University Physics I · unit UPI-13
Gravitation
Connect the ideas, make your working visible, and test the result. Use the hints when you need a starting point; open the full solution after your attempt.
Extra challenging questions
A selection from this pathway. Open a topic below for its complete practice and challenge set.
Newton's Law of Gravitation: infer an input with uncertainty
Newton's law of gravitation states that every pair of point masses attracts along the line joining them with a force F = G m₁ m₂/r², proportional to both masses and to the inverse square of their separation. Use the supplied topic model Y_out = 10.011*hypot((d-cosd(th))/(d*d+1-2*d*cosd(th))^1.5+(d+cosd(th))/(d*d+1+2*d*cosd(th))^1.5, sind(th)*(1/(d*d+1-2*d*cosd(th))^1.5-1/(d*d+1+2*d*cosd(th))^1.5)), where Y_out is EXACT VECTOR SUM (mN). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; hypot(a,b) means √(a²+b²); sind/cosd/tand use degrees. d is Distance D from the pair's centre, measured in km. Keep th (Angle θ from the pair's axis) = 40 ° fixed. For this exercise use only 2.425 km ≤ d ≤ 5.575 km on this continuous branch. A grid of test settings is an input restriction for this exercise, not by itself a claim of physical quantisation. This calculation retains the topic model's stated assumptions; it does not extend them to a different physical setup. A reading gives Y_out = 1.1220717 mN, with a bounded reading uncertainty ±0.06819 mN. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown d by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.000315 km, using a numerical bracket if required. Find the full input interval consistent with the bounded reading uncertainty and compare it with a symmetric input-error assumption. Do not assume that the supplied calibration points prove monotonicity outside this branch.
EXACT VECTOR SUM against d
Predictions from the supplied Newton's Law of Gravitation model, with all other inputs fixed as stated in the question. 5 calibration points support a trend comparison; they are not measured data or a straight-line law.
Vertical axis (y): EXACT VECTOR SUM / mN
Horizontal axis (x): d: Distance D from the pair's centre / km
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| d: Distance D from the pair's centre / km | EXACT VECTOR SUM / mN |
|---|---|
| 2.425 | 3.9914832489979397 |
| 3.2124999999999995 | 2.1427478411783416 |
| 3.9999999999999996 | 1.337673940890353 |
| 4.7875 | 0.9160839743155846 |
| 5.574999999999999 | 0.6674704829991883 |
These are the supplied values, not a worked answer. No fitted trend is drawn.
Show a hint
The model's endpoint predictions bracket the reading. Solve Y_out(d) = Y_measured and then solve again at Y_measured − u and Y_measured + u; a nonlinear inverse can give unequal input deviations.
Reveal the worked solution
- Endpoint predictions are Y_out(2.425) = 3.99148 mN and Y_out(5.575) = 0.66747 mN. The target lies between them. On the selected branch the response is strictly decreasing.
- Endpoint interpolation gives d_line = 2.425 + (1.12207 − 3.99148)(5.575 − 2.425)/(0.66747 − 3.99148) = 5.1442 km. This estimates the inverse of the endpoint chord, rather than assuming that chord is the physical model.
- Use the actual equation, not the chord, and bracket its root. Evaluating the midpoint and retaining the sign-changing half-interval gives d = 4.3465 km; substituting back gives Y_out = 1.1220717 mN.
- The interpolation error in the input is |d_line − d| = 0.797697 km. This is larger than the requested numerical tolerance.
- Invert both limits Y_out = 1.0538817 mN and 1.1902617 mN. Sort the roots even if the response decreases: 4.22666 km ≤ d ≤ 4.47792 km.
- The lower and upper input deviations are 0.119839 km and 0.131423 km. They differ because the inverse response is nonlinear; a symmetric error bar would only be a local approximation. This interval includes reading uncertainty only, not model error or uncertainty in fixed inputs.
Answer d ≈ 4.3465 km; compatible interval [4.22666, 4.47792] km. Endpoint interpolation gives 5.1442 km.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.
Gravitational Field: infer an input with uncertainty
The gravitational field g⃗ at a point is the gravitational force per unit mass on a small test mass placed there, g⃗ = F⃗/m: a vector in N kg⁻¹ that points the way a released body would begin to fall. Use the supplied topic model Y_out = 4004400/xp^2, where Y_out is g₁ TOWARD M₁ (mN kg⁻¹). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. xp is Probe P: distance from M₁, measured in km. Keep q (Mass ratio M₂/M₁) = 0.25 fixed. For this exercise use only 271.625 km ≤ xp ≤ 728.375 km on this continuous branch. A grid of test settings is an input restriction for this exercise, not by itself a claim of physical quantisation. This calculation retains the topic model's stated assumptions; it does not extend them to a different physical setup. A reading gives Y_out = 13.22602 mN kg⁻¹, with a bounded reading uncertainty ±0.85171 mN kg⁻¹. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown xp by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.045675 km, using a numerical bracket if required. Find the full input interval consistent with the bounded reading uncertainty and compare it with a symmetric input-error assumption. Do not assume that the supplied calibration points prove monotonicity outside this branch.
g₁ TOWARD M₁ against xp
Predictions from the supplied Gravitational Field model, with all other inputs fixed as stated in the question. 5 calibration points support a trend comparison; they are not measured data or a straight-line law.
