Weight and Gravitational Field Strength

1Why does everything fall?

A mango comes loose and falls to the ground. A stone thrown upwards slows down, stops and comes back. A 50 kg bag of cement is hard to lift onto a lorry, but an astronaut on the Moon could lift the same bag with one hand. The bag has not changed; something acting on it has. That something is the pull of gravity, and the force it produces is called weight.

2Experiment: how weight depends on mass

You need: a retort stand and clamp, a newton meter (spring balance) reading to 5 N, and a hanger with 100 g slotted masses.

1. Check that the newton meter reads zero with nothing on it. Adjust the zero screw if needed.
2. Hang 100 g on the hook. Wait for the pointer to stop and read the weight with your eye level with the pointer.
3. Add masses 100 g at a time up to 500 g, reading the weight each time.
4. Divide each weight by the mass in kilograms.

Figure 3.1: Measuring the weight of slotted masses. A spring stretches in proportion to the force on it, so the scale can be marked in newtons.

Weight is proportional to mass, and $\frac{W}{m}\backslash frac\{W\}\{m\}$ is close to $9.8\mathrm{N}\mathrm{k}{\mathrm{g}}^{-1}9.8\backslash ,\backslash mathrm\{N\backslash ,kg^\{-1\}\}$ every time. The small differences come from reading a scale that is marked only every 0.5 N.

3Weight and the second law

If a stone falls with no air resistance, its weight is the only force on it, so its weight is the resultant force. By the second law,

The mass cancels. Every body falling freely near the Earth accelerates at the same rate, $g=9.8\mathrm{m}{\mathrm{s}}^{-2}g\; =\; 9.8\backslash ,\backslash mathrm\{m\backslash ,s^\{-2\}\}$, whatever its mass. This is why $gg$ has two names and two equivalent units: the gravitational field strength in $\mathrm{N}\mathrm{k}{\mathrm{g}}^{-1}\backslash mathrm\{N\backslash ,kg^\{-1\}\}$, and the acceleration of free fall in $\mathrm{m}{\mathrm{s}}^{-2}\backslash mathrm\{m\backslash ,s^\{-2\}\}$.

Figure 3.2: In (a) and (c) the forces are balanced, so there is no acceleration. In (b) weight is unbalanced, so the stone accelerates downwards at g.

4Weight on other worlds

A body's mass is the same everywhere, but $gg$, and so its weight, depends on where it is.

5Weight at work

• A porter carrying a 25 kg sack of potatoes up the hills of Nyamagabe must hold up a weight of $25\times 9.8=245\mathrm{N}25\; \backslash times\; 9.8\; =\; 245\backslash ,\backslash mathrm\{N\}$ with every step.
• Builders use a plumb line, a small weight on a string, to check that a wall is vertical: the string hangs along the direction of the weight, towards the centre of the Earth.
• Water flows downhill from the Nyungwe forest into the Nile and Congo basins because its weight pulls it downwards; Unit 5 shows how that weight is turned into electricity.