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SL Paper 3

A radio wave of wavelength $\lambda$ is incident on a conductor. The graph shows the variation with wavelength $\lambda$ of the maximum distance d travelled inside the conductor.

For $\lambda$ = 5.0 x 10⁵ m, calculate the

The graph shows the variation with wavelength $\lambda$ of d ². Error bars are not shown and the line of best-fit has been drawn.

A student states that the equation of the line of best-fit is d ²= a + b $\lambda$ . When d ² and $\lambda$ are expressed in terms of fundamental SI units, the student finds that a = 0.040 x 10^–4 and b = 1.8 x 10^–11.

Suggest why it is unlikely that the relation between d and $\lambda$ is linear.

[1]

fractional uncertainty in d.

[2]

b.i.

percentage uncertainty in d ².

[1]

b.ii.

State the fundamental SI unit of the constant a and of the constant b.

[2]

c.i.

Determine the distance travelled inside the conductor by very high frequency electromagnetic waves.

[2]

c.ii.

The equipment shown in the diagram was used by a student to investigate the variation with volume, of the pressure p of air, at constant temperature. The air was trapped in a tube of constant cross-sectional area above a column of oil.

The pump forces oil to move up the tube decreasing the volume of the trapped air.

The student measured the height H of the air column and the corresponding air pressure p. After each reduction in the volume the student waited for some time before measuring the pressure. Outline why this was necessary.

[1]

The following graph of p versus $\frac{1}{H}$ was obtained. Error bars were negligibly small.

The equation of the line of best fit is $p = a + \frac{b}{H}$ .

Determine the value of b including an appropriate unit.

[3]

Outline how the results of this experiment are consistent with the ideal gas law at constant temperature.

[2]

The cross-sectional area of the tube is 1.3 × 10^–3 $\,$ m² and the temperature of air is 300 K. Estimate the number of moles of air in the tube.

[2]

The equation in (b) may be used to predict the pressure of the air at extremely large values of $\frac{1}{H}$ . Suggest why this will be an unreliable estimate of the pressure.

[2]

An apparatus is used to verify a gas law. The glass jar contains a fixed volume of air. Measurements can be taken using the thermometer and the pressure gauge.

The apparatus is cooled in a freezer and then placed in a water bath so that the temperature of the gas increases slowly. The pressure and temperature of the gas are recorded.

The graph shows the data recorded.

Identify the fundamental SI unit for the gradient of the pressure–temperature graph.

[1]

The experiment is repeated using a different gas in the glass jar. The pressure for both experiments is low and both gases can be considered to be ideal.

(i) Using the axes provided in (a), draw the expected graph for this second experiment.

(ii) Explain the shape and intercept of the graph you drew in (b)(i).

[3]

In a simple pendulum experiment, a student measures the period T of the pendulum many times and obtains an average value T = (2.540 ± 0.005) s. The length L of the pendulum is measured to be L = (1.60 ± 0.01) m.

Calculate, using $g = \frac{{4{\pi ^2}L}}{{{T^2}}}$ , the value of the acceleration of free fall, including its uncertainty. State the value of the uncertainty to one significant figure.

[3]

In a different experiment a student investigates the dependence of the period T of a simple pendulum on the amplitude of oscillations θ. The graph shows the variation of $\frac{T}{{{T_0}}}$ with θ, where T₀ is the period for small amplitude oscillations.

The period may be considered to be independent of the amplitude θ as long as $\frac{{T - {T_0}}}{{{T_0}}} < 0.01$ . Determine the maximum value of θ for which the period is independent of the amplitude.

[2]

The circuit shown may be used to measure the internal resistance of a cell.

M17/4/PHYSI/SP3/ENG/TZ2/02

The ammeter used in the experiment in (b) is an analogue meter. The student takes measurements without checking for a “zero error” on the ammeter.

An ammeter and a voltmeter are connected in the circuit. Label the ammeter with the letter A and the voltmeter with the letter V.

[1]

In one experiment a student obtains the following graph showing the variation with current I of the potential difference V across the cell.

M17/4/PHYSI/SP3/ENG/TZ2/02b

Using the graph, determine the best estimate of the internal resistance of the cell.

[3]

State what is meant by a zero error.

[1]

c.i.

After taking measurements the student observes that the ammeter has a positive zero error. Explain what effect, if any, this zero error will have on the calculated value of the internal resistance in (b).

[2]

c.ii.

A student studies the relationship between the centripetal force applied to an object undergoing circular motion and its period $T$ .

The object (mass $m$ ) is attached by a light inextensible string, through a tube, to a weight $W$ which hangs vertically. The string is free to move through the tube. A student swings the mass in a horizontal, circular path, adjusting the period $T$ of the motion until the radius $r$ is constant. The radius of the circle and the mass of the object are measured and remain constant for the entire experiment.

