Physics · Ch 1 — Units and Measurements

Derived Quantities and Units

1.2.2

Derived Quantities and Units

Beyond the seven fundamental quantities, physics deals with a huge number of other quantities — speed, momentum, resistance, electrical conductivity, and so on — each of which depends on some combination of the fundamental quantities and can be fully expressed in terms of them. These are called derived quantities, and the units used to measure them, built up from the fundamental units, are called derived units.

For example, the SI unit of velocity is obtained from its defining relation, $\text{velocity} = \dfrac{\text{displacement}}{\text{time}}$ , so its unit is $\dfrac{\text{unit of displacement}}{\text{unit of time}} = \dfrac{\text{m}}{\text{s}} = \text{m s}^{-1}$ . Similarly, since momentum is (mass)×(velocity), its unit is (unit of mass)×(unit of velocity) $= \text{kg} \times \text{m s}^{-1} = \text{kg m s}^{-1}$ . Both of these are derived units, built entirely from the fundamental units of mass, length and time.

Besides the seven fundamental units, SI also recognises two supplementary units, for angles: the plane angle $d\theta$ and the solid angle $d\Omega$ . The plane angle is defined as the ratio of the length of a small arc of a circle to the circle's radius, $d\theta = ds/r$ — this is the angle subtended by the arc at the centre of the circle, and it is measured in radian (rad); an angle $\theta$ expressed in radian is often written $\theta^{c}$ . Since the full circumference of a circle of radius $r$ is $2\pi r$ , a complete circle subtends an angle of $2\pi r / r = 2\pi$ radians (i.e. $360° = 2\pi^{c}$ , so $\pi$ radians $= \pi^{c} = 180°$ ). …

Figure 1.1(a)Plane angle dθ

What this figure shows. A small circular arc of length ds lies on a circle of radius r centred at O. Two radii OA and OB are drawn from the centre O to the two ends of the arc ds (labelled A and B on the circle). The angle dθ subtended at the centre O between OA and OB is the plane angle, defined as dθ = ds/r. The figure shows the centre point O, the radius r drawn out to the arc, the short arc segment ds between points A and B on the circumference, and the angle dθ marked at the vertex O between the two radii. This is the geometri …

Figure 1.1(b)Solid angle dΩ

What this figure shows. A sphere of radius r is centred at point O. A small patch of the sphere's surface, of area dA, is shown on the surface, with a line drawn from the centre O out to this patch (the radius r to the patch). The solid angle dΩ subtended at the centre O by this small area dA is defined as dΩ = dA/r², the 3-dimensional analogue of the plane angle. The figure shows the centre O, the small curved surface patch dA on the sphere, and the radius line r connecting O to the patch — illustrating that a solid angle is measured by the ratio of an area on a sphere to the square of the sphe …

Misc Ex 1.1Solid angle subtended by the Moon at the Earth

Worked out. Worked example computing the solid angle subtended by the Moon at a point on Earth, given the Moon's diameter (3474 km) and its mean distance from Earth (3.84×10^8 m). The method treats the Moon's visible disc as a flat circular area of radius equal to half the Moon's diameter, computes that disc's area (πr²), and divides by the square of the Earth-Moon distance (per the dΩ = dA/r² definition from section 1.2.2) to get the solid angle in steradians, obtaining a value of order 6.4×10^-5 sr. This demonstrates a direct numerical application of …

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