Effects of the Gravitational Field: Weight, Apparent Weight, and Weightlessness

An astronaut of $70\ \text{kg}$ is on the surface of the Moon, where the gravitational field is $g_L = 1.62\ \text{m/s}^2$.

1. a) Calculate their weight on the Moon.
2. b) Calculate their weight on Earth ($g = 9.8\ \text{m/s}^2$).
3. c) How many times less do they weigh on the Moon? Has their mass changed? Reason the difference between mass and weight.

The Perseverance rover, of $1025\ \text{kg}$, operates on the surface of Mars, where $g_M = 3.71\ \text{m/s}^2$.

1. a) Calculate its weight on Mars.
2. b) Calculate how much it weighed on Earth before launch ($g = 9.8\ \text{m/s}^2$).
3. c) Reason why such a vehicle can have lighter wheels and suspension on Mars than on Earth.

A precision scale indicates that the weight of a laboratory instrument is $588\ \text{N}$ at a place where $g = 9.8\ \text{m/s}^2$.

1. a) Calculate the mass of the instrument.
2. b) If the instrument were taken to the Moon ($g_L = 1.62\ \text{m/s}^2$), what would its mass be? And its weight?
3. c) Explain which quantity does not change when changing planets, and why.

A satellite of $500\ \text{kg}$ orbits in a region where the Earth's gravitational field is $g = 8.7\ \text{m/s}^2$.

1. a) Calculate the gravitational force exerted by the Earth on it.
2. b) State the direction and sense of that force.
3. c) According to Newton's third law, what force does the satellite exert on the Earth?

The summit of Mount Everest is $8848\ \text{m}$ above sea level.

1. a) Calculate the gravitational field at the summit using $g = G\times\frac{M_T}{(R_T + h)^2}$ and assuming the Earth is spherical ($G = 6.67\times10^{-11}\ \text{N}\times\text{m}^2/\text{kg}^2$, $M_T = 5.97\times10^{24}\ \text{kg}$, $R_T = 6.37\times10^{6}\ \text{m}$).
2. b) Compare the result with $9.8\ \text{m/s}^2$: it hardly changes. Explain why this is so.

The Hubble Space Telescope orbits at an altitude of about $540\ \text{km}$.

1. a) Using the expression $g = g_0\times\left(\frac{R_T}{R_T + h}\right)^2$, with $g_0 = 9.8\ \text{m/s}^2$ and $R_T = 6.37\times10^{6}\ \text{m}$, calculate the gravitational field at that altitude.
2. b) Express the result as a percentage of its surface value.

The San Pedro Mártir Observatory is at an altitude of $3718\ \text{m}$.

1. a) Calculate $g$ at that altitude with $g = g_0\times\left(\frac{R_T}{R_T + h}\right)^2$ ($g_0 = 9.81\ \text{m/s}^2$, $R_T = 6.371\times10^{6}\ \text{m}$).
2. b) Calculate the difference from the value at sea level.
3. c) Express it as a percentage and comment on whether an ordinary measuring instrument would detect it.

The satellites of the GPS system orbit at a distance of $2.66\times10^{7}\ \text{m}$ from the center of the Earth.

1. a) Calculate the Earth's gravitational field in its orbit ($G = 6.67\times10^{-11}\ \text{N}\times\text{m}^2/\text{kg}^2$, $M_T = 5.97\times10^{24}\ \text{kg}$).
2. b) Calculate the gravitational force on a GPS satellite of $1000\ \text{kg}$.
3. c) Compare this force with the weight of the same satellite on the surface ($g = 9.81\ \text{m/s}^2$): how many times smaller is it?

The Apollo 11 command module remained in lunar orbit at an altitude of $110\ \text{km}$ while two astronauts descended to the surface.

1. a) Calculate the lunar gravitational field in that orbit ($M_L = 7.35\times10^{22}\ \text{kg}$, $R_L = 1.74\times10^{6}\ \text{m}$, $G = 6.67\times10^{-11}\ \text{N}\times\text{m}^2/\text{kg}^2$).
2. b) Calculate the gravitational force on the module, which has a mass of $5500\ \text{kg}$.
3. c) Compare this force with the weight of the module on Earth ($g = 9.8\ \text{m/s}^2$).

A probe leaves Earth toward deep space.

