University Physics III · topic UPIII-01.05
Mass–spring oscillators
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.
Mass–spring oscillators: predict a finite change
A mass–spring oscillator is a body of mass m held by a spring whose restoring force is proportional to displacement, so that it oscillates at the single angular frequency ω = √(k/m) whatever the amplitude. Use the supplied topic model Y_out = 1000*m*g/k, where Y_out is STATIC EXTENSION Δl (mm). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; g = 9.81 m s⁻². k is Spring stiffness k, measured in N m⁻¹. Keep m (Load mass m) = 0.3 kg; a (Amplitude A) = 0.05 m fixed. For this exercise use only 23.1875 N m⁻¹ ≤ k ≤ 46.8125 N m⁻¹ 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 k = 28.385 N m⁻¹ and 31.6925 N m⁻¹. Find the signed output change ΔY_out and the average sensitivity ΔY_out/Δk across this interval. State whether the response increases or decreases; distinguish this interval average from a derivative at one point.
STATIC EXTENSION Δl against k
Predictions from the supplied Mass–spring oscillators 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): STATIC EXTENSION Δl / mm
Horizontal axis (x): k: Spring stiffness k / N m⁻¹
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| k: Spring stiffness k / N m⁻¹ | STATIC EXTENSION Δl / mm |
|---|---|
| 23.1875 | 126.92183288409703 |
| 29.09375 | 101.15574650912997 |
| 35 | 84.08571428571429 |
| 40.90625 | 71.9449961802903 |
| 46.8125 | 62.86782376502003 |
These are the supplied values, not a worked answer. No fitted trend is drawn.
Show a hint
Substitute each value of k separately while keeping every other input fixed, then subtract in the same order.
Reveal the worked solution
- At k = 28.385 N m⁻¹, substitute into Y_out = 1000*m*g/k: Y₀ = 103.682 mm.
- At k = 31.6925 N m⁻¹, the same fixed-parameter model gives Y₁ = 92.8611 mm.
- Signed change: ΔY_out = Y₁ − Y₀ = -10.8204 mm; the input change is Δk = 3.3075 N m⁻¹.
- Average sensitivity is ΔY_out/Δk = -3.27148 mm per N m⁻¹. 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 = 103.682 mm, Y₁ = 92.8611 mm; ΔY_out = -10.8204 mm; average sensitivity = -3.27148 mm per N m⁻¹.
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.
Mass–spring oscillators: infer an input with uncertainty
A mass–spring oscillator is a body of mass m held by a spring whose restoring force is proportional to displacement, so that it oscillates at the single angular frequency ω = √(k/m) whatever the amplitude. Use the supplied topic model Y_out = 1000*m*g/k, where Y_out is STATIC EXTENSION Δl (mm). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; g = 9.81 m s⁻². k is Spring stiffness k, measured in N m⁻¹. Keep m (Load mass m) = 0.3 kg; a (Amplitude A) = 0.05 m fixed. For this exercise use only 23.1875 N m⁻¹ ≤ k ≤ 46.8125 N m⁻¹ 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 = 78.273879 mm, with a bounded reading uncertainty ±2.3109 mm. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown k by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.0023625 N m⁻¹, 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.
STATIC EXTENSION Δl against k
Predictions from the supplied Mass–spring oscillators 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): STATIC EXTENSION Δl / mm
Horizontal axis (x): k: Spring stiffness k / N m⁻¹
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| k: Spring stiffness k / N m⁻¹ | STATIC EXTENSION Δl / mm |
|---|---|
| 23.1875 | 126.92183288409703 |
| 29.09375 | 101.15574650912997 |
| 35 | 84.08571428571429 |
| 40.90625 | 71.9449961802903 |
| 46.8125 | 62.86782376502003 |
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(k) = 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(23.1875) = 126.922 mm and Y_out(46.8125) = 62.8678 mm. The target lies between them. On the selected branch the response is strictly decreasing.
- Endpoint interpolation gives k_line = 23.1875 + (78.2739 − 126.922)(46.8125 − 23.1875)/(62.8678 − 126.922) = 41.1303 N m⁻¹. 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 k = 37.5987 N m⁻¹; substituting back gives Y_out = 78.273879 mm.
- The interpolation error in the input is |k_line − k| = 3.53154 N m⁻¹. This is larger than the requested numerical tolerance.
- Invert both limits Y_out = 75.962979 mm and 80.584779 mm. Sort the roots even if the response decreases: 36.5205 N m⁻¹ ≤ k ≤ 38.7426 N m⁻¹.
- The lower and upper input deviations are 1.07821 N m⁻¹ and 1.14381 N m⁻¹. 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 k ≈ 37.5987 N m⁻¹; compatible interval [36.5205, 38.7426] N m⁻¹. Endpoint interpolation gives 41.1303 N m⁻¹.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.