Topic 1 All - Measurements & Uncertainties

Which of the following is not a fundamental SI unit?

• A. $\hspace{1em}$ampere

• B.$\hspace{1em}$ kelvin

• C. $\hspace{1em}$newton

• D.$\hspace{1em}$ mole

A pilot intends to fly a plane from point $A$ to point $B$. Due to strong winds, she points the nose of the plane in the direction shown by the vector $P$. Which of the choices below best represents the direction of the wind encountered by the plane?

The volume of a cylinder is calculated from the product of its base area and height. The measurement of the height has an uncertainty of 5% and the uncertainty of the diameter is 1%. What is the uncertainty of the volume of the cylinder?

• A. $\hspace{1em}$ 1%

• B. $\hspace{1em}$ 5%

• C. $\hspace{1em}$ 6%

• D. $\hspace{1em}$ 7%

Which of the following units is not a unit of power?

• A. $\hspace{1em}$ W

• B. $\hspace{1em}$ J

• C. $\hspace{1em}$ N m s$^{-1}$

• D. $\hspace{1em}$ J s$^{-1}$

A rectangle has length $L = (2\,\pm\,0.1)\,cm$ and width $W = (3\,\pm\,0.2)\,cm$.

What is the uncertainty of the perimeter of the rectangle?

• A. $\hspace{1em}$ 0.08 $cm$

• B. $\hspace{1em}$ 0.2 $cm$

• C. $\hspace{1em}$ 0.3 $cm$

• D. $\hspace{1em}$ 0.6 $cm$

Which of the following is a vector quantity?

• A. $\hspace{1em}$ Speed

• B. $\hspace{1em}$ Electric force

• C. $\hspace{1em}$ Electric current

• D. $\hspace{1em}$ Temperature

Which of the following is an estimate of the order of magnitude of the radius of the Earth?

• A. $\hspace{1em}10^{7} \,m$

• B. $\hspace{1em}10^{11}\, m$

• C. $\hspace{1em}10^{15}\, m$

• D. $\hspace{1em}10^{22} \,m$

A student models the relationship between velocity v of a moving charge in a magnetic field, and the force $F$ acting on it as $F = kv$, with $k$ being a constant

What are the units for $k$ expressed in fundamental SI units?

• A. $\hspace{1em}m \,s$

• B. $\hspace{1em}kg \,s^{-1}$

• C. $\hspace{1em}kg \,s^{-2}$

• D. $\hspace{1em}m \,s^{-2}$

How many significant figures are there in 0.000017000?

• A.$\hspace{1em}$ 2

• B.$\hspace{1em}$ 5

• C.$\hspace{1em}$ 6

• D. $\hspace{1em}$9

Vectors $X$ and $Y$ are in the shown directions.

Which of the following shows the sum of the two vectors?

An experiment requires the measurement of the radius $R$ and height $H$ of a cylindrical battery to determine its volume. The uncertainty in $R$ is $\Delta R$ and the uncertainty in $H$ is $\Delta H$.

What is the fractional uncertainty in the volume of the battery?

• A. $\hspace{1em}$ $2(\dfrac{\Delta R}{R}) \times (\dfrac{\Delta H}{H})$

• B. $\hspace{1em}$ $2(\dfrac{\Delta R}{R}) + (\dfrac{\Delta H}{H})$

• C. $\hspace{1em}$ $(\dfrac{\Delta R}{R}) + (\dfrac{\Delta H}{H})$

• D. $\hspace{1em}$ $(\dfrac{\Delta R}{R}) \times (\dfrac{\Delta H}{H})$

Express 0.04758 to 3 significant figures.

• A. $\hspace{1em}$0.04758

• B. $\hspace{1em}$0.0476

• C. $\hspace{1em}$0.048

• D. $\hspace{1em}$0.05

A student is using the equation $pV = nRT$ to solve for $n$ with the following percentage uncertainties in the measurements.

$\hspace{1em}$ Quantity $\hspace{1em}$	$\hspace{1em}$ Uncertainty $\hspace{1em}$
P	$\pm\,2\,\%$
V	$\pm\,6\,\%$
T	$\pm\,1\,\%$

What is the percentage uncertainty in the calculated value of n?

