Chapter in a nutshell: A machine makes work easier by changing the size, direction or point of application of a force. Its performance is measured by mechanical advantage (MA), velocity ratio (VR) and efficiency (ฮท = MA/VR). Levers (3 classes) and pulleys are the key examples. No machine is 100 % efficient because of friction and the weight of its parts.
1. What a Machine Does
A machine is a device by which we can either overcome a large resistive force (load) by a small applied force (effort), or change the direction/point of application of a force. Four functions:- Force multiplier โ a small effort lifts a large load (e.g. jack, pulley system, crowbar).
- Change the direction of force โ apply effort in a convenient direction (single fixed pulley).
- Speed multiplier โ a small movement of effort gives a large movement of load (e.g. bicycle, our forearm).
- Transfer force to a convenient point (e.g. bicycle chain).
2. Mechanical Advantage, Velocity Ratio, Efficiency
| Quantity | Definition | Formula |
|---|---|---|
| Mechanical Advantage (MA) | ratio of load to effort | $\text{MA}=\dfrac{\text{Load}}{\text{Effort}}=\dfrac{L}{E}$ (no unit) |
| Velocity Ratio (VR) | ratio of effort-distance to load-distance | $\text{VR}=\dfrac{d_E}{d_L}$ (no unit) |
| Efficiency (ฮท) | ratio of useful work output to work input | $\eta=\dfrac{W_{out}}{W_{in}}=\dfrac{L\times d_L}{E\times d_E}=\dfrac{\text{MA}}{\text{VR}}$ |
- Relation: $\eta=\dfrac{\text{MA}}{\text{VR}}$ โ for an ideal machine $\eta=1$ (100 %), so MA = VR.
- For a real machine $\eta<1$ (MA < VR) because of (i) friction between moving parts and (ii) the weight of moving parts (string, pulley blocks).
- VR depends only on the geometry of the machine (fixed); MA depends on friction โ as friction rises, MA (and ฮท) fall while VR stays the same.
- A machine is a force multiplier when MA > 1, a speed multiplier when MA < 1.
3. Lever โ Principle
A lever is a rigid bar capable of turning about a fixed point called the fulcrum. It works on the principle of moments: $$\text{Load} \times \text{load arm} = \text{Effort} \times \text{effort arm}$$ $$\Rightarrow \text{MA}=\frac{L}{E}=\frac{\text{effort arm}}{\text{load arm}}$$ So MA > 1 when the effort arm is longer than the load arm.4. Three Classes of Levers
| Class | Middle component | MA | Examples |
|---|---|---|---|
| Class I | Fulcrum between load & effort | >1, =1 or <1 | seesaw, scissors, pliers, crowbar, beam balance, claw hammer |
| Class II | Load between fulcrum & effort | always > 1 | wheelbarrow, nutcracker, bottle opener, lemon squeezer |
| Class III | Effort between fulcrum & load | always < 1 (speed/range gain) | forceps, tongs, fishing rod, human forearm, sugar tongs |
- Class I can be a force or speed multiplier depending on arm lengths.
- Class II (load arm < effort arm) is always a force multiplier.
- Class III (effort arm < load arm) sacrifices force for larger/faster movement of the load.
5. Pulleys
A pulley is a wheel with a grooved rim over which a string passes.| System | VR | Ideal MA | Effort (ideal) | Use |
|---|---|---|---|---|
| Single fixed | 1 | 1 | $E=L$ | only changes direction (drawing water, flag) |
| Single movable | 2 | 2 | $E=\tfrac{L}{2}$ | force multiplier; effort upward |
| Block & tackle (n strands) | n | n | $E=\tfrac{L}{n}$ | lifting heavy loads (cranes) |
- Single fixed pulley: $\text{VR}=1$; useful because effort can be applied downward (more convenient) though MA โ 1.
- Single movable pulley: the load is shared by two segments of string โ $\text{VR}=2$. Considering the weight w of the pulley, effort $E=\dfrac{L+w}{2}$.
