Uncertainties & Errors | DP IB Physics: SL Structured Questions 2016

1a

3 marks

A student uses a stopwatch to measure the time taken for a pendulum to complete one swing.

The display on the stopwatch after the pendulum completes 10 swings is shown on the diagram.

1-2-q1a-question-stem-easy-sq-sl-phy

For this reading, determine:

(i)

The absolute uncertainty

[1]

(ii)

The fractional uncertainty

[1]

(iii)

The percentage uncertainty

[1]

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1b

4 marks

Calculate the mean time for one complete swing with its absolute uncertainty and a percentage uncertainty.

Give your answer to an appropriate number of significant figures.

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1c

2 marks

Draw lines between the three types of error to show if the error affects the precision or accuracy of a result.

1-2-q1c-question-stem-easy-sq-sl-phy

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1d

4 marks

In order to reduce errors, a different student collected measurements of time over 20 cycles instead of 10.

Complete the following sentences by circling the correct word and placing a tick (✓) next to the correct explanation

Repeated measurements reduce systematic / random errors because...

	using a larger sample to calculate the mean value reduces the uncertainty in the final value
	these cause values to be different by the same amount each time, hence they are not influenced by repetition

Repeated measurements have no effect on systematic / random errors because...

	using a larger sample to calculate the mean value reduces the uncertainty in the final value
	these cause values to be different by the same amount each time, hence they are not influenced by repetition

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3a

4 marks

List the following currents from largest to smallest percentage uncertainty:

4.1 ± 0.2 A

5 ± 1 mA

7.30 ± 0.23 A

0.5 ± 0.05 mA

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3b

4 marks

A circuit is set up to measure the resistance, R, of a resistor. The potential difference (p.d), V, across the resistor and the current, I, are related by the equation:

$V = I R$

The readings for the p.d, V, and the corresponding current, I, are obtained and plotted on a graph with a line of best fit drawn.

1-2-q3b-question-stem-easy-sq-sl-phy

Complete the following sentences by circling the correct words:

Current and potential difference have a directly / inversely proportional relationship.

This means when one quantity is zero, the other will be zero / non-zero.

On the graph, the y-intercept is zero / non-zero, hence, this shows the readings have / have not been affected by systematic / random uncertainties.

The points on the graph are close to / scattered around the line of best fit, hence, this shows the readings have / have not been affected by systematic / random uncertainties.

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3c

6 marks

The student plots error bars on the graph along with lines of maximum and minimum gradient.

1-2-q3c-question-stem-easy-sq-sl-phy

(i)

Determine the percentage uncertainty in the gradient using the following equations of the lines:

Best line	$I = 0.045 V + 0.05$
Maximum line	$I = 0.052 V + 0.03$
Minimum line	$I = 0.036 V + 0.07$

[3]

(ii)

The student suggests the analogue ammeter used to measure the current may have introduced a positive zero error. State what is meant by a zero error.

[1]

(iii)

Outline one way a zero error could affect the results and suggest how this type of error can be fixed.

[2]

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3d

5 marks

In another student's experiment, the resistance of the resistor, R, is determined using the following data:

Current, I	0.74 ± 0.01 A
Potential difference, V	6.5 ± 0.2 V

Calculate the value of R, together with its percentage uncertainty. Give your answer to an appropriate number of significant figures.

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1a

4 marks

One method to determine the acceleration of free fall g involves measuring the time period of a simple pendulum T. It is related to the length of the pendulum l by the equation

$T = 2 π \sqrt{\frac{l}{g}}$

In this method, l was found to be (0.500 ± 0.001) m while the period T was measured to be (1.42 ± 0.02) s.

Based on these measurements, determine the value of g and its absolute uncertainty. Give your final answer to an appropriate degree of precision.

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1b

4 marks

Another method to determine the acceleration of free fall involves timing the descent of a small metal ball bearing, released vertically via an electromagnetic trapdoor. In one particular trial, the displacement of the ball bearing s is measured as (266 ± 1) cm and the time measured t is (0.740 ± 0.005) s.

Determine the value of g using this method, and its absolute uncertainty. Give your final answer to an appropriate degree of precision.

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1a

4 marks

A vernier calliper is used to measure the length of a piece of copper wire.

q1a_uncertainties-and-errors_ib-sl-physics-sq

Determine the full reading of the vernier calliper with its absolute uncertainty.

