Moments, levers and gears

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课程笔记

Forces and Rotation

Forces can cause objects to speed up, slow down, change direction, deform, or rotate.
A pivot (or fulcrum) is a fixed point that an object can rotate around.
Rotation caused by a force can be clockwise or anticlockwise.
Examples of rotation caused by a force include a child on a see-saw, turning the handle of a spanner, and a door opening and closing.
If two forces act on an object without passing through the same point, the object can still rotate.
If the two forces are equal and opposite, this is known as a couple.
Balanced forces that do not act through a common point will still cause the object to rotate.

Clockwise and anticlockwise rotation illustrated by clock faces

The Moment of a Force

A moment is the turning effect of a force about a pivot.
The size of a moment is given by the equation: M = F × d.
M = moment in newton metres (Nm).
F = force in newtons (N).
d = perpendicular distance from the pivot to the line of action of the force, in metres (m).
The moment depends on both the force and the perpendicular distance to the pivot.
A door handle is placed on the opposite side to the hinge so that the perpendicular distance from the pivot is larger, creating a larger moment for the same force.
Opening a door with a handle close to the pivot would be much harder and would require a lot more force.

A spanner turning a bolt, labelled with the pivot, the force, and the perpendicular distance from the force to the pivot.

Units and Conversions

The unit of a moment is the newton metre (N m).
If the distance is measured in centimetres, the moment can be in newton centimetres (N cm).
If the exam question doesn't ask for a specific unit, always convert the distance into metres.
Make sure all distances are in the same units before calculating.

The Principle of Moments

The principle of moments states that if an object is balanced, the total clockwise moment about a pivot equals the total anticlockwise moment about that pivot.
For a balanced horizontal beam: F₂ × d₂ = (F₁ × d₁) + (F₃ × d₃).
The forces should be perpendicular to the distance from the pivot.
On a horizontal beam, the forces that cause a moment are those directed upwards or downwards.
Clockwise is defined as the direction the hands of a clock move; anticlockwise is the opposite.

Principle of moments: a beam balanced on a pivot, with the clockwise and anticlockwise turning directions labelled on each side.

Balancing Moments: Worked Example

A parent weighing 690 N sits 0.3 m from the pivot of a see-saw; a child weighing 140 N sits on the other side.
The anticlockwise moment from the adult is 690 × 0.3 = 207 Nm.
For balance, the clockwise moment from the child must also be 207 Nm.
Using 140 × d = 207, the child must sit 1.48 m from the pivot.

Levers

Levers increase the size of a force acting on an object to make the object turn more easily.
The force applied to a lever must act further from the pivot than the force it has to overcome.
To make a lever work better, increase the size of the force applied or increase the distance of the force from the pivot.
A bottle opener is an example of a lever: a small force applied far from the pivot creates a large force at the bottle cap.
A crowbar is another type of lever used to lift heavy objects; a small downward force far from the pivot creates a large upward force on the load.
A small force applied at a large distance from a pivot can create a large moment.
Levers act as force multipliers by using moments.

Gears

Gears multiply the effect of a turning force using moments.
Gears consist of wheels with toothed edges that rotate on an axle or shaft, which acts as the pivot.
The teeth of one gear fit into the teeth of another gear, letting one gear turn the other.
As one gear turns, the other must also turn; where the gears meet, the teeth move in the same direction.
One gear moves clockwise and the other anticlockwise — they rotate in opposite directions.
Although the force is the same on both gears, the moment is not; it depends on the size of the gear, which changes the distance of the teeth to the pivot.
If a larger gear is driven by a smaller gear, the large gear rotates slower but has a greater moment (e.g. a low gear on a bike or car).
If a smaller gear is driven by a larger gear, the smaller gear rotates quicker but has a smaller moment (e.g. a high gear on a bike or car).

Key Terminology and Exam Tips

Key terms: moment, pivot, fulcrum, lever, gear, principle of moments, clockwise, anticlockwise, perpendicular distance, equilibrium.
Common error: using the distance along the object rather than the perpendicular distance from the pivot.
Common error: forgetting to convert cm to m.
Common error: believing a larger gear always turns faster — it actually turns slower but with a greater moment.
Common error: omitting one of the moments when balancing.
Common error: confusing force multiplication with energy gain — levers and gears do not create energy.

