Force and Equilibrium of Objects

Lessons

Senior 1 · Physics

Force and Equilibrium of Objects

5 lessons

Lesson 3 of 5

Adding Forces: the Resultant

25 min

Who wins the tug of war?

On sports day two teams grip the same rope. Team A pulls with a total force of 1200 N, team B with 1100 N. The rope, and both teams, slide towards team A. When several forces act on one object, what matters is their combined effect, the resultant force. For forces along one line that is simple arithmetic; for forces at an angle, you will need a ruler, a protractor and a scale drawing.

Definition

The resultant force is the single force that has the same effect as all the forces acting on an object together.

The equilibrant is the single force that would balance them: it is equal in size to the resultant and acts in the opposite direction.

Forces along one line

Forces in the same direction add: $R = F_{1} + F_{2}$ .
Forces in opposite directions subtract: $R = F_{1} - F_{2}$ , in the direction of the larger force.

Figure 3.1: Adding parallel forces. Forces in the same direction add; forces in opposite directions subtract.

Worked example 1: forces along one line

(a) Two people push a stalled car forwards with 300 N and 250 N. (b) In a tug of war, team A pulls with 1200 N and team B with 1100 N. Find the resultant in each case.

(a) $R = 300\,\mathrm{N} + 250\,\mathrm{N} = 550\,\mathrm{N}$ forwards.

(b) $R = 1200\,\mathrm{N} - 1100\,\mathrm{N} = 100\,\mathrm{N}$ towards team A, so team A wins.

Experiment: adding forces at an angle

Figure 3.2: Three spring balances pull on one small ring. When the ring is still, the three forces are in equilibrium.

Pin a sheet of paper to a drawing board lying flat on the bench. Tie three strings to a small ring and hook a spring balance to each.
Pull the three balances in different directions and pin them so that the ring stays still over the middle of the paper.
Read the three forces. Mark the direction of each string with two pencil dots, then remove everything.
Draw lines through the dots from the ring's position. Choose a scale, such as 1 cm for 1 N, and draw two of the forces to scale.
Complete the parallelogram and draw its diagonal. Measure it: it should be equal to the third force and point exactly the opposite way. The third force is the equilibrant of the other two.

Important

Safety. Pull the balances gently and never let a stretched balance go: it can spring back into someone's face. Keep drawing pins in a box when you are not using them.

The parallelogram method

To add two forces $F_{1}$ and $F_{2}$ that act at an angle on the same point:

Choose a scale, for example 1 cm for 100 N, and write it down.
From a point O, draw $F_{1}$ to scale in its direction.
From the same point O, draw $F_{2}$ to scale at the correct angle, using a protractor.
Complete the parallelogram with dashed lines parallel to the two forces.
Draw the diagonal from O. Its length, changed back with the scale, is the size of the resultant; measure its angle with a protractor.

Figure 3.3: Forces of 400 N and 300 N at right angles. The diagonal measures 5 squares, so the resultant is 500 N, at about 37° to the 400 N force.

Worked example 2: two ropes at right angles

Two fishermen pull a boat up the beach at Lake Kivu. One rope pulls with 400 N along the beach and the other with 300 N at $90^\circ$ to it. Find the resultant by scale drawing.

Scale: 1 square (1 cm) = 100 N. Draw 4 squares along, 3 squares up, and complete the rectangle (Figure 3.3). The diagonal measures 5.0 cm, so

R = 5.0 \times 100\,\mathrm{N} = 500\,\mathrm{N}, \text{ at about } 37^\circ \text{ to the 400 N rope}

Check, if you know Pythagoras' theorem: $R = \sqrt{400^2 + 300^2} = \sqrt{250\,000} = 500\,\mathrm{N}$ .

Figure 3.4: Forces of 40 N and 30 N at 60° to each other, drawn at 1 cm for 10 N. The diagonal measures about 6.1 cm, so the resultant is about 61 N, at about 25° to the 40 N force.

Worked example 3: forces at 60°

Forces of 40 N and 30 N act on the same point with an angle of $60^\circ$ between them. Find the resultant.

Scale: 1 cm = 10 N, so draw $F_{1}$ 4.0 cm long and $F_{2}$ 3.0 cm long at $60^\circ$ to it. Complete the parallelogram (Figure 3.4). The diagonal measures 6.1 cm and makes $25^\circ$ with $F_{1}$ :

R \approx 6.1 \times 10\,\mathrm{N} = 61\,\mathrm{N}, \text{ at about } 25^\circ \text{ to the 40 N force}

Notice that 61 N is less than $40 + 30 = 70\,\mathrm{N}$ : forces at an angle partly work against each other.

