15.1 Fundamental and derived SI quantities and units — Practise with help

Work each one first, closed book. Before you write a number, decide which quantity it measures and which unit belongs with it.

Warm-up: remember

Work out $1000 \div 3600$ and give the answer as a fraction in its lowest terms.
Work out $2.5 \times 1000$ and $45 \div 1000$ .
A car travels $120$ m in $8$ s. Work out its average speed in m/s.

Answers

$\dfrac{1000}{3600} = \dfrac{5}{18}$ .
$2.5 \times 1000 = 2500$ , and $45 \div 1000 = 0.045$ .
Average speed $= \dfrac{120}{8} = 15$ m/s.

If question 1 was hard, revise simplifying fractions before you continue.

Practise with help

1. Write down the SI unit of length, the SI unit of mass and the SI unit of time.

Hint

These three units are the base of every other unit in mechanics. They measure a distance, an amount of matter and a duration.

Next step

Give the name and the symbol of each unit. The base unit of mass is the kilogram, not the gram.

Full working

Length is measured in metres, m. Mass is measured in kilograms, kg. Time is measured in seconds, s.

2. Write the unit of acceleration in terms of the base units of length and time.

Hint

Acceleration is the change in velocity each second. Build it from the unit of velocity first.

Next step

Velocity is metres divided by seconds, so its unit is m/s. Divide that by seconds again for acceleration.

Full working

\text{acceleration} = \frac{\text{velocity}}{\text{time}} = \frac{\text{m/s}}{\text{s}} = \text{m/s}^2 = \text{m s}^{-2}.

3. Convert $4.5$ km to metres, and $60$ cm to metres.

Hint

The prefix tells you the multiplier. Kilo means $1000$ , and centi means $0.01$ .

Next step

Multiply the kilometres by $1000$ . Divide the centimetres by $100$ .

Full working

4.5 \text{ km} = 4.5 \times 1000 = 4500 \text{ m}, \qquad 60 \text{ cm} = 60 \div 100 = 0.6 \text{ m}.

4. Convert $54$ km/h to m/s.

Hint

Multiply by $\dfrac{5}{18}$ to change km/h into m/s. If you forget the fraction, remember that $1$ km/h is $1000$ m in $3600$ s.

Next step

Work out $54 \times 5$ first, then divide by $18$ .

Full working

54 \times \frac{5}{18} = \frac{270}{18} = 15 \text{ m/s}.

5. Find the weight of a mass of $12$ kg. Take $g = 9.8$ m/s $^2$ .

Hint

Weight is a force, so use $W = mg$ . The answer carries the unit of force.

Next step

Substitute $m = 12$ and $g = 9.8$ into $W = mg$ .

Full working

W = mg = 12 \times 9.8 = 117.6 \text{ N}.

6. Find the moment of a force of $15$ N acting at a perpendicular distance of $0.3$ m from a pivot.

Hint

A moment is a force multiplied by a perpendicular distance. The unit is the newton metre.

Next step

Multiply $15$ by $0.3$ . Keep the unit of force and the unit of distance together as N m.

Full working

\text{moment} = 15 \times 0.3 = 4.5 \text{ N m}.

7. A force of $24$ N acts on a mass of $6$ kg. Find the acceleration, and state its unit.

Hint

Start from $F = ma$ and rearrange it for the acceleration.

Next step

Divide the force by the mass: $a = \dfrac{F}{m}$ .

Full working

a = \frac{F}{m} = \frac{24}{6} = 4 \text{ m/s}^2.

The unit is m/s $^2$ , which comes from dividing a force in N by a mass in kg.

8. A car travels $150$ m in $12$ s. Find its average speed in m/s, then convert that speed to km/h.

Hint

Find the speed first. For the second part, m/s is the smaller unit, so the number in km/h must be larger.

Next step

Divide $150$ by $12$ for the speed. Then multiply the speed by $\dfrac{18}{5}$ .

Full working

\text{speed} = \frac{150}{12} = 12.5 \text{ m/s}.

12.5 \times \frac{18}{5} = 12.5 \times 3.6 = 45 \text{ km/h}.

The answer is larger than $12.5$ , as expected, because one hour is a long time and $1$ km is a long distance.