Vertical axis (y): g₁ TOWARD M₁ / mN kg⁻¹
Horizontal axis (x): xp: Probe P: distance from M₁ / km
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| xp: Probe P: distance from M₁ / km | g₁ TOWARD M₁ / mN kg⁻¹ |
|---|---|
| 271.625 | 54.274767790875295 |
| 385.8125 | 26.902018318461675 |
| 500 | 16.0176 |
| 614.1875 | 10.615379037919764 |
| 728.375 | 7.547921773648051 |
These are the supplied values, not a worked answer. No fitted trend is drawn.
Show a hint
The model's endpoint predictions bracket the reading. Solve Y_out(xp) = Y_measured and then solve again at Y_measured − u and Y_measured + u; a nonlinear inverse can give unequal input deviations.
Reveal the worked solution
- Endpoint predictions are Y_out(271.625) = 54.2748 mN kg⁻¹ and Y_out(728.375) = 7.54792 mN kg⁻¹. The target lies between them. On the selected branch the response is strictly decreasing.
- Endpoint interpolation gives xp_line = 271.625 + (13.226 − 54.2748)(728.375 − 271.625)/(7.54792 − 54.2748) = 672.872 km. This estimates the inverse of the endpoint chord, rather than assuming that chord is the physical model.
- Use the actual equation, not the chord, and bracket its root. Evaluating the midpoint and retaining the sign-changing half-interval gives xp = 550.243 km; substituting back gives Y_out = 13.22602 mN kg⁻¹.
- The interpolation error in the input is |xp_line − xp| = 122.63 km. This is larger than the requested numerical tolerance.
- Invert both limits Y_out = 12.37431 mN kg⁻¹ and 14.07773 mN kg⁻¹. Sort the roots even if the response decreases: 533.338 km ≤ xp ≤ 568.864 km.
- The lower and upper input deviations are 16.9047 km and 18.6212 km. They differ because the inverse response is nonlinear; a symmetric error bar would only be a local approximation. This interval includes reading uncertainty only, not model error or uncertainty in fixed inputs.
Answer xp ≈ 550.243 km; compatible interval [533.338, 568.864] km. Endpoint interpolation gives 672.872 km.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.
Gravitational potential energy: infer an input with uncertainty
Gravitational potential energy U(r) = −GMm/r is the energy of two masses a distance r apart, measured from zero at infinite separation: it equals minus the work gravity does as the masses are brought together from infinitely far apart. Use the supplied topic model Y_out = 62.57*(r2-r1)/r1^2, where Y_out is CONSTANT-FORCE GUESS (MJ kg⁻¹). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. r1 is Start radius r₁, measured in Earth radii. Keep r2 (End radius r₂) = 4 Earth radii fixed. For this exercise use only 1.2125 Earth radii ≤ r1 ≤ 2.7875 Earth radii on this continuous branch. A grid of test settings is an input restriction for this exercise, not by itself a claim of physical quantisation. This calculation retains the topic model's stated assumptions; it does not extend them to a different physical setup. A reading gives Y_out = 24.200587 MJ kg⁻¹, with a bounded reading uncertainty ±2.1655 MJ kg⁻¹. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown r1 by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.0001575 Earth radii, using a numerical bracket if required. Find the full input interval consistent with the bounded reading uncertainty and compare it with a symmetric input-error assumption. Do not assume that the supplied calibration points prove monotonicity outside this branch.
CONSTANT-FORCE GUESS against r1
Predictions from the supplied Gravitational potential energy model, with all other inputs fixed as stated in the question. 5 calibration points support a trend comparison; they are not measured data or a straight-line law.
Vertical axis (y): CONSTANT-FORCE GUESS / MJ kg⁻¹
Horizontal axis (x): r1: Start radius r₁ / Earth radii
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| r1: Start radius r₁ / Earth radii | CONSTANT-FORCE GUESS / MJ kg⁻¹ |
|---|---|
| 1.2125 | 118.63628440854504 |
| 1.6062499999999997 | 58.052197610864695 |
| 1.9999999999999998 | 31.285000000000007 |
| 2.39375 | 17.539681912072485 |
| 2.7874999999999996 | 9.763783707695717 |
These are the supplied values, not a worked answer. No fitted trend is drawn.
Show a hint
The model's endpoint predictions bracket the reading. Solve Y_out(r1) = Y_measured and then solve again at Y_measured − u and Y_measured + u; a nonlinear inverse can give unequal input deviations.