• A. $\hspace{1em}$ 3 %

• B. $\hspace{1em}$ 7 %

• C. $\hspace{1em}$ 9 %

• D. $\hspace{1em}$ 12 %

An object moves $9\,m$ east, then $12\,m$ north.

What is the magnitude of the total displacement of this object?

• A. $\hspace{1em}$ 3 $m$

• B. $\hspace{1em}$ 15 $m$

• C. $\hspace{1em}$ 21 $m$

• D. $\hspace{1em}$ 36 $m$

An electronic balance has a button with “tare” written on it. The reading of the balance is reset to zero when the “tare” button is pressed. Which of the following gives a reason why use of the “tare” button improves the measurement?

• A. $\hspace{1em}$ Reduce systematic error

• B. $\hspace{1em}$ Reduce random error

• C. $\hspace{1em}$ Increase precision

• D. $\hspace{1em}$ Reduce percentage uncertainty of the measurement

The absolute uncertainty in the radius $r$ of a sphere is $\Delta r$.

What is the fractional uncertainty in the volume of the sphere?

• A. $\hspace{1em}$ $\left ( \dfrac{\Delta r}{r} \right )^3$

• B. $\hspace{1em}$ $3 \dfrac{\Delta r}{r}$

• C. $\hspace{1em}$ $\dfrac{4}{3} \pi \left( \dfrac{\Delta r}{r} \right)^3$

• D. $\hspace{1em}$ $4 \pi \dfrac{\Delta r}{r}$

A spring of negligible mass and length of $l_o$ is attached to a fixed point. When the spring is extended by an external force $F$, the length of the spring increases to $l$. The graph shows how the length of the spring varies with the tension in spring. The tension in the spring is equal to $k \Delta l$, where k is a constant and $\Delta l$ is the extension of the spring.

What are the gradient and y-intercept of the graph?

A spring with spring constant $k$ is extended by a distance $x$.

$\small{\textrm{[Created using Geogebra https://www.geogebra.org/m/vUK7gGkN]}}$

The uncertainty of $k$ is 2% and that of $x$ is 3%.

What is the uncertainty of the potential energy stored in the spring?

• A. $\hspace{1em}$ 8%

• B. $\hspace{1em}$ 11%

• C. $\hspace{1em}$ 12%

• D. $\hspace{1em}$ 18%

A ball is projected with an initial speed of $2\,m\,s^{-1}$ making an angle of $35^o$ with the horizontal as shown.

Which expression gives the vertical component of the velocity vector?

• A. $\hspace{1em}$ $2\,m\,s^{-1}$

• B. $\hspace{1em}$ $2\,cos(35^o)\,m\,s^{-1}$

• C. $\hspace{1em}$ $2\,sin(35^o)\,m\,s^{-1}$

• D. $\hspace{1em}$ $2\,tan(35^o)\,m\,s^{-1}$

Two trucks X and Y are pulling a car out of thick mud along a straight path. The only other horizontal force acting on the car is a drag force opposite to the direction of motion. The diagram shows the direction of the force applied by truck $X$ and the direction of motion of the car.

Which of the following vectors shows the direction of the force applied by truck $Y$?

[Maximum mark: 8]

A student uses the shown setup to measure the speed of sound.

$\footnotesize{\textrm{[Source: Image created using Images by OpenClipart-Vectors and Josy Dom Alexis from Pixabay - https ://pixabay.com]}}$

The sound source has a frequency of 590 Hz. The student changes the distance between the two microphones d and records the time difference $\Delta$t between the received sounds from the microphones.

The graph below shows the data from this experiment.

1. Draw the line of best fit. [1]
   

2. Determine the fractional uncertainty of d when the time difference $\Delta$t is 2.2 $ms$. [2]
   

3. Showing your working on the graph, calculate the speed of the sound wave.[3]
   

4. Determine the wavelength of the sound. [2]
   

Which of the following gives the unit for work in fundamental SI units?

• A.$\hspace{1em} m^2\,s^2$

• B.$\hspace{1em} kg\,m\,s^{-1}$

• C.$\hspace{1em} kg\,m^2\,s^{-1}$

• D.$\hspace{1em} kg\,m^2\,s^{-2}$

[Maximum mark: 13]

A student investigates the refraction of light. A light beam emitted from the ray box is incident on a transparent block. The ray trace sheet and the position of the ray box and the transparent block are shown below.