1. a) Calculate at what altitude above the surface the weight of the probe is half of that at launch: set up $G\times\frac{M_T}{(R_T + h)^2} = \frac{g_0}{2}$, note that $G$ and $M_T$ cancel out, and solve for $h$ in terms of $R_T$ ($R_T = 6.37\times10^{6}\ \text{m}$, $g_0 = 9.8\ \text{m/s}^2$).
2. b) At what distance from the center of the Earth would the weight be one tenth of its launch value?
3. c) Express both results in Earth radii and comment on what the rapid decrease of the field implies regarding the validity of assuming $g$ to be constant in projectile problems.

In metrology, mass standards must be weighed with extreme precision.

1. a) Calculate $g$ at the summit of Mont Blanc ($4808\ \text{m}$) with $g = g_0\times\left(\frac{R_T}{R_T + h}\right)^2$ ($g_0 = 9.81\ \text{m/s}^2$, $R_T = 6.371\times10^{6}\ \text{m}$).
2. b) Calculate the weight difference of a standard mass of $1.000\ \text{kg}$ between the summit and sea level.
3. c) How many grams of mass does this difference correspond to? Comment on why calibration laboratories record the altitude of the measurement location.

For small altitudes relative to the radius of the Earth, the approximation $g \approx g_0\times\left(1 - \frac{2\times h}{R_T}\right)$ is used, which comes from $(1 + x)^{-2} \approx 1 - 2\times x$ when $x \ll 1$.

1. a) Starting from $g = g_0\times\left(\frac{R_T}{R_T + h}\right)^2$, justify this approximation.
2. b) Apply it to the summit of Everest ($h = 8848\ \text{m}$, $g_0 = 9.81\ \text{m/s}^2$, $R_T = 6.371\times10^{6}\ \text{m}$) and compare it with the exact calculation.
3. c) Estimate from what altitude the error of the approximation exceeds 1% (hint: the neglected term is of the order of $3\times x^2$) and decide whether the approximation is valid for the orbit of the Space Station ($h = 400\ \text{km}$).

A pendulum clock is calibrated at sea level, where its period is $T_0 = 2.00\ \text{s}$ with $g_0 = 9.81\ \text{m/s}^2$, and is then taken to the summit of Everest, where $g = 9.78\ \text{m/s}^2$. The period of a simple pendulum is $T = 2\times\pi\times\sqrt{\frac{L}{g}}$.

1. a) Without changing the length of the pendulum, reason whether the clock gains or loses time at the summit.
2. b) Calculate the new period.
3. c) Calculate how much time the clock loses in a full day.

Inside a planet of uniform density, the gravitational field increases linearly with distance from the center, such that at a depth $d$ it is equal to $g = g_0\times\left(1 - \frac{d}{R_T}\right)$. The graph shows how $g$ varies with distance $r$ from the center:

1. a) Calculate $g$ at a depth of $3000\ \text{m}$, at the bottom of a mine ($g_0 = 9.81\ \text{m/s}^2$, $R_T = 6.371\times10^{6}\ \text{m}$).
2. b) Calculate the weight there of a loaded mine elevator, with a mass of $800\ \text{kg}$, and the difference from its weight on the surface.
3. c) When going up an altitude of $3000\ \text{m}$, the relative variation of $g$ is double and of opposite sign to that when going down $3000\ \text{m}$: check this using the approximation $g \approx g_0\times\left(1 - \frac{2\times h}{R_T}\right)$ and explain physically why going down produces a smaller effect (think about the mass that remains 'above').

A person is standing on a scale inside a lift.

1. a) With the lift at rest, the scale reads $700\ \text{N}$: what is the person's weight?
2. b) The lift starts moving and accelerates upwards: reason whether the scale will read more, less or the same, with the aid of a force diagram in your notebook.
3. c) What if the lift moves downwards while braking, i.e. also accelerating upwards? Justify your answer.

A person of $60\ \text{kg}$ weighs themselves on a scale inside a lift ($g = 9.8\ \text{m/s}^2$).

1. a) The lift ascends with a constant acceleration of $2.0\ \text{m/s}^2$: draw the forces in your notebook, apply $\sum \vec{F} = m\cdot\vec{a}$ and calculate what the scale reads (the normal force).
2. b) Repeat the calculation when the lift descends with an acceleration of $2.0\ \text{m/s}^2$.
3. c) In which case does the person feel heavier? Does this match what you notice when travelling in a lift?

You can do it on paper and send a photo with your answer.

A person of $70\ \text{kg}$ is standing on a scale in a lift ($g = 9.8\ \text{m/s}^2$).