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练习题

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1.What is the unit of the moment of a force?
Easy
- Anewton metre (N m)
- Bnewton (N)
- Cjoule (J)
- Dnewton per metre (N/m)
2.The turning effect of a force about a pivot is called the moment of the force.
Easy
True or false?
3.Which of these is the correct statement of the principle of moments?
Easy
- AIf an object is balanced, the total clockwise moment about a pivot equals the total anticlockwise moment about that pivot.
- BIf an object is balanced, the total clockwise moment about a pivot is greater than the total anticlockwise moment about that pivot.
- CIf an object is balanced, the total clockwise moment about a pivot is less than the total anticlockwise moment about that pivot.
- DIf an object is balanced, the total clockwise force about a pivot equals the total anticlockwise force about that pivot.
4.Two gears are interlocked. Gear B rotates clockwise. In which direction does gear A rotate?
Easy
- Aanticlockwise
- Bclockwise
- Cit does not rotate
- Ddepends on the number of teeth
5.A force of 500 N acts at right angles to a plank, 4.1 m from a pivot. Calculate the moment of the force about the pivot.
Medium
- A2050 N m
- B205 N m
- C20.5 N m
- D0.0082 N m
6.Which of the following are examples of a force causing rotation? (select all that apply)
Medium
- AA child on a see-saw
- BTurning the handle of a spanner
- CA door opening and closing
- DA book resting on a table
- EA car moving at constant speed in a straight line
7.A student investigates moments using a rod balanced on a pivot. A magnet is fixed to one end and modelling clay to the other. The system is in equilibrium. What is the relationship between the moment of the weight of the magnet and the moment of the weight of the modelling clay about the pivot?
Medium
- AThe moment of the weight of the magnet equals the moment of the weight of the modelling clay.
- BThe moment of the weight of the magnet is greater than the moment of the weight of the modelling clay.
- CThe moment of the weight of the magnet is less than the moment of the weight of the modelling clay.
- DThe moment of the weight of the magnet is twice the moment of the weight of the modelling clay.
8.Match each term with its correct definition.
Medium
- moment
- pivot
- principle of moments
- lever
- the turning effect of a force about a pivot
- a fixed point that an object can rotate around
- if an object is balanced, total clockwise moment equals total anticlockwise moment
- a simple machine that increases the size of a force to make an object turn more easily