$\footnotesize{\textrm{[Source: Created with chemix - https:// chemix.org/]}}$

1. The distances a and b are measured. Then the measurement is repeated for different values for angle $\theta$. The variation of b with a is plotted on the graph below according to obtained data.

   

   Draw the line of best fit for the plotted data on the graph. [1]
   

2. Theory suggests that

   • $n=\dfrac{a}{b}$

   Where $n$ is the refractive index of the material and has a constant value.

   1. Suggest whether the data is consistent with the theoretical prediction. [2]
      

   2. By using the graph calculate n with its absolute uncertainty for the data point when a is measured to be $3.0 \,cm$. [4]
      

   3. The accepted value for the refractive index of the transparent block is 1.71. Comment on your result. [2]
      

3. Suggest why using the line of best fit to find the value for $n$ is better than finding $n$ from an individual data point. [1]
   

[Maximum mark: 13]

In an experiment, a liquid-filled beaker is positioned on an electronic balance. An immersion heater with a power rating of $P$ is introduced into the water. Once the water begins to boil, a researcher monitored the change in mass $m$ of the liquid over time $t$ once the boiling commenced.

$\footnotesize{\textrm{[Source: Created with chemix - https://chemix.org/]}}$

The theoretically predicted relationship between time $t$ and $m$ is $L_v=\dfrac{Pt}{C-m}$

$C$ is the initial mass of the liquid when $t=0\, s$ and $L_v$ is the latent heat of vaporisation of water.

1. Give the units of $L_v$ in fundamental SI units. [2]
   

2. In a particular experiment, the researcher uses a heater of power $65.0\pm0.1\ W$ and measures $m=0.120\pm0.001\, kg$, $C=0.125\pm0.001\ kg$, and $t=600\pm1\ s$.

   1. Calculate the percentage error in the measured value of $L_v$. [3]
      

   2. For this experiment, calculate the value of $L_v$ and its absolute uncertainty. [2]
      

   3. The accepted value of $L_v$ is $2.3 \times 10^6 \, J\,kg^{-1}$. Explain why the experimental value does not match the accepted value. [2]
      

3. The researcher plots a graph to show how m varies with t.

   1. Outline why P must be kept constant during the experiment. [1]
      

   2. Assuming the relationship $L_v=\dfrac{Pt}{C-m}$ is correct, sketch the expected mass-time relationship on the axes below. (There is no need to put numbers to the axes). [2]
      

   

   3. Outline how to obtain the value of $L_v$ from the graph you have drawn in c(ii). [1]
      

[Maximum mark: 11]

1. A student investigates standing waves in strings. A string is connected to an oscillator where a function generator can adjust the frequency. The other end of the string passes through a pulley and supports a mass.

   $\footnotesize{\textrm{[Source: Created with chemix - https ://chemix.org/]}}$

   The student varied the mass $m$ and recorded the values for frequency $f$ for which the first harmonic wass visible on the string. The following graph shows the recorded data.

   

   The uncertainty of the mass is negligible.

   1. Draw the line of best fit for the plotted data on the graph. [1]
      

   2. By referring to the plotted data, explain why it cannot be stated that frequency is directly proportional to mass. [1]
      

   3. Use the graph to determine the value for $f$ with its absolute uncertainty for for $m=1.0\ kg$ [2]
      

2. The theory suggests that

   $f^2=km$

   where k is a constant.

   The variation of $f^2$ with m is seen on the graph below.

   

   1. Show that the absolute uncertainty of $f^2$ is $\pm0.01\ k{Hz}^2$ for $m=1.0\ kg$. [3]
      

   2. The absolute uncertainty of $f^2$ is $0.03\ k{Hz}^2$ for $m=0.25\ kg$. Construct error bars for $f^2$ when $m=0.25\ kg$ and $m=1.00\ kg.$ [1]
      

   3. State and explain whether the theory is supported by data or not. [2]
      

3. Determine the unit of k, in fundamental SI units. [1]
   

Which of the following is $\textbf{not}$ a unit for magnetic flux density?