Warning

Common mistakes. (1) Adding the sizes of forces that act at an angle: 40 N and 30 N at $60^\circ$ do NOT make 70 N. (2) Forgetting to write the scale, or to change the measured length back into newtons. (3) Giving the size of the resultant without its direction. A useful check: the resultant of two forces is always between $F_{1} - F_{2}$ and $F_{1} + F_{2}$ .

Tip

Think of the parallelogram as "two ways round to the same corner": going along $F_{1}$ then along $F_{2}$ ends at the same point as going along $F_{2}$ then $F_{1}$ . The resultant is the straight short-cut from the start to that corner.

Pulling together during umuganda

During umuganda, community work on the last Saturday of the month, neighbours sometimes drag a fallen tree off a road with two ropes. If the two groups pull at an angle to each other, the tree moves along the diagonal between the ropes, and with a force smaller than the two pulls added together. The smaller the angle between the ropes, the larger the resultant, so the groups should pull as nearly in the same direction as they can.

Explore: in the PhET: Forces and Motion: Basics (version 2.6.0) simulation (Net Force screen), put pullers on both sides of the rope and predict the resultant before you press Go. In PhET: Vector Addition (version 1.1.0), add two arrows at any angle and compare the resultant with your scale drawing.

Resultant Forces (FuseSchool). Watch how forces in the same and opposite directions combine, and how the resultant decides what happens to the object.

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Practice

Forces of 15 N and 20 N act on a box. Find the resultant when they act (a) in the same direction; (b) in opposite directions.
In a tug of war, team A pulls with 2400 N to the left and team B with 2250 N to the right. Find the resultant and say which team wins.
Forces of 6 N east and 8 N north act on one point. Using a scale of 1 cm = 1 N, find the resultant by scale drawing.
Two forces of 5 N and 12 N act on an object. Which of these could NOT be their resultant: 6 N, 7 N, 10 N, 17 N, 18 N? Explain.
Two forces of 50 N act at right angles to each other. Find their resultant by scale drawing.
Two ropes pull a boat, each with 300 N, with an angle of $60^\circ$ between them. Find the resultant by scale drawing.
What single force would balance the 400 N and 300 N forces of Worked example 2?

Show answersHide answers

(a) $15 + 20 = 35\,\mathrm{N}$ ; (b) $20 - 15 = 5\,\mathrm{N}$ in the direction of the 20 N force.
$2400 - 2250 = 150\,\mathrm{N}$ to the left; team A wins.
The diagonal measures 10 cm, so $R = 10\,\mathrm{N}$ , at about $53^\circ$ north of east.
6 N and 18 N. The resultant must lie between $12 - 5 = 7\,\mathrm{N}$ and $12 + 5 = 17\,\mathrm{N}$ .
About 71 N, at $45^\circ$ to each force (the diagonal of a square).
About 520 N, along the line half-way between the two ropes (at $30^\circ$ to each).
The equilibrant: 500 N, in exactly the opposite direction to the resultant.

Key points

The resultant is the single force with the same effect as all the forces together.
Along one line: same direction, add; opposite directions, subtract (direction of the larger).
At an angle: draw both forces to scale from one point, complete the parallelogram, and measure the diagonal and its angle.
The equilibrant is equal and opposite to the resultant.

Sources and credits

3.2: Vector Addition and Subtraction: Graphical Methods, from College Physics (1st edition) by Paul Peter Urone and Roger Hinrichs, OpenStax, hosted by LibreTexts, licensed under CC BY 4.0. Adapted by Ganzaa.
Photo: Tug of War - Kids Playing.jpg by Joy Agyepong, Wikimedia Commons, licensed under CC BY-SA 4.0.
Forces and Motion: Basics (version 2.6.0). Simulation by PhET Interactive Simulations, University of Colorado Boulder, licensed under CC BY 4.0 (https://phet.colorado.edu)
Vector Addition (version 1.1.0). Simulation by PhET Interactive Simulations, University of Colorado Boulder, licensed under CC BY 4.0 (https://phet.colorado.edu)
Video: Resultant Forces | Force & Motion | Physics | FuseSchool, FuseSchool - Global Education, embedded from YouTube.

The worked examples, diagrams, Rwandan applications and practice exercises are written by Ganzaa.

1Who wins the tug of war?

2Forces along one line

3Experiment: adding forces at an angle

4The parallelogram method

5Pulling together during umuganda

Who wins the tug of war?

Forces along one line

Experiment: adding forces at an angle

The parallelogram method

Pulling together during umuganda