Machines: mechanical advantage, velocity ratio, levers and pulleys: WASSCE Physics

A woman lifting a 50 kg bag of rice onto a lorry, a mechanic raising a car with a jack, a farmer moving earth in a wheelbarrow: each uses a machine to make a hard job possible. A machine never gives you energy for free, but it lets a small force overcome a large one.

Learning objectives

By the end of this lesson you should be able to:

define a machine, mechanical advantage, velocity ratio and efficiency
explain why the efficiency of a real machine is always less than 100%
classify levers and calculate their mechanical advantage
analyse pulley systems, including the block and tackle

What is a machine?

Definition

A machine is a device by which a force applied at one point (the effort) is used to overcome a force at another point (the load).

Machines make work easier by letting a small effort overcome a large load, changing the direction of a force, or moving a load faster or further.

The three key quantities

Definition

Mechanical advantage (MA) is the ratio of the load to the effort:

\text{MA} = \dfrac{\text{load}}{\text{effort}} = \dfrac{L}{E}

\begin{aligned}\text{MA} &= \dfrac{\text{load}}{\text{effort}} \\ &= \dfrac{L}{E}\end{aligned}

Velocity ratio (VR) is the ratio of the distance moved by the effort to the distance moved by the load (in the same time):

\text{VR} = \dfrac{\text{distance moved by effort}}{\text{distance moved by load}} = \dfrac{d_E}{d_L}

\begin{aligned}\text{VR} &= \dfrac{\text{distance moved by effort}}{\text{distance moved by load}} \\ &= \dfrac{d_E}{d_L}\end{aligned}

Efficiency is the ratio of the useful work output to the work input, as a percentage:

\eta = \dfrac{\text{work output}}{\text{work input}} \times 100\% = \dfrac{L \times d_L}{E \times d_E} \times 100\% = \dfrac{\text{MA}}{\text{VR}} \times 100\%

\begin{aligned}\eta &= \dfrac{\text{work output}}{\text{work input}} \times 100\% \\ &= \dfrac{L \times d_L}{E \times d_E} \times 100\% \\ &= \dfrac{\text{MA}}{\text{VR}} \times 100\%\end{aligned}

MA and VR are ratios, so they have no units.

Why efficiency is less than 100%

Some of the work input is wasted:

overcoming friction at moving parts (turned to heat)
lifting moving parts of the machine itself (such as the lower pulley block)
stretching or bending parts of the machine

So the MA is always less than the VR in a real machine. VR depends only on the machine's geometry; MA depends also on friction and load. Efficiency can be improved by lubrication and by using lighter moving parts.

Worked example

A machine of velocity ratio 5 lifts a load of 400 N with an effort of 100 N. Find its MA and efficiency.

\text{MA} = \dfrac{400}{100} = 4 \qquad \eta = \dfrac{4}{5} \times 100\% = 80\%

\begin{gathered}\begin{aligned}\text{MA} &= \dfrac{400}{100} \\ &= 4\end{aligned} \\[6pt] \begin{aligned}\eta &= \dfrac{4}{5} \times 100\% \\ &= 80\%\end{aligned}\end{gathered}

Levers

A lever is a rigid bar that turns about a fixed point, the fulcrum (pivot). By the principle of moments,

\text{effort} \times \text{effort arm} = \text{load} \times \text{load arm}

\begin{aligned}&\text{effort} \times \text{effort arm} \\ &= \text{load} \times \text{load arm}\end{aligned}

so for an ideal lever $\text{MA} = \text{VR} = \dfrac{\text{effort arm}}{\text{load arm}}$ .

The three classes of lever: class 1 with the fulcrum between effort and load, class 2 with the load in the middle, class 3 with the effort in the middle

Figure 1. The three classes of lever. WASSCE Tutor diagram.

Class	Arrangement	MA	Examples
First	fulcrum between effort and load	can be more or less than 1	crowbar, see-saw, pliers, scissors, claw hammer
Second	load between fulcrum and effort	always more than 1	wheelbarrow, bottle opener, nutcracker
Third	effort between fulcrum and load	always less than 1	human forearm, tweezers, fishing rod, broom

Third-class levers make the effort larger than the load, but they move the load further and faster: your biceps moves your hand quickly through a large distance.

Worked example

A wheelbarrow carrying 600 N of sand has the load 0.4 m from the wheel axle (fulcrum); the handles are 1.2 m from the axle. Find the effort needed.

E \times 1.2 = 600 \times 0.4 \Rightarrow E = 200\ \text{N} \qquad \text{MA} = 3

\begin{gathered}\begin{aligned}&E \times 1.2 \\ &= 600 \times 0.4\end{aligned} \\[6pt] \Rightarrow E = 200\ \text{N} \\[6pt] \text{MA} = 3\end{gathered}

Pulleys

Single fixed pulley: changes only the direction of the force. VR = 1, so MA ≤ 1. Used to raise a flag or draw water from a well.

Single movable pulley: the load hangs on the pulley, supported by two strands. VR = 2.

Block and tackle: two blocks, each with one or more pulleys. The upper block is fixed; the lower block carries the load.

A block and tackle with two pulleys in the upper fixed block and two in the lower block carrying the load; four rope strands support the lower block and the effort pulls on the free end

Figure 2. Count the strands that support the lower block to find the VR. WASSCE Tutor diagram.

Definition

For a pulley system, the velocity ratio = the number of rope strands supporting the movable (lower) block, which equals the total number of pulleys in both blocks for a block and tackle.

If the load rises 1 m, each of the 4 supporting strands must shorten by 1 m, so the effort end must be pulled 4 m: VR = 4.

Worked example

A block and tackle has 3 pulleys in each block. An effort of 250 N lifts a load of 1200 N. Find (a) the VR, (b) the MA, (c) the efficiency, (d) the work wasted when the load is raised 2 m.

(a) VR = 3 + 3 = 6. (b) MA = $\dfrac{1200}{250} = 4.8$ . (c) $\eta = \dfrac{4.8}{6} \times 100\% = 80\%$ .

(d) Work output $= 1200 \times 2 = 2400$ J. Effort moves $6 \times 2 = 12$ m, so work input $= 250 \times 12 = 3000$ J. Wasted: $3000 - 2400 = 600$ J.

Key points

MA = load ÷ effort; VR = effort distance ÷ load distance; efficiency = MA ÷ VR × 100%.
Real machines: MA < VR and efficiency < 100%, because of friction and the weight of moving parts.
Levers: class 1 (fulcrum in the middle), class 2 (load in the middle, MA > 1), class 3 (effort in the middle, MA < 1).
Pulley system: VR = number of strands supporting the movable block.

WASSCE examination focus

WAEC tests this

Define MA, VR and efficiency and show that efficiency = MA/VR × 100%. Explain why efficiency is less than 100%. Draw a block and tackle with a stated VR (a favourite: "draw a pulley system of velocity ratio 4") and calculate MA, efficiency and wasted work. Objective items ask you to classify levers.

Quick check

Which class of lever is a pair of tweezers?
A machine has VR 4 and efficiency 75%. What effort lifts a 600 N load?
Why is the MA of a pulley system less than its VR?

Answers: 1. Third class. 2. MA $= 0.75 \times 4 = 3$ , so $E = 600 \div 3 = 200$ N. 3. Some effort is used to overcome friction and to lift the movable pulley block.

Machines: mechanical advantage, velocity ratio, levers and pulleys

Learning objectives

What is a machine?

The three key quantities

Why efficiency is less than 100%

Levers

Pulleys

Key points

WASSCE examination focus

Quick check

Keep going with simple machines

More lessons in Simple Machines