The student collects the measurements of $T$ five times, for weight $W$ . The weight is then doubled ( $2 W$ ) and the data collection repeated. Then it is repeated with $3 W$ and $4 W$ . The results are expected to support the relationship

$W = \frac{4 π^{2} m r}{T^{2}} .$

In reality, there is friction in the system, so in this case $W$ is less than the total centripetal force in the system. A suitable graph is plotted to determine the value of $m r$ experimentally. The value of $m r$ was also calculated directly from the measured values of $m$ and $r$ .

State why the experiment is repeated with different values of $W$ .

[1]

Predict from the equation whether the value of $m r$ found experimentally will be larger, the same or smaller than the value of $m r$ calculated directly.

[2]

The measurements of $T$ were collected five times. Explain how repeated measurements of $T$ reduced the random error in the final experimental value of $m r$ .

[2]

c(i).

Outline why repeated measurements of $T$ would not reduce any systematic error in $T$ .

[1]

c(ii).

A student carries out an experiment to determine the variation of intensity of the light with distance from a point light source. The light source is at the centre of a transparent spherical cover of radius C. The student measures the distance x from the surface of the cover to a sensor that measures the intensity I of the light.

M18/4/PHYSI/SP3/ENG/TZ2/02

The light source emits radiation with a constant power P and all of this radiation is transmitted through the cover. The relationship between I and x is given by

$I = \frac{P}{{4\pi {{(C + x)}^2}}}$

The student obtains a set of data and uses this to plot a graph of the variation of $\frac{1}{{\sqrt I }}$ with x.

This relationship can also be written as follows.

$\frac{1}{{\sqrt I }} = Kx + KC$

Show that $K = 2\sqrt {\frac{\pi }{P}}$ .

[1]

Estimate C.

[2]

b.i.

Determine P, to the correct number of significant figures including its unit.

[4]

b.ii.

Explain the disadvantage that a graph of I versus $\frac{1}{{{x^2}}}$ has for the analysis in (b)(i) and (b)(ii).

[2]

In an experiment to measure the specific latent heat of vaporization of water L_v, a student uses an electric heater to boil water. A mass m of water vaporizes during time t. L_v may be calculated using the relation

${L_v} = \frac{{VIt}}{m}$

where V is the voltage applied to the heater and I the current through it.

Outline why, during the experiment, V and I should be kept constant.

[1]

Outline whether the value of L_v calculated in this experiment is expected to be larger or smaller than the actual value.

[2]

A student suggests that to get a more accurate value of L_v the experiment should be performed twice using different heating rates. With voltage and current V₁, I₁ the mass of water that vaporized in time t is m₁. With voltage and current V₂, I₂ the mass of water that vaporized in time t is m₂. The student now uses the expression

${L_v} = \frac{{\left( {{V_1}{I_1} - {V_2}{I_2}} \right)t}}{{{m_1} - {m_2}}}$

to calculate L_v. Suggest, by reference to heat losses, why this is an improvement.

[2]

An experiment to find the internal resistance of a cell of known emf is to be set. The following equipment is available:

M18/4/PHYSI/SP3/ENG/TZ1/02

Draw a suitable circuit diagram that would enable the internal resistance to be determined.

[1]

It is noticed that the resistor gets warmer. Explain how this would affect the calculated value of the internal resistance.

[3]

Outline how using a variable resistance could improve the accuracy of the value found for the internal resistance.

[2]

A magnetized needle is oscillating on a string about a vertical axis in a horizontal magneticfield B. The time for 10 oscillations is recorded for different values of B.

M18/4/PHYSI/SP3/ENG/TZ1/01_01

The graph shows the variation with B of the time for 10 oscillations together with the uncertainties in the time measurements. The uncertainty in B is negligible.

Draw on the graph the line of best fit for the data.

[1]

Write down the time taken for one oscillation when B = 0.005 T with its absolute uncertainty.

[1]

b.i.

A student forms a hypothesis that the period of one oscillation P is given by:

$P = \frac{K}{{\sqrt B }}$

where K is a constant.

Determine the value of K using the point for which B = 0.005 T.

State the uncertainty in K to an appropriate number of significant figures.

[3]

b.ii.

State the unit of K.

[1]

b.iii.

The student plots a graph to show how P² varies with $\frac{1}{B}$ for the data.

Sketch the shape of the expected line of best fit on the axes below assuming that the relationship $P = \frac{K}{{\sqrt B }}$ is verified. You do not have to put numbers on the axes.

