr}{\text{corr}}$

$\newcommand{\len}[1]{\left|#1\right|}$

$\newcommand{\bbar}{\overline{\bvec}}$

$\newcommand{\bhat}{\widehat{\bvec}}$

$\newcommand{\bperp}{\bvec^\perp}$

$\newcommand{\xhat}{\widehat{\xvec}}$

$\newcommand{\vhat}{\widehat{\vvec}}$

$\newcommand{\uhat}{\widehat{\uvec}}$

$\newcommand{\what}{\widehat{\wvec}}$

$\newcommand{\Sighat}{\widehat{\Sigma}}$

$\newcommand{\lt}{<}$

$\newcommand{\gt}{>}$

$\newcommand{\amp}{&}$

$\definecolor{fillinmathshade}{gray}{0.9}$

Learning Objectives

To use the force table to experimentally determine the force that balances two or more forces. This result is checked by analytically adding two or more forces using their horizontal and vertical vector components, and then by graphically adding the force vectors on the force table.

Theory

If several forces are acting on a point, their resultant is given as

R_x = A_x + B_x + C_x

R_y = A_y + B_y + C_y

R =

Then if the equilibrant is a force that brings the system to equilibrium

, this means

(E = R, θ_E = θ_R+180°)

This means E_x = -R_x and E_y = -R_y

Notes for today's lab

(1) Read the details, discuss with your group, and follow the instructions systematically. You have done several of these questions in class so now work by yourselves. If you want more details, look up your textbook or online.

(2) Every time to determine a resultant vector, assess whether it’s magnitude and direction makes sense before you proceed.

(3) When working with the force table, do not hang any masses until after you have completed the calculations for the Resultant and Equilibrant.

Method

You will hang some mass on the pulley hangers that are attached by a thread. This means the weight of that mass is a force vertically down. However, the string is attached to the central ring of the force table, and this means a tension equal to the weight of the mass is the force acting on the central ring. This means you can set up one or more forces acting on the central ring, calculate their resultant force (resultant, ). Then you can determine what force (Equilibrant, ) would balance these forces to bring the system to equilibrium.

Apparatus

Force table, 4 pulley clamps, 3 mass hangers, 1 mass set, string (or spool of thread)

Force table, an introduction

A force table is a simple set up that can be used to observe vector addition and equilibrium. You can attach a (one or more) pulley at the edge of the table, and hang a mass on a string that goes through this pulley. Hanging mass means a weight is acting downward and the tension on the hanging string is acting upward. However, on the top of the table, the string is attached to a central ring. This string applies a horizontal tension to the ring. The central ring is our object of interest and we will observe the effect of various forces on this ring. You can change the magnitude of the force by changing the hanging mass.

The table surface has a protractor so you can set up vectors in specific directions.

You can find more information online on how a force table works.

If a mass “m” is hanging over the pulley, the mass has a force downward (= the weight of the mass, mg). And the tension on the string is upward. The magnitude of the tension = mg)

Figure $\PageIndex{1}$ : Side and top view of the force table (image credit: CCNY CUNY)

Set up the force table such that 0 of the table protractor is on your right (just like x-axis on a Cartesian coordinate system. This means 0°, 90°, 180°, and 270° should be along +x, +y, -x, -y of your coordinate system.

Figure $\PageIndex{1}$ : a: 3D view of force table with four equal weights, b: Top view of the protractor, c: Vector diagram of the forces

Resultant vs. Equilibrant

The Resultant force is the sum of the individual forces acting on the ring. It means, you can replace all the vectors by the Resultant to get the same effect. The equilibrant is the force that brings the system to equilibrium.

Macintosh HD:Users:nthakur:Desktop:Screen Shot 2019-10-09 at 5.21.56 AM.png — Figure $\PageIndex{1}$ : Left: 3D view of the force table with two weights hanging at 90^o from each other. Right: Top view showing the vector forces and the resultant and the equilibrant (image credit: CCNY CUNY)

Precautions:

(1) Ensure that the central pin on the force table is always attached in place before and while you hang any mass unless otherwise specified. Otherwise the mass can suddenly drop and hurt someone (and also mess your experiment).