- Block and tackle: with n segments of string supporting the load, $\text{VR}=n$. Accounting for friction and the weight of the lower block, $\eta=\dfrac{\text{MA}}{n}<1$.
6. Other Machines
| Machine | Velocity Ratio |
|---|---|
| Inclined plane | $\text{VR}=\dfrac{l}{h}$ (length รท height) |
| Wheel & axle | $\text{VR}=\dfrac{R}{r}$ (radii of wheel & axle) |
| Gears | ratio of number of teeth |
| Screw jack | $\dfrac{2\pi l}{p}$ (l = arm, p = pitch) |
7. Worked Numerical Examples (ICSE pattern)
Q1. A lever has a load of 200 N at 20 cm from the fulcrum; effort acts at 80 cm. Find the effort and MA. Solution: $L\times d_L=E\times d_E \Rightarrow 200\times20 = E\times80 \Rightarrow E=\mathbf{50\ N}$. $\text{MA}=200/50=\mathbf{4}$.Q2. A machine has VR = 4 and efficiency 80 %. Find its MA. Solution: $\eta=\dfrac{\text{MA}}{\text{VR}} \Rightarrow \text{MA}=\eta\times\text{VR}=0.8\times4=\mathbf{3.2}$.
Q3. In a block-and-tackle with 5 pulleys, an effort of 120 N lifts a 500 N load. Find MA, VR and efficiency. Solution: $\text{MA}=500/120=\mathbf{4.17}$; $\text{VR}=5$; $\eta=\dfrac{4.17}{5}=0.83=\mathbf{83\%}$.
Q4. A single movable pulley (pulley weight 5 N) lifts a 95 N load. Find the effort (ideal string). Solution: $E=\dfrac{L+w}{2}=\dfrac{95+5}{2}=\mathbf{50\ N}$.
Q5. A 2.5 m inclined plane is used to raise a load to a height of 0.5 m. Find the VR; if the load is 600 N and effort 150 N, find efficiency. Solution: $\text{VR}=l/h=2.5/0.5=5$; $\text{MA}=600/150=4$; $\eta=4/5=\mathbf{80\%}$.
Q6. Why is the efficiency of a single fixed pulley not 100 %? Solution: Because some effort is wasted against friction at the axle and in bending the (slightly stiff) string.
8. Key Terms โ Quick Glossary
| Term | One-line definition |
|---|---|
| Machine | device that makes work easier (force/direction/speed). |
| Load | the resistance/weight overcome by the machine. |
| Effort | the force applied to operate the machine. |
| Mechanical Advantage | Load รท Effort. |
| Velocity Ratio | effort-distance รท load-distance (geometry only). |
| Efficiency | useful output รท input = MA/VR (always < 1 in practice). |
| Lever | rigid bar turning about a fulcrum (principle of moments). |
| Block & tackle | pulley system with n strands; VR = n. |
9. Common Mistakes to Avoid
- Confusing MA (depends on friction) with VR (geometry only) โ they are equal only for an ideal machine.
- Thinking efficiency can exceed 100 % โ it never can.
- Identifying lever class by the wrong component โ always look at the middle one (Fulcrum/Load/Effort).
- Forgetting the weight of the pulley in single-movable-pulley numericals.
- Assuming a single fixed pulley gives a force advantage โ it only changes direction (MA โ 1).
10. Likely Exam Questions (with crisp answers)
- Define MA, VR and efficiency. โ L/E; effort-dist/load-dist; MA/VR.
- Why is a machine's efficiency always less than 100 %? โ Friction and the weight of moving parts waste some input energy.
- On what does VR depend? โ Only on the geometry/construction of the machine, not on friction.
- Give an example each of class I, II, III levers. โ Seesaw; wheelbarrow; forceps.
- Which class of lever always has MA > 1? Why? โ Class II โ its load arm is always shorter than the effort arm.
- Why is the human forearm a class III lever โ what is gained? โ Effort (muscle) is between fulcrum (elbow) and load (hand); we gain speed/range at the cost of force.
- What is the VR of a single movable pulley? โ 2.
- State the use of a single fixed pulley. โ To change the direction of the effort (apply it downward).