• A. $\hspace{1em}$ $T$

• B. $\hspace{1em}$ $Wb\,m^2$

• C. $\hspace{1em}$ $kg\,s^{-2}\,A^{-1}$

• D. $\hspace{1em}$ $N\,A^{-1}\,m^{-1}$

A box moves a distance of ($5.0 \pm 0.5)\ m$ due to a horizontal force of ($20 \pm 1)\ N$.

What is the percentage uncertainty in the work done on the box?

• A. $\hspace{1em}$ $0.15\%$

• B. $\hspace{1em}$ $1.5\%$

• C. $\hspace{1em}$ $9\%$

• D. $\hspace{1em}$ $15\%$

The area of one of the faces of a cube is $(10.0\pm0.4)\, cm^2$. What is the percentage uncertainty in the volume of the cube?

• A. $\hspace{1em}$ 2%

• B. $\hspace{1em}$ 4%

• C. $\hspace{1em}$ 6%

• D. $\hspace{1em}$ 8%

[Maximum mark: 11]

1. A student investigates the variation of gas pressure with volume at a constant temperature. Air is trapped inside a gas syringe with a constant cross sectional area of $A$. The plunger of the gas syringe is tied to a mass holder. The student measures the volume $V$ of the trapped gas for different hanging masses with the external constant air pressure, $P_A$, remaining constant.

   $\footnotesize{\textrm{[Source: Created with chemix - https:// chemix.org/]}}$

   1. After each increase in mass, the student waits for some time to measure the volume of the gas. Outline why this is necessary. [1]
      

      The recorded data is plotted on a graph:

      

   2. Draw the line of best fit for the plotted data. [1]
      

   3. Determine the gradient of the graph. [2]
      

2. Theory suggests that

   • $(P_A-\dfrac{mg}{A})V=k$;

   where m is the total mass of the plunger and hanging masses, g is the acceleration of free fall and k is a constant.

   1. Determine the unit of k in SI base units. [2]
      

   2. Calculate the percentage uncertainty of the volume V when the mass m is 700 g. [2]
      

3. In a particular experiment, the researcher measures

   • $m=0.700\pm0.005\ kg$,
     $A=0.00010\pm0.00001{\ m}^2$ and
     $g=9.81\pm0.01\ ms^{-2}$.

Calculate the absolute uncertainty of $\dfrac{mg}{A}$. [3]

Which of the following is equivalent to the unit of electric potential difference?

• A. $\hspace{1em} N \,m \,C^{-1}$

• B. $\hspace{1em} N \,m^{-1} \,C^{-1}$

• C. $\hspace{1em} J \,C$

• D. $\hspace{1em} J\, m^{-1} \,C^{-1}$

[Maximum mark: 9]

1. In an experiment, a beaker containing liquid of mass $m$ is heated with a heater of power P. Then the variation of temperature T of the liquid with time t is recorded.

   $\footnotesize{\textrm{[Source: Created with chemix - https:// chemix.org/]}}$

   The data from the experiment is shown on the graph below.

   

   1. On the graph, draw the line of best fit for the data. [1]
      

   2. Explain why it can not be concluded that the relationship between temperature and time is directly proportional. [1]
      

2. Theory suggests that

   • $\dfrac{T-T_0}{t}=k$

   where $T_0$ is the temperature when $t=0$ s.

   1. Using the data given, calculate the percentage uncertainty of $(T-T_0)$ at time $25\,s$. [2]
      

   2. Use the graph to determine constant k. [2]
      

   3. Theory also suggests that another expression for k is given by

      • $k=\dfrac{P}{mc}$, where c is another constant.

      Determine the units of constant c, expressing your answer in fundamental SI units. [2]
      

A student investigates gas behaviour. A volume of gas is trapped in a capillary tube by a thin layer of oil. The tube, a thermometer and a ruler are immersed in water.

$\footnotesize{\textrm{[Source: Created with chemix - https:// chemix.org]}}$

The student slowly added boiling water and recorded the values of the temperature T of the water and the height h of the gas trapped inside the capillary tube. The recorded data is plotted on a graph as shown.

1. Draw the best fit line for the graph plotted data. [1]
   

2. The theory suggests that

   • $h=mT+C$

   where m and C are constant. Temperature T is in $\degree$C.