更多题目

9.A spanner is used to tighten a nut. A force F is applied at the midpoint of the spanner, 12 cm from the centre of the nut. Calculate the force needed to produce a moment of 6.0 N m about the centre of the nut.
Medium
- A50 N
- B0.72 N
- C72 N
- D500 N
10.Place the steps in the correct order to calculate the distance of a child from the pivot so that a see-saw balances. The adult weighs 690 N and sits 0.3 m from the pivot; the child weighs 140 N.
Medium
- Calculate the anticlockwise moment of the adult: 690 × 0.3 = 207 N m
- Write the principle of moments: total clockwise moment = total anticlockwise moment
- Substitute into the equation: 140 × d = 207
- Rearrange for d: d = 207 ÷ 140 = 1.48 m
11.A lever is used to lift a heavy load. The weight of the load is 650 N and its centre is 0.75 m from the pivot. Calculate the moment of the load about the pivot.
Medium
- A487.5 N m
- B866.7 N m
- C48.75 N m
- D4.875 N m
12.An effort of 160 N is applied to the end of a lever to balance a load that produces a moment of 487.5 N m. Calculate the distance between the effort and the pivot.
Medium
- A3.05 m
- B0.33 m
- C78 000 m
- D30.5 m
13.A larger gear always turns faster than a smaller gear that drives it.
Easy
True or false?
14.Which of these is the correct equation for the moment of a force?
Medium
- AM = F × d
- BM = F ÷ d
- CM = d ÷ F
- DM = F + d
15.A rod can rotate around an axle. Four forces, P, Q, R and S, act on the rod as shown. Which force will make the rod rotate about the axle?
Medium
- AP
- BQ
- CR
- DS
16.A person tries to lift a rock using a metal bar. The rock weighs 1800 N and the person can produce a downwards force of 600 N. The person cannot lift the rock. Which change to the arrangement would make it possible for this person to lift the rock?
Medium
- AMove the pivot closer to the rock so the effort acts further from the pivot.
- BMove the pivot closer to the person so the effort acts closer to the pivot.
- CUse a shorter metal bar.
- DApply the effort closer to the pivot.
17.Two interlocking gear wheels, P and Q, are shown. P has 20 teeth and rotates clockwise. Q has 40 teeth. Which of these describes how Q moves?
Medium
- Aanticlockwise and slower than gear P
- Bclockwise and faster than gear P
- Cclockwise and slower than gear P
- Danticlockwise and faster than gear P
18.Which of these gives the ratio of the number of teeth on gear P to the number of teeth on gear Q? Gear P has 20 teeth and gear Q has 40 teeth.
Medium
- A1 : 2
- B2 : 1
- C1 : 4
- D4 : 1
19.Gear X is rotated clockwise at 1.0 rotations per second. Gear X has 20 teeth and gear Y has 40 teeth. Which row describes the movement of gear Y?
Medium
- Aanticlockwise, 0.5 rotations per second
- Banticlockwise, 2.0 rotations per second
- Cclockwise, 0.5 rotations per second
- Dclockwise, 2.0 rotations per second
20.Which of these statements about the moment of a force are correct? (select all that apply)
Medium
- AThe moment is measured in newton metres (N m)
- BThe moment depends on the perpendicular distance from the pivot to the line of action of the force
- CDoubling the distance from the pivot halves the moment for the same force
- DThe moment of a force is the turning effect of the force about a pivot
- EA moment can be measured in newtons (N)
21.Match each term with its correct description.
Medium
- Pivot
- Perpendicular distance
- Couple
- Equilibrium
- The fixed point that an object rotates around
- The shortest distance from the pivot to the line of action of the force
- Two equal and opposite forces that do not act through the same point
- The state of a system when the total clockwise moment equals the total anticlockwise moment
22.Place these steps in the correct order to find the unknown distance d at which a 200 N load must sit on the left of a pivot to balance a 500 N load sitting 0.4 m to the right of the pivot.
Medium
- State the principle of moments: total anticlockwise moment = total clockwise moment
- Calculate the clockwise moment: 500 × 0.4 = 200 N m
- Write the anticlockwise moment as 200 × d
- Set 200 × d equal to 200 N m
- Solve to get d = 1.0 m
23.A lever is being used to lift a heavy load. Which of these changes would increase the moment produced by the effort? (select all that apply)
Medium
- AIncreasing the size of the effort force
- BMoving the effort further from the pivot
- CMoving the effort closer to the pivot
- DReducing the size of the effort force
- EMaking the effort act at right angles to the lever at a greater distance from the pivot
24.Match each gear situation with the correct outcome.
Medium
- A smaller gear drives a larger gear
- A larger gear drives a smaller gear
- Two gears meshing together
- The teeth of two meshing gears at the point where they meet
- The larger gear turns slower but with a greater moment
- The smaller gear turns faster but with a smaller moment
- They rotate in opposite directions
- They move in the same direction
25.When a smaller gear drives a larger gear, the larger gear rotates more slowly than the smaller gear but has a greater moment.
Easy
True or false?
26.Which arrow correctly shows the force that will cause the biggest rotation of the rod about the axle?
Medium
- AAn arrow at the far end of the rod, perpendicular to the rod
- BAn arrow at the far end of the rod, along the rod
- CAn arrow close to the axle, perpendicular to the rod
- DAn arrow close to the axle, along the rod
27.State the direction of gear C during this motion.
Medium
- Aclockwise
- Banticlockwise
- Cupwards
- Ddownwards
28.A student reverses the direction of the current in the coil. Describe how the student can bring the system back into equilibrium without making any changes to the magnet.
Medium
- AMove the hanging weight to the opposite side of the pivot
- BIncrease the size of the hanging weight
- CMove the hanging weight closer to the pivot
- DReverse the direction of the current again
29.Which row describes the movement of gear Y?
Medium
- Adirection of rotation: anticlockwise, rotations per second: 0.5
- Bdirection of rotation: anticlockwise, rotations per second: 2.0
- Cdirection of rotation: clockwise, rotations per second: 0.5
- Ddirection of rotation: clockwise, rotations per second: 2.0
30.Which of these gives the ratio of the number of teeth on gear X to the number of teeth on gear Y?
Medium
- A1 : 2
- B2 : 1
- C1 : 4
- D4 : 1
31.A student builds a gear system with three interlocking gears in a line: gear A (12 teeth), gear B (36 teeth) and gear C (12 teeth). Gear A is turned clockwise. Which statement describes the motion of gear C?
Medium
- AGear C rotates clockwise at the same speed as gear A
