University Physics III · topic UPIII-11.01
Nuclear constituents, size, and density
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.
Further practice
Work through a fresh case before tackling the extension.
Nuclear constituents, size, and density: predict a finite change
A nucleus is a bound cluster of Z protons and N neutrons whose radius, measured by scattering, grows as R = R₀A^(1/3) with R₀ ≈ 1.2 fm, so that its volume, not its radius, is proportional to the nucleon number A. Use the supplied topic model Y_out = (2.88*(anum/(1.98+0.0155*anum^(2/3)))*(anum+4)/(kin*anum)), where Y_out is HEAD-ON DISTANCE d (before display rounding) (fm). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. kin is Alpha kinetic energy K (lab), measured in MeV. Keep anum (Target nucleon number A) = 197; reach (Reach of the attraction a) = 1.5 fm fixed. For this exercise use only 6.0375 MeV ≤ kin ≤ 35.9625 MeV 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. Calculate Y_out at kin = 12.621 MeV and 16.8105 MeV. Find the signed output change ΔY_out and the average sensitivity ΔY_out/Δkin across this interval. State whether the response increases or decreases; distinguish this interval average from a derivative at one point.
HEAD-ON DISTANCE d (before display rounding) against kin
Predictions from the supplied Nuclear constituents, size, and density 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): HEAD-ON DISTANCE d (before display rounding) / fm
Horizontal axis (x): kin: Alpha kinetic energy K (lab) / MeV
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| kin: Alpha kinetic energy K (lab) / MeV | HEAD-ON DISTANCE d (before display rounding) / fm |
|---|---|
| 6.0375 | 38.27913481910075 |
| 13.518749999999999 | 17.095535938627517 |
| 21 | 11.005251260491463 |
| 28.481249999999996 | 8.114470975477579 |
| 35.9625 | 6.426424093717643 |
These are the supplied values, not a worked answer. No fitted trend is drawn.
Show a hint
Substitute each value of kin separately while keeping every other input fixed, then subtract in the same order.
Reveal the worked solution
- At kin = 12.621 MeV, substitute into Y_out = (2.88*(anum/(1.98+0.0155*anum^(2/3)))*(anum+4)/(kin*anum)): Y₀ = 18.3116 fm.
- At kin = 16.8105 MeV, the same fixed-parameter model gives Y₁ = 13.748 fm.
- Signed change: ΔY_out = Y₁ − Y₀ = -4.56359 fm; the input change is Δkin = 4.1895 MeV.
- Average sensitivity is ΔY_out/Δkin = -1.08929 fm per MeV. Its negative sign means that this response decreases over the specified interval.
- This is a secant slope between two settings. It equals a point derivative only for a locally linear relation or in a justified small-interval limit.
Answer Y₀ = 18.3116 fm, Y₁ = 13.748 fm; ΔY_out = -4.56359 fm; average sensitivity = -1.08929 fm per MeV.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.
Extra challenging questions
Work through the linked stages and explain what each result means.
Nuclear constituents, size, and density: infer an input with uncertainty
A nucleus is a bound cluster of Z protons and N neutrons whose radius, measured by scattering, grows as R = R₀A^(1/3) with R₀ ≈ 1.2 fm, so that its volume, not its radius, is proportional to the nucleon number A. Use the supplied topic model Y_out = (2.88*(anum/(1.98+0.0155*anum^(2/3)))*(anum+4)/(kin*anum)), where Y_out is HEAD-ON DISTANCE d (before display rounding) (fm). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. kin is Alpha kinetic energy K (lab), measured in MeV. Keep anum (Target nucleon number A) = 197; reach (Reach of the attraction a) = 1.5 fm fixed. For this exercise use only 6.0375 MeV ≤ kin ≤ 35.9625 MeV 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 = 9.513941 fm, with a bounded reading uncertainty ±0.46313 fm. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown kin by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.0029925 MeV, 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.
HEAD-ON DISTANCE d (before display rounding) against kin
Predictions from the supplied Nuclear constituents, size, and density 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): HEAD-ON DISTANCE d (before display rounding) / fm
Horizontal axis (x): kin: Alpha kinetic energy K (lab) / MeV
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| kin: Alpha kinetic energy K (lab) / MeV | HEAD-ON DISTANCE d (before display rounding) / fm |
|---|---|
| 6.0375 | 38.27913481910075 |
| 13.518749999999999 | 17.095535938627517 |
| 21 | 11.005251260491463 |
| 28.481249999999996 | 8.114470975477579 |
| 35.9625 | 6.426424093717643 |
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(kin) = 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(6.0375) = 38.2791 fm and Y_out(35.9625) = 6.42642 fm. The target lies between them. On the selected branch the response is strictly decreasing.
- Endpoint interpolation gives kin_line = 6.0375 + (9.51394 − 38.2791)(35.9625 − 6.0375)/(6.42642 − 38.2791) = 33.0618 MeV. 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 kin = 24.2918 MeV; substituting back gives Y_out = 9.513941 fm.
- The interpolation error in the input is |kin_line − kin| = 8.77009 MeV. This is larger than the requested numerical tolerance.
- Invert both limits Y_out = 9.050811 fm and 9.977071 fm. Sort the roots even if the response decreases: 23.1641 MeV ≤ kin ≤ 25.5348 MeV.
- The lower and upper input deviations are 1.12761 MeV and 1.24301 MeV. 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 kin ≈ 24.2918 MeV; compatible interval [23.1641, 25.5348] MeV. Endpoint interpolation gives 33.0618 MeV.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.