Compound Units

Determine the unit of measuring a quantity \(H\) which is a quotient of distance and time.

Solution:

Here, the quantity \(H\) is derived from dividing distance by time.

Step 1: Find the unit of component quantities. Component quantities are quantities that are used in deriving a compound quantity. Here, the component quantities are distance and time. The unit of distance metres, \(m\), and the unit of time is seconds, \(s\).

Step 2: Apply the operation needed. In this case, it is the quotient of distance and time. Thus, we are dividing so that \[H=\frac{m}{s}\]

Hence, the unit of the quantity, \(H\) is \(ms^{-1}\).

Work is the product of force and distance. Find the unit of work.

Solution:

We have been told that work is the product of force and distance.

Step 1: Find the unit of component quantities. Here, the component quantities are force and distance. The unit of force is newtons, \(N\), and the unit of distance is meters, \(m\).

Step 2: Apply the operation needed. In this case, it is the product of force and distance. Thus, we are multiplying so that if work is \(W\), then, \[W=N\times m\]

Hence, the unit of the quantity, \(W\) is \(Nm\).

If the density of a material is \(5\, kgdm^{-3}\), find the mass when its volume is \(20\, dm^3\).

Solution:

Step 1: Look through the units given, and classify the units into compound and standard units. Note sometimes, both units may be compound units. In such cases, determine which compound unit is more complex. This is often easy to spot because the more elements in a unit, the more complex it is.

This question provides two units, a compound unit, \(kgdm^{-3}\) and a standard unit, \(dm^3\).

Step 2: Determine the relationship between both units by comparing both. Try to see differences. If the standard (or less complex compound) unit is seen to be part of the more complex unit, and it is not different in its exponentials, then, you divide. Otherwise, you should multiply both quantities. When doing this, do so with the units because you wish to determine the unit of the unknown too.

In this case, the unit \(dm^3\) which is found in \(kgdm^{-3}\) has a different exponential. Because the standard unit has an exponential of \(3\), while the compound unit has an exponential of \(-3\) for \(dm\). This suggests you are to multiply.

Step 3: Carryout the operation explicitly. Thus, \[5\, kgdm^{-3}\times 20\, dm^3=5\times20\times kg\times dm^{-3}\times dm^{3}\]

Hence, the mass \(m\) of the material is \[m=5\times 20\times kg\times1\] This gives \[m=100\, kg\]

The power produced by a machine is \(15\, kW\). Find the power of the same machine in \(MW\).

Solution:

Step 1: Identify the units involved. Your value has been given in kilowatts, \(kW\), while your answer is in megawatts, \(MW\).

Step 2: Define the relationship between units. If \[1\, kW=10^3\, W\] and \[1\, MW=10^6\, W\] with \[10^6=10^3\times10^3\] we can say \[1\, MW=10^3\times10^3\, W\]

Since \[1\, kW=10^3\, W\] it surely means that

\[1\, MW=10^3\times1\, kW\]

Hence,

\[1\, MW=10^3\, kW\]

or

\[1\, MW=1000\, kW\]

Step 3: So we would have to convert \(15\, kW\) to \(MW\) which is

\[15\, kW=\frac{15}{1000}\]

So our answer is \(0.015\, MW\)

If \[1\, Pa=7.5\times10^{-3}\, mmHg\]

and the pressure exerted on a body is \(4\, Pa\), express the pressure in \(mmHg\).

Solution:

Since

\[1\, Pa=7.5\times10^3\, mmHg\]

then,

\[4\, Pa=4\times7.5\times10^{-3}\, mmHg\]

So we have

\[4\, Pa=3\times10^{-2}\, mmHg\]

A man is paid \(£120\) for \(6\) hours of labour. What is his hourly wage?

Solution:

We intend on knowing how much he is paid for 1 hr. So, \[£120=6\, hrs\] Divide both sides by \(6\) to arrive at \[£20=1\, hr\]

Now to derive our compound unit, divide, by \(1\, hr\) to get \[£20\, hr^{-1}=1\]

The \(1\) signifies \(1\) unit of labour which is interpreted as \(1\) unit of labor at the rate of \(£20\, hr^{-1}\).

Classify the following quantities in a table based on the compound and standard units; displacement, velocity, volume, area, force, pressure, electromagnetic induction, and electric current.

Solution:

The compound and standard units can be arranged as

Imisi covers a distance of \(500\, m\) in \(160\, s\), find the Imisi's speed in miles per hour.

Solution:

We know speed is usually measured in \(ms^{-1}\), but our answer this time is to be in another compound unit that measures speed.

Step 1: Convert the related components. Here, we have two components, distance and time. So, convert the distance in meters to miles. If \[1\, m=6.2\times10^{-4}\, mi\]

Then,

\[500\, m=500\times6.2\times10^{-4}\, mi\]

\[500\, m=3.1\times10^{-1}\, mi\]

Likewise, convert seconds to hours. If

\[1\, s=2.78\times10^{-4}\, hr\]

Then,

\[160\, s=160\times2.78\times10^{-4}\, hr\]

\[160\, s=4.448\times10^{-2}\, hr\]

Step 2: Calculate the speed with new values.

\[\frac{3.1\times10^{-1}\, mi}{4.448\times10^{-2}\, hr}\]

Hence the speed in \(mihr^{-1}\) is \(7\, mihr^{-1}\).