Reveal the worked solution
- Endpoint predictions are Y_out(1.2125) = 118.636 MJ kg⁻¹ and Y_out(2.7875) = 9.76378 MJ kg⁻¹. The target lies between them. On the selected branch the response is strictly decreasing.
- Endpoint interpolation gives r1_line = 1.2125 + (24.2006 − 118.636)(2.7875 − 1.2125)/(9.76378 − 118.636) = 2.57865 Earth radii. This estimates the inverse of the endpoint chord, rather than assuming that chord is the physical model.
- Use the actual equation, not the chord, and bracket its root. Evaluating the midpoint and retaining the sign-changing half-interval gives r1 = 2.17325 Earth radii; substituting back gives Y_out = 24.200587 MJ kg⁻¹.
- The interpolation error in the input is |r1_line − r1| = 0.405401 Earth radii. This is larger than the requested numerical tolerance.
- Invert both limits Y_out = 22.035087 MJ kg⁻¹ and 26.366087 MJ kg⁻¹. Sort the roots even if the response decreases: 2.11502 Earth radii ≤ r1 ≤ 2.23727 Earth radii.
- The lower and upper input deviations are 0.0582327 Earth radii and 0.0640206 Earth radii. They differ because the inverse response is nonlinear; a symmetric error bar would only be a local approximation. This interval includes reading uncertainty only, not model error or uncertainty in fixed inputs.
Answer r1 ≈ 2.17325 Earth radii; compatible interval [2.11502, 2.23727] Earth radii. Endpoint interpolation gives 2.57865 Earth radii.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.
Circular Orbital Motion: infer an input with uncertainty
A circular orbit is uniform circular motion in which gravity from the central body is the only force and supplies the entire centripetal force, so the speed at radius r must be v = √(GM/r), whatever the satellite's mass. Use the supplied topic model Y_out = 9.8202/(1+h/6371)^2, where Y_out is g AT THE ORBIT (m s⁻²). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. h is Altitude above surface h, measured in km. Keep t (Elapsed time t) = 3 h fixed. For this exercise use only 4003.75 km ≤ h ≤ 32196.2 km on this continuous branch. A grid of test settings is an input restriction for this exercise, not by itself a claim of physical quantisation. This calculation retains the topic model's stated assumptions; it does not extend them to a different physical setup. A reading gives Y_out = 0.52431646 m s⁻², with a bounded reading uncertainty ±0.038451 m s⁻². Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown h by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 2.81925 km, using a numerical bracket if required. Find the full input interval consistent with the bounded reading uncertainty and compare it with a symmetric input-error assumption. Do not assume that the supplied calibration points prove monotonicity outside this branch.
g AT THE ORBIT against h
Predictions from the supplied Circular Orbital Motion model, with all other inputs fixed as stated in the question. 5 calibration points support a trend comparison; they are not measured data or a straight-line law.
Vertical axis (y): g AT THE ORBIT / m s⁻²
Horizontal axis (x): h: Altitude above surface h / km
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| h: Altitude above surface h / km | g AT THE ORBIT / m s⁻² |
|---|---|
| 4003.75 | 3.7032263869113584 |
| 11051.875 | 1.3130942554138314 |
| 18100 | 0.665628806811917 |
| 25148.124999999996 | 0.4012243090499102 |
| 32196.249999999996 | 0.2679774171316045 |
These are the supplied values, not a worked answer. No fitted trend is drawn.
Show a hint
The model's endpoint predictions bracket the reading. Solve Y_out(h) = Y_measured and then solve again at Y_measured − u and Y_measured + u; a nonlinear inverse can give unequal input deviations.
Reveal the worked solution
- Endpoint predictions are Y_out(4003.75) = 3.70323 m s⁻² and Y_out(32196.2) = 0.267977 m s⁻². The target lies between them. On the selected branch the response is strictly decreasing.
- Endpoint interpolation gives h_line = 4003.75 + (0.524316 − 3.70323)(32196.2 − 4003.75)/(0.267977 − 3.70323) = 30092.5 km. This estimates the inverse of the endpoint chord, rather than assuming that chord is the physical model.
- Use the actual equation, not the chord, and bracket its root. Evaluating the midpoint and retaining the sign-changing half-interval gives h = 21201.2 km; substituting back gives Y_out = 0.52431646 m s⁻².
- The interpolation error in the input is |h_line − h| = 8891.34 km. This is larger than the requested numerical tolerance.
- Invert both limits Y_out = 0.48586546 m s⁻² and 0.56276746 m s⁻². Sort the roots even if the response decreases: 20242.6 km ≤ h ≤ 22271.4 km.
- The lower and upper input deviations are 958.596 km and 1070.25 km. They differ because the inverse response is nonlinear; a symmetric error bar would only be a local approximation. This interval includes reading uncertainty only, not model error or uncertainty in fixed inputs.
Answer h ≈ 21201.2 km; compatible interval [20242.6, 22271.4] km. Endpoint interpolation gives 30092.5 km.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.