The Concept of Weight

A rucksack full of books has a mass of $6\ \text{kg}$, as shown in the figure.

Calculate its weight on Earth using $P = m\times g$, taking $g = 9.8\ \text{m/s}^2$.

A bag of cement weighs $196\ \text{N}$ on Earth ($g = 9.8\ \text{m/s}^2$). Rearrange for mass in the expression $P = m\times g$ and calculate it.

A car with a mass of $1200\ \text{kg}$ carries four passengers of $70\ \text{kg}$ each. Take $g \approx 10\ \text{m/s}^2$.

1. a) What is the total mass of the combination?
2. b) What total weight is supported between the four tyres?
3. c) How many newtons of weight does each passenger add?

An astronaut with all their equipment has a mass of $90\ \text{kg}$ and is on the Moon, where $g = 1.6\ \text{m/s}^2$. Calculate their weight there. Has their mass changed compared to what it was on Earth? Justify your answer.

State whether each of these statements refers to mass or weight, and correct any that are incorrectly phrased:

1. a) "This watermelon weighs $4\ \text{kg}$."
2. b) "My mass is $540\ \text{N}$."
3. c) "With a force meter we measure mass in newtons."

The rover of a mission to Mars has a mass of $200\ \text{kg}$, as shown in the figure.

1. a) Calculate its weight on Earth ($g = 9.8\ \text{m/s}^2$).
2. b) Calculate its weight on Mars ($g = 3.7\ \text{m/s}^2$).
3. c) By how many newtons does the rover become "lighter" when travelling to Mars? Has its mass changed? Justify your answer.

A space probe of $15\ \text{kg}$ arrives at Jupiter, where $g = 24.8\ \text{m/s}^2$. Calculate its weight there and compare it with its weight on Earth ($g = 9.8\ \text{m/s}^2$): how many extra newtons does it weigh?

Gravity at a height $h$ above the Earth's surface is calculated using $g(h) = \frac{G\times M}{(R + h)^2}$. Use $M = 5.97\times10^{24}\ \text{kg}$, $R = 6.37\times10^{6}\ \text{m}$, $G = 6.67\times10^{-11}\ \text{N}\times\text{m}^2/\text{kg}^2$ and $h = 4\times10^{5}\ \text{m}$, the height at which the International Space Station orbits.

1. a) Calculate the value of $g$ at that height.
2. b) Compare it with $9.8\ \text{m/s}^2$ at the surface: has it decreased much?

The BepiColombo probe is travelling towards Mercury.

1. a) Calculate the surface gravity of Mercury using $g = \frac{G\times M}{R^2}$, with $M = 3.30\times10^{23}\ \text{kg}$, $R = 2.44\times10^{6}\ \text{m}$ and $G = 6.67\times10^{-11}\ \text{N}\times\text{m}^2/\text{kg}^2$.
2. b) How much will the probe weigh on Mercury if its mass is $300\ \text{kg}$?
3. c) Compare that weight with its weight on Earth: where does it weigh more and what is the difference in newtons?

A newly discovered planet has twice the mass of the Earth and twice the radius.

1. a) By comparing with the expression $g = \frac{G\times M}{R^2}$ for Earth, deduce the value of $g$ on the surface of that planet (take Earth's surface gravity as $9.8\ \text{m/s}^2$).
2. b) How much would an automated probe of $80\ \text{kg}$ weigh there?

The height reached in a vertical jump with initial velocity $v_0$ is $h = \frac{v_0^2}{2\times g}$. An athlete pushes off with $v_0 = 3\ \text{m/s}$, as shown in the figure.

1. a) What height does it reach on Earth ($g = 9.8\ \text{m/s}^2$)?
2. b) What height would it reach on the Moon ($g = 1.6\ \text{m/s}^2$)?
3. c) How many times higher does the athlete jump on the Moon? Relate that factor to the ratio of the two gravities.

An astronaut of $60\ \text{kg}$ travels through the Solar System and notices that on one of the worlds they visit they weigh about $2.5$ times more than on Earth.

1. a) How much do they weigh on Earth ($g = 9.8\ \text{m/s}^2$)?
2. b) What is the approximate value of $g$ on that world?
3. c) From this list, deduce which world it is: Moon $1.6\ \text{m/s}^2$, Mars $3.7\ \text{m/s}^2$, Venus $8.9\ \text{m/s}^2$ and Jupiter $24.8\ \text{m/s}^2$.
4. d) Calculate their exact weight on that world.