University Physics II · topic UPII-E3.01
Internal energy and the first law
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.
Internal Energy & the First Law: predict a finite change
Internal energy is the total microscopic kinetic and potential energy a system holds in its own rest frame, and the first law says its change equals the heat added minus the work the system does: ΔU = Q − W. Use the supplied topic model Y_out = q-w, where Y_out is STORED ΔU = Q − W (J). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. q is Heat into the system Q, measured in J. Keep w (Work done by the system W) = 250 J fixed. For this exercise use only -708.75 J ≤ q ≤ 708.75 J 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 q = -396.9 J and -198.45 J. Find the signed output change ΔY_out and the average sensitivity ΔY_out/Δq across this interval. State whether the response increases or decreases; distinguish this interval average from a derivative at one point.
STORED ΔU = Q − W against q
Predictions from the supplied Internal Energy & the First Law 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): STORED ΔU = Q − W / J
Horizontal axis (x): q: Heat into the system Q / J
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| q: Heat into the system Q / J | STORED ΔU = Q − W / J |
|---|---|
| -708.75 | -958.75 |
| -354.37500000000006 | -604.375 |
| -1.1368683772161603e-13 | -250.0000000000001 |
| 354.3749999999998 | 104.37499999999977 |
| 708.7499999999998 | 458.7499999999998 |
These are the supplied values, not a worked answer. No fitted trend is drawn.
Show a hint
Substitute each value of q separately while keeping every other input fixed, then subtract in the same order.
Reveal the worked solution
- At q = -396.9 J, substitute into Y_out = q-w: Y₀ = -646.9 J.
- At q = -198.45 J, the same fixed-parameter model gives Y₁ = -448.45 J.
- Signed change: ΔY_out = Y₁ − Y₀ = 198.45 J; the input change is Δq = 198.45 J.
- Average sensitivity is ΔY_out/Δq = 1 J per J. Its positive sign means that this response increases 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 = -646.9 J, Y₁ = -448.45 J; ΔY_out = 198.45 J; average sensitivity = 1 J per J.
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.
Internal Energy & the First Law: infer an input with uncertainty
Internal energy is the total microscopic kinetic and potential energy a system holds in its own rest frame, and the first law says its change equals the heat added minus the work the system does: ΔU = Q − W. Use the supplied topic model Y_out = q-w, where Y_out is STORED ΔU = Q − W (J). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. q is Heat into the system Q, measured in J. Keep w (Work done by the system W) = 250 J fixed. For this exercise use only -708.75 J ≤ q ≤ 708.75 J 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 = -94.075 J, with a bounded reading uncertainty ±82.924 J. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown q by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.14175 J, 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.
STORED ΔU = Q − W against q
Predictions from the supplied Internal Energy & the First Law 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): STORED ΔU = Q − W / J
Horizontal axis (x): q: Heat into the system Q / J
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| q: Heat into the system Q / J | STORED ΔU = Q − W / J |
|---|---|
| -708.75 | -958.75 |
| -354.37500000000006 | -604.375 |
| -1.1368683772161603e-13 | -250.0000000000001 |
| 354.3749999999998 | 104.37499999999977 |
| 708.7499999999998 | 458.7499999999998 |
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(q) = 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(-708.75) = -958.75 J and Y_out(708.75) = 458.75 J. The target lies between them. On the selected branch the response is strictly increasing.
- Endpoint interpolation gives q_line = -708.75 + (-94.075 − -958.75)(708.75 − -708.75)/(458.75 − -958.75) = 155.925 J. 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 q = 155.925 J; substituting back gives Y_out = -94.075 J.
- The interpolation error in the input is |q_line − q| = 5.68434e-14 J. This is within the requested numerical tolerance.
- Invert both limits Y_out = -176.999 J and -11.151 J. Sort the roots even if the response decreases: 73.001 J ≤ q ≤ 238.849 J.
- The lower and upper input deviations are 82.924 J and 82.924 J. They agree on this branch; this does not establish linearity for changes in other inputs or outside the interval. This interval includes reading uncertainty only, not model error or uncertainty in fixed inputs.
Answer q ≈ 155.925 J; compatible interval [73.001, 238.849] J. Endpoint interpolation gives 155.925 J.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.