[2]

State how the value of K can be obtained from the graph.

[1]

The resistance R of a wire of length L can be measured using the circuit shown.

In one experiment the wire has a uniform diameter of d = 0.500 mm. The graph shows data obtained for the variation of R with L.

The gradient of the line of best fit is 6.30 Ω m^–1.

Estimate the resistivity of the material of the wire. Give your answer to an appropriate number of significant figures.

[2]

a(i).

Explain, by reference to the power dissipated in the wire, the advantage of the fixed resistor connected in series with the wire for the measurement of R.

[3]

a(ii).

The experiment is repeated using a wire made of the same material but of a larger diameter than the wire in part (a). On the axes in part (a), draw the graph for this second experiment.

[2]

In an investigation to measure the acceleration of free fall a rod is suspended horizontally by two vertical strings of equal length. The strings are a distance d apart.

When the rod is displaced by a small angle and then released, simple harmonic oscillations take place in a horizontal plane.

The theoretical prediction for the period of oscillation T is given by the following equation

$T = \frac{c}{{d\sqrt g }}$

where c is a known numerical constant.

In one experiment d was varied. The graph shows the plotted values of T against $\frac{1}{d}$ . Error bars are negligibly small.

State the unit of c.

[1]

A student records the time for 20 oscillations of the rod. Explain how this procedure leads to a more precise measurement of the time for one oscillation T.

[2]

Draw the line of best fit for these data.

[1]

c.i.

Suggest whether the data are consistent with the theoretical prediction.

[2]

c.ii.

The numerical value of the constant c in SI units is 1.67. Determine g, using the graph.

[4]

A student investigates how the period T of a simple pendulum varies with the maximum speed v of the pendulum’s bob by releasing the pendulum from rest from different initial angles. A graph of the variation of T with v is plotted.

Suggest, by reference to the graph, why it is unlikely that the relationship between T and v is linear.

[1]

Determine the fractional uncertainty in v when T = 2.115 s, correct to one significant figure.

[2]

The student hypothesizes that the relationship between T and v is T = a + bv², where a and b are constants. To verify this hypothesis a graph showing the variation of T with v² is plotted. The graph shows the data and the line of best fit.

Determine b, giving an appropriate unit for b.

[3]

The lines of the minimum and maximum gradient are shown.

Estimate the absolute uncertainty in a.

[2]

In an experiment to measure the acceleration of free fall a student ties two different blocks of masses m₁ and m₂ to the ends of a string that passes over a frictionless pulley.

The student calculates the acceleration a of the blocks by measuring the time taken by the heavier mass to fall through a given distance. Their theory predicts that $a = g\frac{{{m_1} - {m_2}}}{{{m_1} + {m_2}}}$ and this can be re-arranged to give $g = a\frac{{{m_1} + {m_2}}}{{{m_1} - {m_2}}}$ .

In a particular experiment the student calculates that a = (0.204 ±0.002) ms^–2 using m₁ = (0.125 ±0.001) kg and m₂ = (0.120 ±0.001) kg.

Calculate the percentage error in the measured value of g.

[3]

a.i.

Deduce the value of g and its absolute uncertainty for this experiment.

[2]

a.ii.

There is an advantage and a disadvantage in using two masses that are almost equal.

State and explain the advantage with reference to the magnitude of the acceleration that is obtained.

[2]

b.i.

There is an advantage and a disadvantage in using two masses that are almost equal.

State and explain the disadvantage with reference to your answer to (a)(ii).

[2]

b.ii.

A student measures the refractive index of water by shining a light ray into a transparent container.

IO shows the direction of the normal at the point where the light is incident on the container. IX shows the direction of the light ray when the container is empty. IY shows the direction of the deviated light ray when the container is filled with water.

The angle of incidence θ is varied and the student determines the position of O, X and Y for each angle of incidence.

The table shows the data collected by the student. The uncertainty in each measurement of length is ±0.1 cm.

(i) Outline why OY has a greater percentage uncertainty than OX for each pair of data points.

(ii) The refractive index of the water is given by $\frac{{{\rm{OX}}}}{{{\rm{OY}}}}$ when OX is small.

Calculate the fractional uncertainty in the value of the refractive index of water for OX = 1.8 cm.

[3]

A graph of the variation of OY with OX is plotted.

(i) Draw, on the graph, the error bars for OY when OX = 1.8 cm and when OY = 5.8 cm.

(ii) Determine, using the graph, the refractive index of the water in the container for values of OX less than 6.0 cm.

(iii) The refractive index for a material is also given by $\frac{{\sin i}}{{\sin r}}$ where i is the angle of incidence and r is the angle of refraction.

Outline why the graph deviates from a straight line for large values of OX.