- BGear C rotates anticlockwise at one third the speed of gear A
- CGear C rotates clockwise at one third the speed of gear A
- DGear C rotates anticlockwise at the same speed as gear A
32.A uniform metre rule is balanced on a pivot at the 50 cm mark. A 2.0 N weight is hung at the 20 cm mark. Which of the following statements are correct? (select all that apply)
Hard
- AThe moment of the 2.0 N weight about the pivot is 0.60 N m
- BTo balance the rule, a 2.0 N weight must be hung at the 80 cm mark
- CTo balance the rule, a 4.0 N weight must be hung at the 65 cm mark
- DThe rule will rotate anticlockwise
- EThe total clockwise moment must equal the total anticlockwise moment
33.When a smaller gear drives a larger gear, the larger gear rotates more slowly and has a greater moment.
Easy
True or false?
34.Match each situation with the correct effect on the moment.
Medium
- Doubling the force applied at the same distance
- Moving the force closer to the pivot
- Increasing the perpendicular distance of the force from the pivot
- Applying the force at an angle so it is no longer perpendicular
- moment doubles
- moment decreases
- moment increases
- effective perpendicular distance is reduced
35.Place the steps in the correct order to calculate the force needed at the end of a lever to balance a load. The load is 300 N at 0.20 m from the pivot, and the effort is applied 0.60 m from the pivot.
Medium
- Calculate the moment of the load: 300 N × 0.20 m = 60 N m
- State the principle of moments: total clockwise moment = total anticlockwise moment
- Set the effort moment equal to the load moment: F × 0.60 m = 60 N m
- Rearrange to find F: F = 60 N m ÷ 0.60 m = 100 N
36.A student investigates how the moment of a force depends on distance using a rod pivoted at its centre, hanging weights at different distances. Which of these are correct improvements to the method? (select all that apply)
Hard
- AMeasure the perpendicular distance from the pivot to the line of action of the force each time
- BUse a set square to check that the force acts at right angles to the rod
- CRepeat each reading and calculate a mean
- DRecord the distance in centimetres and the moment in newton metres without converting
- EChange the mass of the rod between readings
37.If a force acts at right angles to a spanner, moving the force further from the nut increases the moment produced.
Easy
True or false?
38.Match each practical situation with the correct effect on the moment about the pivot.
Medium
- Force doubled, distance unchanged
- Distance halved, force unchanged
- Force halved and distance doubled
- Force acts along the line through the pivot
- Moment doubles
- Moment halves
- Moment stays the same
- Moment is zero
39.A student finds that a lever lifts a 600 N load using only a 150 N effort. Which statement about this lever is correct?
Hard
- AThe lever multiplies the force but the work done by the effort is still at least as large as the work done on the load
- BThe lever creates extra energy so more work is done on the load than by the effort
- CThe lever multiplies both the force and the energy transferred
- DThe lever reduces the total moment needed to lift the load
40.State the equation for the moment of a force.
Easy
41.Gear A has 10 teeth and gear B has 50 teeth. How many revolutions does gear A make after 1 revolution of gear B?
Medium
42.Complete the sentence about the moment of a force.
Medium
The moment of a force is the turning effect of a force about a ____, and it is calculated using the equation M = F × ____, where d is the ____ distance from the pivot to the line of action of the force.
43.Complete the sentence about gears.
Easy
Two interlocking gears rotate in ____ directions. If a larger gear is driven by a smaller gear, the larger gear rotates ____ than the smaller gear but has a greater ____.
44.A bicycle has a large gear wheel with 48 teeth and a small gear wheel with 12 teeth. The large gear wheel turns 2 times each second. Calculate the number of times the small gear wheel turns each second.
Medium
45.A force of 250 N acts at right angles to a lever, 0.60 m from the pivot. Calculate the moment of the force about the pivot.
Medium
46.A see-saw is balanced. An adult weighing 690 N sits 0.30 m from the pivot. Calculate the distance from the pivot at which a child weighing 140 N must sit to balance the see-saw. Give your answer to 2 significant figures.
Hard
47.Complete the sentence about the principle of moments.
Medium
If an object is balanced, the total ____ moment about a pivot equals the total ____ moment about that pivot.
48.A bottle opener is an example of a lever. State the name given to a lever that increases the size of the force applied to it.
Medium
49.Calculate the size of force F, in newtons, that produces the maximum moment of 3.6 N m on the 12 cm rod.
Medium
50.After 1 complete revolution of B, determine the distance moved by gear C.
Medium
51.Calculate the size of the force between the magnet and the coil.
Medium
52.Explain why a door handle is placed on the opposite side of the door to the hinge.
Hard
53.Explain how a lever acts as a force multiplier.
Hard
54.Explain how a crowbar acts as a force multiplier when lifting a heavy object.
Hard
55.Describe how the student could develop the investigation to determine if the size of the force between the magnet and the coil is directly proportional to the size of the current in the coil.
Hard
56.Explain, using calculations, why the person cannot lift the rock.
Hard
57.Gear A has 10 teeth and gear B has 50 teeth. How many revolutions does gear A make after 1 revolution of gear B?
Medium
- A7 revolutions
- B5 revolutions
- C4 revolutions
- D6 revolutions
58.A bicycle has a large gear wheel with 48 teeth and a small gear wheel with 12 teeth. The large gear wheel turns 2 times each second. Calculate the number of times the small gear wheel turns each second.
Medium
- A10 turns per second
- B7 turns per second
- C9 turns per second
- D8 turns per second
59.A force of 250 N acts at right angles to a lever, 0.60 m from the pivot. Calculate the moment of the force about the pivot.
Medium
- A150 N m
- B135 N m
- C180 N m
- D165 N m
60.A see-saw is balanced. An adult weighing 690 N sits 0.30 m from the pivot. Calculate the distance from the pivot at which a child weighing 140 N must sit to balance the see-saw. Give your answer to 2 significant figures.
Hard
- A2.5 m
- B0.5 m
- C1.5 m
- D3.5 m
61.Calculate the size of force F, in newtons, that produces the maximum moment of 3.6 N m on the 12 cm rod.
Medium
- A36 N
- B30 N
- C27 N
- D33 N
62.After 1 complete revolution of B, determine the distance moved by gear C.
Medium
- A3 mm
- B5 mm
- C2 mm
- D4 mm