- In a block-and-tackle with n strands, what are the ideal VR and effort? โ VR = n, effort = Load/n.
- How can the MA of a lever be increased? โ By increasing the length of the effort arm relative to the load arm.
- What is the relation between MA, VR and efficiency? โ ฮท = MA/VR.
- Name a machine that is a speed multiplier. โ Bicycle / a class III lever (forearm).
11. How the Velocity Ratios Arise (reasoning)
- Single fixed pulley: when the load rises by 1 m, the effort end of the string also moves 1 m โ $d_E=d_L$ โ VR = 1. It cannot multiply force; it only redirects it.
- Single movable pulley: the load hangs from two segments of string. To raise the load by 1 m, both segments must shorten by 1 m, so the effort end moves 2 m โ $d_E=2d_L$ โ VR = 2.
- Block and tackle (n strands): the load is supported by n string segments; raising it 1 m needs the effort to pull n metres โ VR = n. More pulleys โ larger VR โ smaller effort.
12. Combined / Practical Pulley Systems
- A single fixed + single movable combination gives VR = 2 and the convenience of a downward effort (fixed pulley changes direction, movable gives the force advantage).
- In a real block-and-tackle, account for friction + lower-block weight: actual effort $E>\dfrac{L}{n}$, so $\eta=\dfrac{\text{MA}}{n}<1$. Efficiency improves with a smoother system and a lighter lower block.
13. Ideal vs Practical Machine
| Feature | Ideal machine | Practical machine |
|---|---|---|
| Friction | none | present |
| Weight of parts | weightless | has weight |
| Efficiency | 100 % | < 100 % |
| MA vs VR | MA = VR | MA < VR |
| Work | $W_{out}=W_{in}$ | $W_{out}<W_{in}$ |
14. More Worked Numericals
Q7. A crowbar of length 1.2 m is used to lift a stone by placing a fulcrum 0.2 m from the stone. Find the MA. Solution: effort arm = 1.0 m, load arm = 0.2 m โ $\text{MA}=\dfrac{1.0}{0.2}=\mathbf{5}$.Q8. In a machine an effort of 25 kgf moves through 80 cm to raise a load of 100 kgf through 10 cm. Find MA, VR and efficiency. Solution: $\text{MA}=100/25=4$; $\text{VR}=80/10=8$; $\eta=4/8=\mathbf{50\%}$.
Q9. A pulley system has VR = 4. A load of 400 N is raised by an effort of 150 N. (i) MA (ii) efficiency (iii) work wasted per 1 m of load lift. Solution: (i) $\text{MA}=400/150=2.67$. (ii) $\eta=2.67/4=\mathbf{66.7\%}$. (iii) effort moves 4 m: $W_{in}=150\times4=600$ J, $W_{out}=400\times1=400$ J โ 200 J wasted.
Q10. A wheel-and-axle has wheel radius 40 cm and axle radius 10 cm. Find the VR and the load liftable by 30 N (ideal). Solution: $\text{VR}=40/10=4$; ideal MA = 4 โ Load $=4\times30=\mathbf{120\ N}$.
15. More Exam Questions (with crisp answers)
- Why can a single fixed pulley not act as a force multiplier? โ Its VR = 1, so ideal MA = 1; effort equals load.
- How does friction affect MA and efficiency (VR unchanged)? โ Both decrease.
- Why is it easier to lift a load with more pulleys in a block-and-tackle? โ Larger VR โ effort = Load/n is smaller.
- State two ways to increase a machine's efficiency. โ Reduce friction (oiling); reduce the weight of moving parts.
- A machine has MA > VR. Is it possible? โ No โ it would mean ฮท > 100 %, impossible.
- Name a lever with MA always less than 1, and state what is gained. โ Class III lever; gains speed/range of movement.
- Define 1 unit of efficiency / express it as %. โ ฮท = MA/VR, expressed as a percentage = (MA/VR) ร 100.
- Why are scissors used to cut cloth long-bladed but those to cut metal short-bladed with long handles? โ Long blades = speed (cloth, less force); long handles + short blades = more force (metal).