Equilibrium of a rigid body
Introduced| English | Meaning |
|---|---|
| turning effect/ˈtɜːnɪŋ ɪˈfekt/ | The tendency of a force to cause rotation about an axis or pivot, measured by its moment or torque. |
| moment/ˈməʊmənt/ | The turning effect of a force about a pivot. |
| perpendicular/ˌpɜːpənˈdɪkjʊlə/ | Meeting at a right angle of 90 degrees. |
| equilibrium/ˌiːkwɪˈlɪbrɪəm/ | A state with no resultant force or resultant moment. |
| pivot/ˈpɪvət/ | A fixed point about which an object can turn. |
| rigid body/ˈrɪdʒɪd ˈbɒdi/ | An idealised object whose shape and distances between its particles do not change. |
| resultant force/rɪˈzʌltənt fɔːs/ | The single force equivalent to all the forces acting together. |
| centre of mass/ˈsentə ɒv mæs/ | The point representing an object's or system's mass-weighted average position. |
| topple/ˈtɒpl/ | topple |
| limiting friction/ˈlɪmɪtɪŋ ˈfrɪkʃn/ | limiting friction |
Why a heavy door is easy to open
- Push near the hinge and a door barely moves; push at the handle and it swings easily.
- Same force, different turning effect 转动效应 — the moment 力矩. It's what keeps beams, bridges and see-saws balanced.
The moment of a force about a point is:
Difficulty 1/5 · 100 coins + a small bonus on first completion today
Moment = force × perpendicular distance — the turning effect.
Moments
- The moment of a force about a point = force × perpendicular 垂直 distance from the point.
- It measures the turning effect; a bigger distance means a bigger turn.
Equilibrium 平衡 of a beam: take moments about the pivot 支点
Equilibrium of forces
resultant = 0
A body is in equilibrium when the forces add tip-to-tail back to zero.
For a rigid body in equilibrium, both the resultant force and the total moment must be zero.
Difficulty 1/5 · 100 coins + a small bonus on first completion today
Force balance stops translation; moment balance stops rotation.
For a uniform beam, the weight is taken to act at the ______ of mass.
Difficulty 1/5 · 100 coins + a small bonus on first completion today
A uniform body's weight acts at its centre of mass (its midpoint).
Equilibrium of a rigid body 刚体
- A rigid body is in equilibrium when both conditions hold:
- the resultant force 合力 is zero (it won't accelerate), and
- the total moment about any point is zero (it won't rotate).
- Weight acts at the centre of mass 质心.
Taking moments about the pivot A removes its (unknown) reaction from the equation.
A uniform 4 m beam of weight 100 N is pivoted at A; a force F at B (4 m away) balances it. Taking moments about A (weight acts at the 2 m centre), find F (N).
Difficulty 2/5 · 140 coins + a small bonus on first completion today
F × 4 = 100 × 2, so F = 200/4 = 50 N.
Worked example — a beam
- A uniform 4 m beam of weight $100$ N is pivoted at A; a force $F$ at B (4 m away) holds it level.
- Weight acts at the centre, 2 m from A. Moments about A: $F \times 4 = 100 \times 2$, so $F = 50$ N.
Tip — choose the pivot wisely. Taking moments about a point where an unknown force acts makes that force's moment zero, so it drops out of the equation.
A body topples when the line of action of its weight falls:
Difficulty 1/5 · 100 coins + a small bonus on first completion today
Once the weight line passes outside the base, the moment tips it over.
Toppling 倾倒 and sliding
- On a slope or with a horizontal push, a body may topple (rotate about an edge) or slide (slip).
- It topples when the line of its weight falls outside the base; it slides when the push exceeds limiting friction 最大静摩擦.
- Find the centre of mass of a lamina or a composite body; a rigid body under coplanar forces is in equilibrium when forces and moments both sum to zero.
You've got it
- moment = force × perpendicular distance
- equilibrium: resultant force zero and total moment zero
- weight acts at the centre of mass; take moments about an unknown to remove it