(2) Measure/note the mass of each hanger before you use it.

(3) The force needed to balance the force table is not the resultant force but the equilibrant force, which is negative of the resultant.

Part I: Experimental Procedure I, Use of One Force

Step 1: Calculation only. Do not hang any mass yet; you will do that in Step 2 after you finish your data table below.

You will hang a mass (an example: 100 g) on a hanger at 0°. Fill out the table below using these values.

Force	Mass m [g]	Mass m [kg]	Magnitude mg [N]	Angle θ[°]	x-component [N]	y-component [N]


Resultant

Then we can write the resultant and the equilibrant below

Force	Magnitude	Angle
Resultant
Equilibrant
Mass that you need to hang for this Equilibrant: _______________ g

Step 2: now hang the mass for force . You will notice that the central ring seems to be pulled in the direction of the Resultant force. What should you do to bring it in equilibrium? Write your answer below (one brief sentence):

Then apply the equilibrant force as you determined in your data table above.

To check if the system is actually in equilibrium, remove the central pin (at the center of the ring). If your system is actually in equilibrium, the ring will stay in place otherwise the masses will fall off in the direction on any net force.

Explain your observations.

Part II: Experimental Procedure II, Use of Two Forces

Step 1: Calculation only. Do not hang any mass yet; you will do that in Step 2 after you finish your data table below.

You will hang two masses (an example: 100 g) on two separate hangers. Choose your masses and angles; ensure that individual hanger should not exceed a mass of 250 grams. Fill out the table below using these values.

Force	Mass m [g]	Mass m [kg]	Magnitude mg [N]	Angle θ[°]		x-component [N]	y-component [N]


Resultant

Then we can write the resultant and the equilibrant below

Force	Magnitude	Angle
Resultant
Equilibrant
Mass that you need to hang for this Equilibrant: _______________ g

Step 2: now hang the masses for forces and Then apply the equilibrant force as you determined in your data table above.

Explain your observations.

Part II: Experimental Procedure III, Use Of Three Forces

Step 1: Calculation only. Do not hang any mass yet; you will do that in Step 2 after you finish your data table below.

You will hang three masses (an example: 100 g) on three separate hangers. Choose your masses and angles; ensure that individual hanger should not exceed a mass of 250 grams. Fill out the table below using these values.

Force	Mass m [g]	Mass m [kg]	Magnitude mg [N]	Angle θ[°]	x-component [N]	y-component [N]



Resultant

Then we can write the resultant and the equilibrant below

Force	Magnitude	Angle
Resultant
Equilibrant
Mass that you need to hang for this Equilibrant: _______________ g

Step 2: now hang the masses for forces and and Then apply the equilibrant force as you determined in your data table above.

Explain your observations.

_______________________________________________________________________________________

Part IV: Wrap up and Lab End Checklist

1. Remove all masses and place them properly in the box/carrier. Double check to ensure no mass is left behind.

2. Remove all mass hangers.

3. DO NOT remove the central ring and strings.

4. Place your Force Table, masses, and strings back on the cart. Please take a moment to ensure that nothing is left behind.

5. If you have used a college laptop to do your calculations, do log out before you leave the lab.

What to include in your lab report:

(1) Your data tables and observations, comments, and analysis for three procedures you performed.

(2) Draw a free body diagram for the ring in each case.

(3) Explain why the forces on the central ring can be measured using the hanging masses.

(4) Names of lab partners and specific contributions each person made.

Where to submit the lab report: on Canvas

Search

Text Color

Text Size

Margin Size

Font Type

Notes for today's lab

Learning Objectives

Theory

Notes for today's lab

Method

Apparatus

Force table, an introduction

Resultant vs. Equilibrant

Precautions:

Part I: Experimental Procedure I, Use of One Force

Part II: Experimental Procedure II, Use of Two Forces

Part II: Experimental Procedure III, Use Of Three Forces

Part IV: Wrap up and Lab End Checklist

What to include in your lab report:

Support Center

How can we help?