Exercises: 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 person has a mass of $75\ \text{kg}$.

1. a) Make a quick estimate of their weight taking $g \approx 10\ \text{m/s}^2$.
2. b) Refine the result using the exact value $g = 9.8\ \text{m/s}^2$.

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.

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 value $g = 9.8\ \text{m/s}^2$ is also expressed as $9.8\ \text{N/kg}$. Explain in your own words what it means that "each kilogram weighs $9.8\ \text{N}$" and calculate the weight of a box containing $12\ \text{kg}$ of oranges.

In the laboratory, you want to measure the mass and weight of a bag of potatoes.

1. a) Which instrument would you use for each quantity and in which unit would you give each result?
2. b) Draw in your notebook the setup you would use with a force meter to measure the weight.

You can do it on paper and send a photo with 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?

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.

The gravity on the surface of some bodies in the Solar System is: Venus $8.9\ \text{m/s}^2$, Mars $3.7\ \text{m/s}^2$, Jupiter $24.8\ \text{m/s}^2$ and the Moon $1.6\ \text{m/s}^2$, as shown in the chart (Earth is added as a reference).

1. a) Calculate the weight of a $60\ \text{kg}$ person on each of them.
2. b) Order them from smallest to largest weight.
3. c) On which one would they weigh almost the same as on Earth ($g = 9.8\ \text{m/s}^2$)?

In a lunar module, a force meter reads $16\ \text{N}$ when hanging a rock from it ($g_{\text{Luna}} = 1.6\ \text{m/s}^2$), as shown in the figure.

1. a) What is the mass of the rock?
2. b) What would the same force meter read with the same rock on Earth ($g = 9.8\ \text{m/s}^2$)?

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?

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?

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?

In a gym, an Olympic weight plate has a mass of $25\ \text{kg}$.

1. a) How much does it weigh on Earth ($g = 9.8\ \text{m/s}^2$)?
2. b) How much would it weigh in a gym set up on a lunar base ($g = 1.6\ \text{m/s}^2$)?
3. c) Would it take an athlete the same effort to lift it on the Moon as on Earth? Give reasons for your answer, thinking about the force they have to overcome.

How many $70\ \text{kg}$ people would have to step onto a platform together to equal the weight of a $1400\ \text{kg}$ car? Solve it first by comparing masses and then check it using weights, taking $g \approx 10\ \text{m/s}^2$.

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?

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.

A school laboratory takes a beam balance with its weights and a force meter to the Moon, and measures a rock with them, as shown in the figure.

1. a) The balance balances with weights totalling $60\ \text{kg}$. What is the mass of the rock? Would it balance with the same weights on Earth?
2. b) What does the force meter read when the rock is hung from it on the Moon ($g = 1.6\ \text{m/s}^2$)? What about on Earth ($g = 9.8\ \text{m/s}^2$)?
3. c) Explain why one of the instruments gives the same result in both worlds and the other does not.

Due to the flattening of the Earth and its rotation, $g$ is $9.83\ \text{m/s}^2$ at the poles and $9.78\ \text{m/s}^2$ at the equator. A ship carries $20\,000\ \text{kg}$ of scientific equipment from the pole to the equator, as shown in the figure.

1. a) Calculate the weight of the cargo in each of the two locations.
2. b) By how many newtons does the weight change during the voyage?
3. c) Has the mass of the cargo changed? If weighed with a beam balance at both ports, what would it read at each?

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.

A bathroom scale reads $55\ \text{kg}$ when a person steps onto it. In reality, the scale measures the force with which the person pushes down on it and converts it to kilograms assuming that $g = 9.8\ \text{m/s}^2$.

1. a) What force, in newtons, is the scale measuring?
2. b) If we took that scale to the Moon ($g = 1.6\ \text{m/s}^2$), what force would it measure with the same person?
3. c) What "mass" would the scale read then, continuing to convert using $g = 9.8\ \text{m/s}^2$?
4. d) Explain why the scale gives an incorrect reading on the Moon even though the person has not changed.

A crane can exert a maximum force of $20\,000\ \text{N}$.

1. a) What maximum mass can it lift on Earth ($g = 9.8\ \text{m/s}^2$)?
2. b) What maximum mass could it lift on the Moon ($g = 1.6\ \text{m/s}^2$)?
3. c) An advertisement claims: "Our crane would lift six times more mass on the Moon than on Earth." Check using your results whether the claim is correct and justify your answer.

At an altitude of $400\ \text{km}$, where the International Space Station orbits, Earth's gravity is still $g \approx 8.7\ \text{m/s}^2$.

1. a) Calculate the actual weight of a $70\ \text{kg}$ astronaut at that height.
2. b) Compare it with their weight on the surface of the Earth ($g = 9.8\ \text{m/s}^2$).
3. c) If they weigh almost the same in the station as on the ground, why do they float inside the spacecraft? Explain this thinking about how the station moves around the Earth.

Unit challenge. Does your body weigh the same in your bedroom as it would on Mars? In this challenge, you are going to determine the gravity of your surroundings and compare it with Newton's theory.

1. Home measurement: Choose a dense object (such as a ball) and drop it from three heights: $1.00\text{ m}$, $1.50\text{ m}$, and $2.00\text{ m}$ (uncertainty $\pm 0.01\text{ m}$). Time each drop with your phone (recording in slow motion and counting frames, or using a sensor app such as phyphox, to avoid reaction time), performing three trials to calculate the mean time $\bar{t}$ and its uncertainty.

2. Data processing: Create a table of $h$ versus $t^2$. Plot the graph and calculate the local acceleration $g_{\text{exp}}$ from the relationship $h = \frac{1}{2}g t^2$.

3. Calculation and real data: Look up the altitude of your municipality on the IGN geoportal. Apply the law of universal gravitation ($g = G \frac{M}{(R+h)^2}$) using Earth's data ($M_T = 5.97 \times 10^{24}\text{ kg}$; $R_T = 6.37 \times 10^6\text{ m}$) and calculate your theoretical weight ($P = m \times g$). Repeat the calculation simulating your weight on the surface of Mars ($M_M = 6.42 \times 10^{23}\text{ kg}$; $R_M = 3.40 \times 10^6\text{ m}$).

4. Report: Write a brief document with your table, graph, and a reasoned comparison between your experimental, theoretical, and Martian $P$.

Evaluation criteria:

• Rigor in collecting data, calculating uncertainties, and plotting graphs.

• Correct application of the law of universal gravitation and conceptual distinction between mass and weight.

• Critical analysis of experimental errors compared to theoretical values.