Cambridge International AS & A Level · P5 · structured
Select the axes and recover two constants
A2 practical 5: choose transformed axes before fitting · Original GioPhysics question with a detailed, mark-by-mark answer guide.
- Demand
- demanding
- Marks
- 12
- Topics
- 5
- Answer
- Complete
A mass on a spring follows T=2pi sqrt((m+m0)/k), where T is period, m is total suspended mass, k is spring constant and m0 is an unknown effective mass. In these constructed readings, t20 is the time for 20 complete oscillations. Raw pairs (m/kg, t20/s): (0.100,9.8), (0.150,11.6), (0.200,13.3), (0.250,14.7), (0.300,15.9), (0.350,17.2). Choose for yourself which quantities should be the graph axes to obtain a straight line; construct the required derived columns from the raw data, draw and fit all points, and find k and m0 using a large fitted-line triangle. Separately, for positive dimensionless x and y, derive the straight-line logarithmic forms of y=A x^n and y=B exp(qx), identifying the axes, slope and intercept in each case. These are constructed data, not observed experimental evidence.
- (a)
Independently select straight-line axes and construct the needed derived table from raw mass and 20-cycle time.
4 marks - (b)
Draw and analyse the transformed fit.
4 marks - (c)
Derive the physical constants, then linearise the separate power and exponential models and interpret their lines.
4 marks
Ready to self-mark?Reveal the detailed answer guide
(a)
Independently select straight-line axes and construct the needed derived table from raw mass and 20-cycle time.
List the given quantities with units, identify the required quantity, write the governing relationship before substituting, and keep extra digits until the final line so rounding does not distort the result.
- 1
(a) Square the spring model and independently choose T² against m: 2 marks. Calculate periods from t20/20 and squared periods with justified precision: 2 marks.
(b)
Draw and analyse the transformed fit.
List the given quantities with units, identify the required quantity, write the governing relationship before substituting, and keep extra digits until the final line so rounding does not distort the result.
- 1
(b) Plot all six transformed pairs with correct labels/scales and fit a reasonable straight line: 2 marks. Read a triangle longer than half the fitted line and recover its slope/intercept: 2 marks. Reference unrounded fit M=1.9899286 s² kg^-1, C=0.04069524 s².
(c)
Derive the physical constants, then linearise the separate power and exponential models and interpret their lines.
List the given quantities with units, identify the required quantity, write the governing relationship before substituting, and keep extra digits until the final line so rounding does not distort the result.
- 1
(c) Derive and use k=4pi²/M≈19.8 N m^-1 and m0=C/M≈0.0205 kg: 2 marks; accept a consistent independently drawn fit, including about 0.021 kg from rounded graph values. Taking common logs gives lg y=n lg x+lg A, with horizontal lg x, vertical lg y, slope n and intercept lg A: 1 mark. Taking natural logs gives ln y=qx+ln B, with horizontal x, vertical ln y, slope q and intercept ln B: 1 mark.
Complete marking guidance
Use these worked steps alongside the part-by-part answers. They explain the method and mark allocation; they do not add extra marks.
- (a) Square the spring model and independently choose T² against m: 2 marks. Calculate periods from t20/20 and squared periods with justified precision: 2 marks.
- (b) Plot all six transformed pairs with correct labels/scales and fit a reasonable straight line: 2 marks. Read a triangle longer than half the fitted line and recover its slope/intercept: 2 marks. Reference unrounded fit M=1.9899286 s² kg^-1, C=0.04069524 s².
- (c) Derive and use k=4pi²/M≈19.8 N m^-1 and m0=C/M≈0.0205 kg: 2 marks; accept a consistent independently drawn fit, including about 0.021 kg from rounded graph values. Taking common logs gives lg y=n lg x+lg A, with horizontal lg x, vertical lg y, slope n and intercept lg A: 1 mark. Taking natural logs gives ln y=qx+ln B, with horizontal x, vertical ln y, slope q and intercept ln B: 1 mark.
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