1. a) The cable snaps and the lift falls with acceleration $g$: calculate what the scale reads and explain the result.
2. b) In a less severe breakdown, the lift descends with an acceleration of $2.0\ \text{m/s}^2$: calculate the reading on the scale.
3. c) Reason whether it would be possible for the scale to read a negative value during a descent, and what condition would have to be met.

A person of $70\ \text{kg}$ is travelling in a lift, standing on a scale ($g = 9.8\ \text{m/s}^2$). The upward journey consists of three stages: starting with an acceleration of $1.5\ \text{m/s}^2$ upwards, a central stage at constant velocity, and final braking with an acceleration of $1.5\ \text{m/s}^2$ directed downwards.

1. a) Calculate the reading on the scale in each of the three stages.
2. b) Draw in your notebook the graph of the scale reading against time and indicate in which stage the person feels heaviest and in which lightest.
3. c) What would the scale read during the entire journey if the cable snapped just as it started moving? Justify your answer.

You can do it on paper and send a photo with your answer.

In a drop tower at an amusement park, the carriage with a passenger of $60\ \text{kg}$ falls freely for $2.0\ \text{s}$ and then brakes with constant acceleration until coming to a stop ($g = 9.8\ \text{m/s}^2$).

1. a) What is the apparent weight of the passenger during free fall? Justify your answer.
2. b) The braking takes place with an acceleration of magnitude $2\times g$ directed upwards: calculate the apparent weight in this phase and compare it with the real weight.
3. c) Calculate the velocity at the end of the free fall and the distance travelled during braking.

In videos of the International Space Station, astronauts "float". However, at the altitude of $400\ \text{km}$ at which they orbit, the Earth's gravitational field is about $8.7\ \text{m/s}^2$, almost the same as on the surface.

1. a) Explain why astronauts and the station are said to be in permanent "free fall" around the Earth.
2. b) Reason why, under these conditions, the apparent weight inside the station is zero even though the actual weight is not.

The International Space Station orbits $400\ \text{km}$ above the Earth's surface.

1. a) Calculate the Earth's gravitational field at that altitude ($G = 6.67\times10^{-11}\ \text{N}\times\text{m}^2/\text{kg}^2$, $M_T = 5.97\times10^{24}\ \text{kg}$, $R_T = 6.37\times10^{6}\ \text{m}$).
2. b) Calculate the gravitational force exerted by the Earth on an astronaut of $75\ \text{kg}$ inside the station.
3. c) If the Earth pulls on them with that force, why does the astronaut float inside the station? Give a detailed explanation using the concepts of free fall and apparent weight.

Parabolic flight aircraft are used to train astronauts and test medical experiments in weightlessness. The diagram shows such a flight profile:

1. a) Explain how the aircraft must move during the parabola so that the passengers experience zero apparent weight: what must its acceleration be?
2. b) What is the apparent weight of a female researcher of $65\ \text{kg}$ during the weightlessness phase? What about her real weight ($g = 9.8\ \text{m/s}^2$)?
3. c) Why do you think space medicine uses these flights before sending experiments to the Space Station?

The Apollo 11 ascent module, with a mass of $4700\ \text{kg}$, took off from the lunar surface ($g_L = 1.62\ \text{m/s}^2$).

1. a) Calculate the minimum thrust its engine had to produce to ascend vertically with an acceleration of $2.0\ \text{m/s}^2$ (apply $\sum \vec{F} = m\cdot\vec{a}$ in the vertical direction).
2. b) Calculate the thrust that would be needed to perform the same manoeuvre from the Earth's surface ($g = 9.8\ \text{m/s}^2$).
3. c) In light of both results, explain why the lunar module was able to take off with a small engine while the Saturn V rocket required a thrust of $3.4\times10^{7}\ \text{N}$.

An astronaut practices vertical jumps with an initial velocity of $3.0\ \text{m/s}$.

1. a) Calculate the maximum height they would reach on Earth ($g = 9.8\ \text{m/s}^2$) using $h = \frac{v_0^2}{2\times g}$.
2. b) Calculate the maximum height on the Moon ($g_L = 1.62\ \text{m/s}^2$).
3. c) Calculate the total time of flight (ascent and descent) in both places using $t = \frac{2\times v_0}{g}$.
4. d) Reason, using the two formulas, why both the height and time of flight are multiplied by the same factor when changing planet, and calculate this factor.