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1b

3 marks

The reading from part (a) is taken after a mass has been added to the copper wire of length L and the wire extends.

The original length of the wire L was 14.9 ± 0.05 mm.

Calculate the extension ∆L of the copper wire after the mass has been added. Give the range of the uncertainty of this extension.

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1c

4 marks

Tensile strain is a measure of the deformation of an object and is defined as the ratio between the extension of the wire and its original length.

Tensile Strain, ε = $\frac{Δ L}{L}$

Deduce the tensile strain of the copper wire and its percentage uncertainty.

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1d

2 marks

State two ways to reduce the systematic error in this experiment.

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5a

2 marks

The decay of a radioactive substance can be represented by the equation:

$C = C_{0} e^{- λ t}$

where C is the count rate of the sample at time t, C₀ is the initial count rate at time t = 0 and λ is the decay constant.

The half-life, t_½ of the radioactive substance is given by

$t_{1 / 2}$ = $\frac{\ln 2}{λ}$

An experiment was performed to determine the half-life of a radioactive substance which was a beta emitter. The radioactive source was placed close to a detector.

The results in the table show the total count for exactly 5 minutes, repeated at 15 minute intervals.

time, t / minutes	total count, recorded in 5 minutes	Count rate, C / counts minute^–1	ln (C / minute^–1)
0	1016	183	5.21
15	920	164	5.10
30	835	147	4.99
45	758	132	4.88
60	665	113	4.73
75	623	105	4.65
90	568	94	4.54
105	520	84	4.43
120	476	75	4.32
135	437	67	4.21

The uncertainty in the count rate, C, is given by

ΔC = ± $\sqrt{C}$

Calculate the uncertainty in each value of ln C.

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5b

3 marks

Draw a line of best fit and error bars for each point on the graph.

q5b_uncertainties--errors_ib-sl-physics-sq-medium

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5c

5 marks

The activity of the sample λ= – $\frac{\ln C}{t}$

Calculate the activity of the sample and its percentage uncertainty.

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5d

3 marks

Another student performed the same experiment with identical equipment but took total counts over a 1-minute period rather than a 5-minute period. The total count, C, at 140 minutes was equal to 54 counts.

Use the relationship

ln(x) = y so x = e^y

to estimate the percentage uncertainty in this total count and explain the advantage of using a larger time.

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Time, t₁ / s	Time, t₂ / s	Time, t₃ / s	Time, t₄ / s	Time, t₅ / s	Time, t₆ / s
0.423	0.422	0.424	0.421	0.423	0.424

Measured time to fall	0.322 ± 0.002 s
Distance between the point of release and the trapdoor	543 ± 1 mm
Diameter of the metal sphere	10.0 ± 0.1 mm

Operation	Example	Propagation Rule
Addition & Subtraction	$y = a \pm b$	$Δ y = Δ a + Δ b$ The sum of the absolute uncertainties
Multiplication & Division	$y = a \times b$ or $y = \frac{a}{b}$	$\frac{Δ y}{y} = \frac{Δ a}{a} + \frac{Δ b}{b}$ The sum of the fractional uncertainties
Power	$y = a^{\pm n}$	$\frac{Δ y}{y} = n (\frac{Δ a}{a})$ The magnitude of n times the fractional uncertainty

Load / N	Length / mm	Corrected Length / mm
1.00	3.00
1.50	3.54
2.00	4.02
2.50	4.61
3.00	4.99

DP IB Physics: SL

Topic Questions

1.2 Uncertainties & Errors

s / m	t₁ / s	t₂ / s	t₃ / s	mean time t / s	t² / s²
0.100	0.141	0.138	0.144	0.141	0.020
0.200	0.201	0.205	0.209	0.205	0.042
0.300	0.240	0.246	0.250	0.245	0.0600
0.400	0.285	0.288	0.284	0.286	0.0818
0.500	0.315	0.319	0.323	0.319	0.102
0.600	0.345	0.349	0.354	0.349	0.122
0.700	0.376	0.379	0.382	0.379	0.144
0.800	0.399	0.405	0.407	0.404	0.163
0.900	0.426	0.428	0.432	0.429	0.184