The shape of the lesson
By the end of the lesson, learners can
- The washing-line surprise: A weight hangs from two strings.
- Two conditions, one state: A neat trick: you may take moments about any point.
- A method that always works.
- Forces balance: total up = total down, total left = total right.
- Moments balance: clockwise = anticlockwise about any point.
- Smart pivot: take moments about an unknown force to remove it.
One picture every idea returns to
A body is in equilibrium when two conditions both hold: the resultant force is zero (in every direction), and the resultant moment about any point is zero. The principle of moments is the second condition applied to a pivot: the total clockwise moment equals the total anticlockwise moment. For exactly three coplanar forces in equilibrium, drawing them nose to tail produces a closed vector triangle.
Forty-five minutes, phase by phase
| Time | Phase | What happens in the room | Grouping |
|---|---|---|---|
| 0 to 5 min | Hook: predict | Open with the question in the title, Going nowhere, turning nowhere.. A body in equilibrium is not just standing still in one sense but in two: it has no tendency to accelerate and no tendency to rotate. Put the figure on the board: it is the one picture every idea in the lesson returns to. | Whole class, then pairs |
| 5 to 13 min | Teach and check: The washing-line surprise | A weight hangs from two strings. Project tab 2 of the lab (Equilibrium of three forces) and let a student drive it while the class predicts. | Whole class, then pairs |
| 13 to 21 min | Teach and check: Two conditions, one state | A neat trick: you may take moments about any point. | Pairs |
| 21 to 30 min | Teach and check: A method that always works | A method that always works. | Connect-Extend-Challenge |
| 30 to 36 min | Check | The four Stage 1 questions from the student lesson, answered on mini whiteboards or by show of fingers. Re-teach the one most get wrong. | Individual, then whole class |
| 36 to 43 min | Plenary | The exam-style questions from Stage 2, one at a time: thirty seconds alone, then a random call. The feedback lines in the key give the examiner's wording. | Individual, random call |
| 43 to 45 min | Exit and homework | Set the worksheet below (6 questions, 16 marks). Learners self-assess against the objectives before they leave. | Individual |
Protect the parts that carry the learning
The sequence carries about fifty minutes of material, so the lab is driven by one student while the class predicts rather than explored freely. Times are a guide.
Protected: the check at 30 minutes. It is the only individual read on whether the key idea landed, and it is not cut.
Cut first if the build overruns: the last check in A method that always works, never the check or the plenary.
In a 60 minute block: run worksheet questions 1 to 3 in class as a Rally Coach set and model both the hardest worked example and its check live.
Connect-Extend-Challenge
After the new idea, each student writes three lines: how it connects to what they knew, how it extends their thinking, and what still challenges them. Pairs compare, then the challenges become the plenary.
The full structure, with its variations and the other twelve, is on the cooperative-learning page.
What to head off, and how
| Trap learners fall into | Teaching move that pre-empts it |
|---|---|
| Equilibrium needs both conditions. | A pair of equal and opposite forces that are not in line (a couple) has no resultant force, yet it still turns the body, so it is not in equilibrium. Conversely, balanced moments about one point do not guarantee equilibrium unless the forces also balance. When taking moments, always use the perpendicular distance, and remember you can choose the most convenient pivot. |
Equilibrium of forces: worksheet
6 questions, 16 marks. Written in the style of the papers; no past paper question is reproduced.
State the two conditions that must both be satisfied for a body to be in equilibrium under a system of coplanar forces.
A uniform metre rule of weight 1.2 N is pivoted at the 30 cm mark. A weight is hung at the 10 cm mark to balance it. Find the weight needed. (The rule's weight acts at the 50 cm mark.)
A uniform beam of weight 80 N and length 6.0 m rests horizontally on two supports, one at each end. A 120 N load is placed 2.0 m from the left support. By taking moments about the left support, find the upward force at the right support.
A 50 N picture hangs from a single nail by two wires that each make 50° with the horizontal. Find the tension in each wire.
Explain why a pair of equal and opposite forces that are not in the same line is not in equilibrium, even though the resultant force is zero.
A street lamp of weight 200 N is held out from a wall by a horizontal rod, with a wire from the wall to the end of the rod making 35° above the rod. Treating the lamp as hanging from the end of the rod, find the tension in the wire. (The wire's vertical component supports the lamp.)
Answer key and marking notes
- The resultant force in any direction is zero. ✓
- The resultant moment about any point is zero. ✓
- The rule's weight acts at 50 cm, which is 20 cm to the right of the pivot: clockwise moment = 1.2 × 0.20 = 0.24 N m. ✓
- The hung weight is at 10 cm, 20 cm to the left: anticlockwise moment = W × 0.20. ✓
- Balance: W × 0.20 = 0.24, so W = 1.2 N. ✓
- Moments about the left support: R × 6.0 = (80 × 3.0) + (120 × 2.0). ✓
- R × 6.0 = 240 + 240 = 480, so R = 80 N. ✓
- (The left support then carries 80 + 120 − 80 = 120 N.) ✓
- Vertical balance: 2T sin 50° = 50. ✓
- 2T × 0.766 = 50, so 1.532T = 50. ✓
- T = 33 N in each wire. ✓
- The resultant force is zero, so the first condition is met. ✓
- But the forces form a couple with a non-zero resultant moment, so there is a turning effect and the second condition fails: it is not in equilibrium. ✓
- The wire's vertical component must balance the lamp's weight: T sin 35° = 200. ✓
- T × 0.574 = 200, so T = 200 / 0.574. ✓
- T = 350 N (to 2 significant figures). ✓
17 multiple-choice questions on this lesson
From the topic question bank. Assign them in the app for a marked score in your gradebook, or run the topic as a class game.
