A mass oscillates on the end of a spring. The motion of the mass is described by the equation: .
Where is the frequency of the oscillations, is the spring constant of the spring and is the mass hanging on the spring. The period of the spring is measured for a number of different masses. The period is related to the frequency of oscillations through the relationship: .
A student is trying to determine the spring constant, , of the spring. In order to do this they plan to plot a graph. Which of the following statements about this graph are correct?

Question

BlackTom AI · Accepted Answer

From 2π f = sqrt(k/m) and T = 1/f we can derive a linear relation suitable for graphing. Start by substituting f = 1/T:

2π(1/T) = sqrt(k/m).

Square both sides to remove the square root:
(2π)² (1/T)² = k/m
4π² / T² = k/m.

Rearrange to express T² in terms of m (a straight-line form y = mx):
T² = (4π² / k) m.
This is linear with T² as the dependent variable (y) and m as the independent variable (x). The gradient (slope) is 4π² / k. So plotting x-axis: m and y-axis: T² gives gradient = 4π² / k. From that gradient you can solve for k: k = 4π² / (gradient).

Alternatively rearrange the squared equation to put 4π² m on one side:
4π² m = k T².
This is linear with x = T² and y = 4π² m, and the gradient is k. Thus plotting x-axis: T² and y-axis: 4π² m yields a slope equal to the spring constant k directly.

Why these are correct: the algebraic manipulations above show T² is proportional to m with constant 4π²/k, so T² vs m is linear; multiplying both sides by 4π² produces a direct proportionality between 4π² m and T² with proportionality constant k.

Why other graph choices are incorrect (briefly):
- Plotting T vs m is not linear because T = 2π sqrt(m/k) (T ∝ sqrt(m)), so the graph would be a square-root curve, not a straight line.
- Any choice that puts T (not T²) on the axis expecting a simple linear gradient will yield the wrong slope and relation.

Conclusion: The two correct plotting choices are (1) x = m, y = T² with gradient 4π²/k, and (2) x = T², y = 4π² m with gradient k — which matches the correct answers.

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