[5]

A spherical soap bubble is made of a thin film of soapy water. The bubble has an internal air pressure $P_{i}$ and is formed in air of constant pressure $P_{o}$ . The theoretical prediction for the variation of $(P_{i} - P_{o})$ is given by the equation

$(P_{i} - P_{o}) = \frac{4 g}{R}$

where $γ$ is a constant for the thin film and $R$ is the radius of the bubble.

Data for $(P_{i} - P_{o})$ and $R$ were collected under controlled conditions and plotted as a graph showing the variation of $(P_{i} - P_{o})$ with $\frac{1}{R}$ .

Suggest whether the data are consistent with the theoretical prediction.

[2]

Show that the value of $γ$ is about 0.03.

[2]

b(i).

Identify the fundamental units of $γ$ .

[1]

b(ii).

In order to find the uncertainty for $γ$ , a maximum gradient line would be drawn. On the graph, sketch the maximum gradient line for the data.

[1]

b(iii).

The percentage uncertainty for $γ$ is $15 %$ . State $γ$ , with its absolute uncertainty.

[2]

b(iv).

The expected value of $γ$ is $0.027$ . Comment on your result.

[1]

b(v).

To determine the acceleration due to gravity, a small metal sphere is dropped from rest and the time it takes to fall through a known distance and open a trapdoor is measured.

M18/4/PHYSI/SP3/ENG/TZ2/01

The following data are available.

$\begin{array}{*{20}{l}} {{\text{Diameter of metal sphere}}}&{ = 12.0 \pm 0.1{\text{ mm}}} \\ {{\text{Distance between the point of release and the trapdoor}}}&{ = 654 \pm 2{\text{ mm}}} \\ {{\text{Measured time for fall}}}&{ = 0.363 \pm 0.002{\text{ s}}} \end{array}$

Determine the distance fallen, in m, by the centre of mass of the sphere including an estimate of the absolute uncertainty in your answer.

[2]

Using the following equation

${\text{acceleration due to gravity}} = \frac{{2 \times {\text{distance fallen by centre of mass of sphere}}}}{{{{{\text{(measured time to fall)}}}^{\text{2}}}}}$

calculate, for these data, the acceleration due to gravity including an estimate of the absolute uncertainty in your answer.

[4]

A student investigates the electromotive force (emf) ε and internal resistance r of a cell.

The current I and the terminal potential difference V are measured.

For this circuit V = ε - Ir .

The table shows the data collected by the student. The uncertainties for each measurement
are shown.

The graph shows the data plotted.

The student has plotted error bars for the potential difference. Outline why no error bars are shown for the current.

[1]

Determine, using the graph, the emf of the cell including the uncertainty for this value. Give your answer to the correct number of significant figures.

[3]

Outline, without calculation, how the internal resistance can be determined from this graph.

[2]

A student uses a Young’s double-slit apparatus to determine the wavelength of light emitted by a monochromatic source. A portion of the interference pattern is observed on a screen.

The distance D from the double slits to the screen is measured using a ruler with a smallest scale division of 1 mm.

The fringe separation s is measured with uncertainty ± 0.1 mm.

The slit separation d has negligible uncertainty.

The wavelength is calculated using the relationship $\lambda = \frac{{sd}}{D}$ .

When d = 0.200 mm, s = 0.9 mm and D = 280 mm, determine the percentage uncertainty in the wavelength.

[2]

Explain how the student could use this apparatus to obtain a more reliable value for λ.

[2]

An experiment is conducted to determine how the fundamental frequency f of a vibrating wire varies with the tension T in the wire.

The data are shown in the graph, the uncertainty in the tension is not shown.

It is proposed that the frequency of oscillation is given by f² = kT where k is a constant.

Draw the line of best fit for the data.

[1]

Determine the fundamental SI unit for k.

[1]

bi.

Write down a pair of quantities that, when plotted, enable the relationship f² = kT to be verified.

[1]

bii.

Describe the key features of the graph in (b)(ii) if it is to support this relationship.

[2]

biii.

In an investigation a student folds paper into cylinders of the same diameter D but different heights. Beginning with the shortest cylinder they applied the same fixed load to each of the cylinders one by one. They recorded the height H of the first cylinder to collapse.

They then repeat this process with cylinders of different diameters.

The graph shows the data plotted by the student and the line of best fit.

Theory predicts that H = $c{D^{\frac{2}{3}}}$ where c is a constant.

Suggest why the student’s data supports the theoretical prediction.

[2]

Determine c. State an appropriate unit for c.

[3]

Determine c. State an appropriate unit for c.

[3]

Identify one factor that determines the value of c.

[1]