闪卡

What is the moment of a force?

The turning effect of a force about a pivot.

State the equation for the moment of a force.

M = F × d, where M is the moment in newton metres (Nm), F is the force in newtons (N), and d is the perpendicular distance from the pivot in metres (m).

What is the unit of a moment?

Newton metre (Nm), or newton centimetre (N cm) if the distance is in centimetres.

What is the principle of moments?

If an object is balanced, the total clockwise moment about a pivot equals the total anticlockwise moment about that pivot.

What is a pivot?

A fixed point that an object can rotate around.

Give three examples of forces causing rotation.

A child on a see-saw, turning the handle of a spanner, and a door opening or closing.

What is a couple?

Two equal and opposite forces that act on an object without passing through the same point, causing rotation.

Why is a door handle placed far from the hinge?

To increase the perpendicular distance from the pivot (the hinge), so a given force creates a larger moment, making the door easier to open.

How does a lever act as a force multiplier?

A small force applied far from the pivot creates a large moment, which can overcome a larger force closer to the pivot.

How can you make a lever work better?

Increase the size of the force applied, or increase the distance of the force from the pivot.

Give an example of a lever used as a force multiplier.

A bottle opener or a crowbar.

How do gears transmit rotational effects?

The teeth of one gear fit into the teeth of another, so one gear turns the other, allowing one axle or shaft to turn another.

In a pair of meshing gears, what is the relationship between their directions of rotation?

They rotate in opposite directions: if one turns clockwise, the other turns anticlockwise.

If a larger gear is driven by a smaller gear, how does the larger gear move?

The larger gear rotates slower than the smaller gear but has a greater moment.

If a smaller gear is driven by a larger gear, how does the smaller gear move?

The smaller gear rotates quicker than the larger gear but has a smaller moment.

Why does a larger gear have a greater moment than a smaller gear?

The turning force on the larger gear acts further from its pivot (axle), so it creates a larger moment.

A force of 500 N acts at right angles to a plank, 4.1 m from the pivot. Calculate the moment.

Moment = 500 N × 4.1 m = 2050 N m.

A moment of 3.6 N m is produced by a force at a perpendicular distance of 0.12 m. Calculate the force.

F = M ÷ d = 3.6 N m ÷ 0.12 m = 30 N.

A force of 160 N balances a load with a moment of 487.5 N m. Calculate the distance between the effort and the pivot.

d = M ÷ F = 487.5 N m ÷ 160 N ≈ 3.05 m.

A spanner is 12 cm from the centre of a nut. What force is needed to produce a moment of 6.0 N m?

F = M ÷ d = 6.0 N m ÷ 0.12 m = 50 N.

How can the same spanner be used to loosen a nut that requires a moment of 12.0 N m?

Apply the force further from the pivot (e.g. at the end of the spanner) to increase the perpendicular distance, or increase the force applied.

What happens to the moment if the force moves closer to the pivot?

The moment decreases because the perpendicular distance from the pivot is smaller.

Gear A has 10 teeth and gear B has 50 teeth. How many revolutions does gear A make for 1 revolution of gear B?

Gear A makes 5 revolutions (50 ÷ 10 = 5).

A large gear wheel has 48 teeth and turns 2 times per second. The small gear wheel has 12 teeth. How many times does the small gear wheel turn each second?

The small gear wheel turns 8 times per second (48 ÷ 12 = 4; 4 × 2 = 8).

What is the ratio of the number of teeth on gear P (24 teeth) to gear Q (16 teeth)?

3 : 2.

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