5.4 Mass, Weight, Levers, Gears & Pulleys

5.4 Mass, Weight, Levers, Gears & Pulleys

Unit 202 learning outcome 3 (7 of the 40 test questions) includes mass and weight and "the principles of basic mechanics as they apply to levers, gears and pulleys", including class I, II and III levers. Electricians use these simple machines every day: pliers and crowbars are levers, motors drive loads through gearboxes and belts, and pulley blocks lift cable drums and heavy luminaires.

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Mass and Weight

• Mass ($m$) is the amount of matter in an object, measured in kilograms (kg). It does not change with location.
• Weight ($W$) is the force of gravity on that mass, measured in newtons (N):

$W = m \times g \qquad (g = 9.81\ \text{m/s}^2 \text{ on Earth})$

A 25 kg cable drum has a weight of $25 \times 9.81 = 245.25$ N. On the Moon, where gravity is about one-sixth as strong, its weight would be much smaller but its mass would still be 25 kg.

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Moments and the Principle of Moments

The turning effect of a force about a pivot is its moment:

$\text{Moment} = \text{Force} \times \text{perpendicular distance from the pivot}$

with the moment in newton metres (N·m), the force in newtons and the distance in metres.

The principle of moments states that when an object is balanced, the sum of the clockwise moments about the pivot equals the sum of the anticlockwise moments. A 40 N force acting 0.5 m from a pivot (20 N·m) balances a 20 N force acting 1.0 m from the pivot on the other side (20 N·m).

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The Three Classes of Lever

Every lever has a fulcrum (the pivot), an effort (the force you apply) and a load (the force you overcome). The class depends on which one is in the middle:

Class I:    EFFORT ----------- [FULCRUM] ----- LOAD
Class II:   [FULCRUM] ----- LOAD ----------- EFFORT
Class III:  [FULCRUM] ----- EFFORT ----------- LOAD

Mechanical advantage, velocity ratio and efficiency

$\text{Mechanical advantage (MA)} = \frac{\text{Load}}{\text{Effort}}$

$\text{Velocity ratio (VR)} = \frac{\text{Distance moved by the effort}}{\text{Distance moved by the load}}$

$\text{Efficiency } (\eta) = \frac{\text{MA}}{\text{VR}} \times 100\%$

For an ideal lever (no friction), the velocity ratio equals the effort arm divided by the load arm, and MA = VR.

Worked example 1: a crowbar (class I)

A crowbar 1.5 m long has its fulcrum 0.25 m from the end touching a 600 N load, so the effort arm is $1.5 - 0.25 = 1.25$ m. Taking moments about the fulcrum:

$\text{Effort} \times 1.25 = 600 \times 0.25 \implies \text{Effort} = \frac{150}{1.25} = 120\ \text{N}$

$\text{MA} = \frac{600}{120} = 5$

Worked example 2: a wheelbarrow (class II)

A wheelbarrow carries a 900 N load whose centre is 0.4 m from the wheel axle (the fulcrum), and the handles are 1.2 m from the axle:

$\text{Effort} = \frac{900 \times 0.4}{1.2} = 300\ \text{N}, \qquad \text{MA} = \frac{900}{300} = 3$

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Gears

Gears transmit rotation between shafts. For two meshing gears:

$\text{Gear ratio} = \frac{\text{Teeth on the driven gear}}{\text{Teeth on the driver gear}}$

$N_{\text{driven}} = N_{\text{driver}} \times \frac{T_{\text{driver}}}{T_{\text{driven}}}, \qquad \tau_{\text{driven}} \approx \tau_{\text{driver}} \times \frac{T_{\text{driven}}}{T_{\text{driver}}}$

where $N$ is speed in rev/min, $T$ is the number of teeth and $\tau$ is torque (ignoring losses). A gear train that reduces speed increases torque by the same ratio, which is why motors drive slow, heavy loads through reduction gearboxes.

Worked example 3: a reduction gear

A 15-tooth motor pinion running at 1,440 rev/min drives a 45-tooth gear. The gear ratio is $45 \div 15 = 3:1$, so the output speed is $1{,}440 \div 3 = 480$ rev/min and the output torque is about three times the motor torque (less frictional losses).

Points to remember

• Two meshing gears turn in opposite directions. An idler gear placed between them makes the driven gear turn the same way as the driver without changing the ratio.
• Spur gears link parallel shafts, bevel gears turn the drive through 90°, and a worm and wheel gives a large reduction in one stage (a single-start worm driving a 30-tooth wheel gives 30:1) and often stops the load running back.

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Pulleys and Belt Drives

In real pulley systems, friction and the weight of the lower block mean the actual mechanical advantage is lower than the velocity ratio, so the efficiency is below 100%.

Worked example 4: a block and tackle

A block and tackle with four rope sections supporting the lower block (VR = 4) lifts an 800 N luminaire with an effort of 250 N:

$\text{MA} = \frac{800}{250} = 3.2, \qquad \eta = \frac{3.2}{4} \times 100\% = 80\%$

To raise the load 1 m the effort rope must be pulled 4 m. Check by work done: work in $= 250 \times 4 = 1{,}000$ J and work out $= 800 \times 1 = 800$ J, so the efficiency is again 80%.

Belt drives

For a belt drive between two pulleys, speed is inversely proportional to diameter:

$N_1 \times D_1 = N_2 \times D_2$

A motor pulley of 100 mm diameter running at 1,450 rev/min drives a 250 mm fan pulley at $1{,}450 \times 100 \div 250 = 580$ rev/min.