   1. Suggest whether the data are consistent with the theory. [2]
      

   2. By using the graph drawn in (a), determine the constants m and C. [4]
      

   3. State the units of the gradient in fundamental SI units. [1]
      

3. Following the experiment, the student tested their thermometer in an ice bath and in boiling water. The temperatures were 2$\degree$C and 102$\degree$C respectively.

   1. Explain how this finding will affect how the student interprets their results. [2]
      

   2. The student also goes on to find the intercept on the temperature axis. Will this error cause their intercept value to be higher or lower than expected? [1]
      

[Maximum mark: 9]

1. The circuit shown is used to measure the variation of the resistance R of a wire with the length L of the wire. The wire is connected across points $X$ and $Y$.

   $\footnotesize{\textrm{[Source: Created with GeoGebra by Tom Walsh - http:// ophysics.com/t1.html]}}$

   A voltmeter and an ammeter are connected to the circuit in order to investigate the relationship.

   1. Draw the ammeter and voltmeter on the given circuit with their correct connections. [2]
      

   2. State the purpose of connecting a resistor in series with the wire as shown in the diagram. [1]
      

2. The diameter d of the wire is kept constant at $d=0.58\pm0.01 \ mm$. Determine the percentage uncertainty of the cross sectional area A of the wire. [2]
   

3. The graph shows the variation of resistance of the wire R with the wire's length L.

   

   1. Draw the line of best fit. [1]
      

   2. Use the graph to estimate the resistivity of the wire. [3]
      

[Maximum mark: 7]

In an experiment to determine the speed of sound in air, some students used a number of tuning forks of known frequencies and a tube with a moveable piston arranged as shown below.

For each tuning fork they used the setup shown in the diagram.

$\footnotesize{\textrm{[Source: Created with chemix - https:// chemix.org/]}}$

The students would sound the tuning fork, then move the piston slowly downwards until a loud sound is first heard. The length L was then measured. This was repeated for different tuning forks.

The graph shows the variation of $\dfrac{1}{L}$ with frequency.

1. Draw the line of best fit. [1]
   

2. Calculate the fractional uncertainty of $\frac{1}{L}$ when the length is $20\, cm$. [2]
   

3. 1. Determine the gradient, expressing your answer with the correct SI units. [2]
      

   2. Use the gradient calculated in (i) to determine the speed of sound. [2]
      

A rectangular prism is obtained by combining two identical cubes as shown in the figure.

The length of one side of a cube is $10.0 \pm 0.2\, cm$.

What is the ratio

$\dfrac{\textrm{absolute uncertainty of volume of the rectangular prism}}{\textrm{absolute uncertainty of volume of one cube}}$?

• A. $\hspace{1em}$ $\dfrac{1}{2}$

• B. $\hspace{1em}$ 1

• C. $\hspace{1em}$ 2

• D. $\hspace{1em}$ 5

Vector $d$ makes an angle $\theta$ with the horizontal as shown. Its magnitude is $5\,m$ and the magnitude of its horizontal component $d_x$ is $3\,m$.

Which of the following is correct about the magnitude and direction of its vertical component?

A body moves a distance of 1.5 $km$ in 10 $ks$.

Which of the following gives the correct magnitudes of both of distance and time?

[Maximum mark: 12]

In an experiment, the free-fall of a ball bearing is used to measure the acceleration due to gravity. The ball bearing is suspended with an electromagnet. Then it is released from a height of $h$ above a table.

$\footnotesize{\textrm{[Source: Created with chemix - https:// chemix.org/]}}$

The time t needed for the ball bearing to reach the ground after release is measured. The experiment is repeated for different values of $h$. The theoretically predicted relationship between time t and h is

$t^2=\dfrac{2h}{g}$

1. State why there is a need to collect data for a range of values of $h$ to verify this relationship. [1]
   

2. The experimental data for $t^2$ and h are plotted on the graph below.

   

   1. Draw the line of best fit for the plotted data on the graph. [1]
      

   2. Suggest whether the data is consistent with the theoretical prediction. [2]
      

   3. By using the line of best fit, determine a value for g and state the units. [4]
      

3. The theoretical relationship is for an object falling in a vacuum.

   1. Compare the experimental result to the accepted value of g and suggest a reason for the discrepancy. [2]
      

   2. State and explain whether the source of error in (c)(i) is a systematic or random error. [2]