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9702 · 1.3

Errors and Uncertainties in Paper 5 — practice questions

Practice and worked examples for 9702 Errors and Uncertainties in Paper 5. Short previews only — attempt the full question in MarkScheme against the official scheme.

Worked example 1

Fig. 1.1 shows a thin cylindrical metal rod of length LL.

[Figure: Fig. 1.1 shows a thin cylindrical metal rod of length L.]

One end of the rod is hit with a hammer. A stationary sound wave is set up within the rod. The rod vibrates at its resonant frequency ff.

A microphone placed at the other end of the rod detects the sound wave emitted from the rod. The frequency of the detected sound is also ff.

A number of rods of different length are available.

It is suggested that ff is related to LL by the relationship 2fLn=Eρ2fL^n = \sqrt{\frac{E}{\rho}} where ρ\rho is the density of the metal, and EE and nn are constants.

Plan a laboratory experiment to test the relationship between ff and LL.

Draw a diagram showing the arrangement of your equipment.

Explain how the results could be used to determine values for EE and nn.

In your plan you should include:

  • the procedure to be followed
  • the measurements to be taken
  • the control of variables
  • the analysis of the data
  • any safety precautions to be taken.
Show solution outline
  1. LL is the independent variable and ff is the dependent variable, or vary LL and measure ff.
  2. Keep ρ\rho constant
  3. Labelled diagram of workable experiment including: • rod supported by string / elastic bands from a clamp • clamp attached to stand, with stand on bench • two labels from stand, clamp, hammer, microphone, rod, string.
  4. Diagram showing labelled microphone connected to labelled oscilloscope.
  5. Method to measure LL, e.g. use a metre rule
  6. Method to measure mass (mm) (of metal rod), e.g. use a (top-pan) balance
  7. Plots a graph of log⁡f\log f against log⁡L\log L or equivalent e.g. log⁡f\log f against log⁡1L\log \frac{1}{L}
  8. n=−gradientn = -\text{gradient}
  9. E=4ρ×102×y-interceptE = 4\rho \times 10^{2 \times y\text{-intercept}}
  10. Any six from the following points:

Worked example 2

Values of RR and II are given in Table 2.1.

R/kΩR/\text{k}\OmegaI/μAI/\mu\text{A}1I/A−1\frac{1}{I}/\text{A}^{-1}
1.25225±5225 \pm 5
------
2.55185±5185 \pm 5
3.90160±5160 \pm 5
5.25140±5140 \pm 5
6.55125±5125 \pm 5
7.80115±5115 \pm 5

Calculate and record values of 1I/A−1\frac{1}{I}/\text{A}^{-1} in Table 2.1.

Include the absolute uncertainties in 1I\frac{1}{I}.

R/kΩR/\text{k}\OmegaI/μAI/\mu\text{A}1I/A−1\frac{1}{I}/\text{A}^{-1}
1.25225±5225 \pm 5
------
2.55185±5185 \pm 5
3.90160±5160 \pm 5
5.25140±5140 \pm 5
6.55125±5125 \pm 5
7.80115±5115 \pm 5
Show solution outline
  1. Correctly calculated values for 1I/A−1\frac{1}{I} / \text{A}^{-1} in the table.
  2. Correctly calculated uncertainties in 1I\frac{1}{I}.

Worked example 3

Values of LL and TT are given in Table 2.1.

Table 2.1

L/cmL/\text{cm}T/10−5sT/10^{-5}\text{s}lg⁡(L/cm)\lg (L/\text{cm})lg⁡(T/10−5s)\lg (T/10^{-5}\text{s})
5424±124 \pm 1
------
7032±132 \pm 1
8639±139 \pm 1
10849±249 \pm 2
14064±264 \pm 2
16774±274 \pm 2

Calculate and record values of lg⁡(L/cm)\lg (L/\text{cm}) and lg⁡(T/10−5s)\lg(T/10^{-5}\text{s}) in Table 2.1. Include the absolute uncertainties in lg⁡T\lg T.

L/cmL/\text{cm}T/10−5sT/10^{-5}\text{s}lg⁡(L/cm)\lg (L/\text{cm})lg⁡(T/10−5s)\lg (T/10^{-5}\text{s})
5424±124 \pm 1
------
7032±132 \pm 1
8639±139 \pm 1
10849±249 \pm 2
14064±264 \pm 2
16774±274 \pm 2
Show solution outline
  1. Values of lg (LL / cm) and lg (T/10−5T/10^{-5} s) correct as shown above.
  2. Uncertainties in lg (T/10−5T/10^{-5} s) correct as shown above.

