1 Using formulae and substitution

In the study of engineering, physical quantities can be related to each other using a formula. The formula will contain variables and constants which represent the physical quantities. To evaluate a formula we must substitute numbers in place of the variables.

For example, Ohm’s law provides a formula for relating the voltage, $v$ , across a resistor with resistance value, $R$ , to the current through it, $i$ . The formula states

$v=iR$

We can use this formula to calculate $v$ if we know values for $i$ and $R$ . For example, if $i=13$ A, and $R=5\Omega$ , then

The voltage is 65 V.

Note that it is important to pay attention to the units of any physical quantities involved. Unless a consistent set of units is used a formula is not valid.

Example 62

The kinetic energy, $K$ , of an object of mass $M$ moving with speed $v$ can be calculated from the formula, $K=\frac{1}{2}M{v}^2$ .

Calculate the kinetic energy of an object of mass 5 kg moving with a speed of $2{\text{m s}}^{-1}$ .

Solution

In this example $M=5$ and $v=2$ . Substituting these values into the formula we find

In the SI system the unit of energy is the joule. Hence the kinetic energy of the object is 10 joules.

Task!

The area, $A$ , of the circle of radius $r$ can be calculated from the formula $A=\pi r^2$ .

If we know the diameter of the circle, $d$ , we can use the equivalent formula $A=\frac{\pi d^2}{4}$ . Find the area of a circle having diameter 0.1 m. Your calculator will be preprogrammed with the value of $\pi$ .

Answer

Example 63

The volume, $V$ , of a circular cylinder is equal to its cross-sectional area, $A$ , times its length, $h$ .

Find the volume of a cylinder having diameter 0.1 m and length 0.3 m.

Solution

We can use the result of the previous Task to obtain the cross-sectional area $A=\frac{\pi d^2}{4}$ . Then

The volume is $0.0024{\text{m}}^3$ .