A body is in equilibrium only if:
- Athe resultant force is zero only
- Bit is stationary
- Cthe resultant force is zero and the resultant moment is zero
- Dthe resultant moment is zero only
The principle of moments states that, in equilibrium, about any point the total clockwise moment equals the:
- Atotal force
- Btotal anticlockwise moment
- Ctotal weight
- Dtotal upward force
If three coplanar forces keep an object in equilibrium, drawing them nose to tail gives:
- Aa straight line
- Ban open V shape
- Cthree parallel arrows
- Da closed triangle
A couple acts on a free body. Is the body in equilibrium?
- Ano, because there is a resultant moment
- Byes, because the forces cancel
- Cyes, because there is no resultant force
- Donly if the forces are large
A uniform beam of weight 60 N rests on two supports, one at each end. The force on each support is:
- A60 N
- B15 N
- C30 N
- D120 N
A light rod is pivoted at one end. A downward 24 N force acts 0.10 m from the pivot. An upward force F at 0.30 m balances the rod. F is:
- A72 N
- B8.0 N
- C24 N
- D2.4 N
When solving an equilibrium problem, moments may be taken about:
- Aonly the centre of gravity
- Bonly a support
- Conly the midpoint
- Dany point you choose
A non-uniform plank balances on a single pivot. This shows that:
- Aits centre of gravity is directly above the pivot
- Bit is uniform
- Cthe pivot is at its midpoint
- Dits weight is zero
A 40 N weight hangs from two strings that each make 45 degrees with the horizontal. The tension in each string is:
- A20 N
- B40 N
- C28 N
- D57 N
A sign of weight 80 N hangs from the middle of a horizontal wire fixed at both ends. As the wire is pulled tighter and straighter, the tension:
- Adecreases
- Bincreases
- Cstays at 40 N
- Dbecomes zero
The single force that would hold a system in equilibrium is equal in size and opposite in direction to the:
- Alargest single force
- Bweight only
- Cnormal contact force
- Dresultant of the other forces
Three non-parallel coplanar forces in equilibrium must:
- Apass through a single point
- Bbe equal in size
- Cbe perpendicular
- Dall point upward
A uniform beam of weight 100 N and length 4.0 m is supported at its left end and by a vertical string at its right end. The tension in the string is:
- A100 N
- B25 N
- C50 N
- D200 N
For a body in equilibrium under several forces, resolving shows that:
- Aonly the vertical forces matter
- Bthe forces balance in two perpendicular directions
- Conly the horizontal forces matter
- Dthe forces must be equal
A uniform plank of weight 50 N and length 2.0 m is pivoted 0.80 m from the left end. A child stands on the left end. The childs weight needed to balance the plank is:
- A50 N
- B25 N
- C40 N
- D12.5 N
Taking moments about a pivot is convenient because it:
- Aremoves the unknown force at the pivot from the equation
- Bdoubles the moments
- Cignores the weight
- Dmakes all forces equal
A body in equilibrium:
- Amust be at rest
- Bmust be accelerating
- Cmay be moving at constant velocity
- Dmust have zero weight
Quiz key
Equilibrium needs zero resultant force (no acceleration) and zero resultant moment (no rotation).
Sum of clockwise moments = sum of anticlockwise moments about any chosen point.
A zero resultant means the three vectors close into a triangle when drawn nose to tail.
Although the resultant force is zero, the couple gives a resultant moment, so the body is not in equilibrium.
By symmetry the weight is shared equally, so each support provides 30 N.
24 x 0.10 = F x 0.30, so F = 2.4 / 0.30 = 8.0 N.
In equilibrium the resultant moment about every point is zero, so you may take moments about any convenient point.
A body balances on a single pivot only when its centre of gravity is directly above the pivot.
2T sin 45 = 40, so 2T x 0.707 = 40, giving T = 28 N.
The flatter the wire, the smaller the angle to the horizontal, so 2T sin theta = 80 requires a larger T.
The equilibrant is equal and opposite to the resultant of all the other forces.
For three non-parallel forces in equilibrium, their lines of action must meet at one point, otherwise there is a net moment.
Taking moments about the left support: T x 4.0 = 100 x 2.0, so T = 50 N.
Zero resultant force means the components balance both horizontally and vertically.
The planks weight acts at its midpoint 1.0 m, which is 0.20 m to the right of the pivot, giving a moment of 50 x 0.20 = 10 N m. The child at the left end is 0.80 m from the pivot, so W x 0.80 = 10, giving W = 12.5 N.
The reaction at the pivot has zero moment about the pivot, so taking moments there removes that unknown.
Equilibrium means zero resultant force, so zero acceleration. The body may be at rest or moving at constant velocity.
What to project
The student lesson embeds these tabs. Open them full screen on the projector; every slider has a drag handle on the stage.
Slides for this lesson
A projectable deck for every pack is on the way. Until then, present the student lesson full screen: its sections, figure and lab tabs follow the same order as the plan above.