Past-paper practice

February/March · Paper 52 · Q1 · 15 marks

Fig. 1.1 shows a thin cylindrical metal rod of length $L$. [Figure: Fig. 1.1 shows a thin cylindrical metal rod of length L.] One end of the rod is hit with a hammer. A stationary sound wave is set up within the rod. The rod vibrates at its resonant frequency $f$. …

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May/June · Paper 52 · Q2(b) · 2 marks

Values of $L$ and $T$ are given in Table 2.1. **Table 2.1** | $L/\text{cm}$ | $T/10^{-5}\text{s}$ | $\lg (L/\text{cm})$ | $\lg (T/10^{-5}\text{s})$ | …

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February/March · Paper 52 · Q1 · 15 marks

An electric pump is placed in a container of liquid. A model wind turbine is connected to the pump by a cable, as shown in Fig. 1.1. [Figure: Fig. 1.1 shows a model wind turbine with blades connected by a cable to a pump submerged in a container of liquid. The pump pushes liquid up a vertical pipe to a height h. Moving air causes the turbine to rotate.] The turbine is placed in moving air. As the turbine blades turn, electricity is generated and the pump pushes liquid through a vertical pipe. …

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February/March · Paper 52 · Q2(b) · 2 marks

Values of $R$ and $I$ are given in Table 2.1. | $R/\text{k}\Omega$ | $I/\mu\text{A}$ | $\frac{1}{I}/\text{A}^{-1}$ | |---|---|---| …

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February/March · Paper 52 · Q2(c)(i) · 2 marks

Plot a graph of $\frac{1}{I}/\text{A}^{-1}$ against $R/\text{k}\Omega$. Include error bars for $\frac{1}{I}$.

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February/March · Paper 52 · Q2(c)(ii) · 2 marks

Draw the straight line of best fit and a worst acceptable straight line on your graph. Label both lines.

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February/March · Paper 52 · Q2(c)(iii) · 2 marks

Determine the gradient of the line of best fit. Include the absolute uncertainty in your answer.

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February/March · Paper 52 · Q2(c)(iv) · 2 marks

Determine the y-intercept of the line of best fit. Include the absolute uncertainty in your answer.

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  • February/March 2024 · 9702/52 · Q1

    Fig. 1.1 shows a thin cylindrical metal rod of length $L$. <img src="9702_52_F_M_24_1.png" alt="Fig. 1.1 shows a thin cylindrical metal rod of length L." /> One end of the rod is hit with a hammer. A stationary sound wave is set up within the rod. The rod vibrates at its resonant frequency $f$. A microphone placed at the other end of the rod detects the sound wave emitted from the rod. The frequency of the detected sound is also $f$. A number of rods of different length are available. It is

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  • February/March 2024 · 9702/52 · Q2(b)

    Values of $R$ and $I$ are given in Table 2.1. | $R/\text{k}\Omega$ | $I/\mu\text{A}$ | $\frac{1}{I}/\text{A}^{-1}$ | |---|---|---| | 1.25 | $225 \pm 5$ | | | 2.55 | $185 \pm 5$ | | | 3.90 | $160 \pm 5$ | | | 5.25 | $140 \pm 5$ | | | 6.55 | $125 \pm 5$ | | | 7.80 | $115 \pm 5$ | | Calculate and record values of $\frac{1}{I}/\text{A}^{-1}$ in Table 2.1. Include the absolute uncertainties in $\frac{1}{I}$. | $R/\text{k}\Omega$ | $I/\mu\text{A}$ | $\frac{1}{I}/\text{A}^{-1}$ | | --- | --- | ---

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  • May/June 2024 · 9702/52 · Q2(b)

    Values of $L$ and $T$ are given in Table 2.1. **Table 2.1** | $L/\text{cm}$ | $T/10^{-5}\text{s}$ | $\lg (L/\text{cm})$ | $\lg (T/10^{-5}\text{s})$ | |:---:|:---:|:---:|:---:| | 54 | $24 \pm 1$ | | | | 70 | $32 \pm 1$ | | | | 86 | $39 \pm 1$ | | | | 108 | $49 \pm 2$ | | | | 140 | $64 \pm 2$ | | | | 167 | $74 \pm 2$ | | | Calculate and record values of $\lg (L/\text{cm})$ and $\lg(T/10^{-5}\text{s})$ in Table 2.1. Include the absolute uncertain

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