Tool 3: Mathematics (DP IB Physics: HL): Exam Questions

Derive the units of pressure in terms of fundamental SI units.

At sea level, the pressure due to the atmosphere is 101 kPa. In the Mariana Trench, at the bottom of the Pacific Ocean, the pressure is around 110 MPa.

Determine how many orders of magnitude greater the pressure at the bottom of the Mariana Trench is than the pressure at sea level.

The density$\rho$ and pressure $P$ of a gas are related by the expression

 $x=\sqrt{\frac{\gamma P}{\rho }}$

where $\gamma$ is a dimensionless constant.

(i) Determine the unit of $x$ in terms of fundamental SI units.

[2]

(ii) Suggest the physical quantity represented by $x$.

[2]

Estimate the time it takes light to cross the nucleus of a hydrogen atom.

Estimate the order of magnitude and give an appropriate SI unit for:

(i) the mass of an aeroplane

[1]

(ii) the current through an LED

[1]

(iii) the time between two heartbeats.

[1]

Determine the value of 0.01 kW h in PeV.

Tensile stress $\sigma$ is defined as the force applied per unit cross-sectional area on a material. The tensile strength is the maximum amount of tensile stress a material can be subjected to before fracturing, meaning that it is equivalent to the tensile stress at the breaking point.

The humerus bone is approximately cylindrical and has a tensile strength of 0.17 GPa and a diameter of 20 mm. 

Calculate the maximum force that can be applied to the humerus bone before it fractures.

The femur bone is the strongest bone in the body. It has a tensile strength of 0.135 kN mm^–2. 

Calculate the tensile strength of the femur bone in GPa.

Derive the units of resistance in terms of fundamental SI units.

Explain why potential difference is not defined as the product of current and resistance.

Two students, A and B, investigate the extension of a wire when put under various loads. The wire is clamped horizontally at one end and a weight $W$ is attached to the other end and allowed to hang vertically. For each value of $W$, the group measures the vertical length $L$ of the wire.

Student A measures $L$ using vernier callipers and records their data in the following table.

The vernier calliper has a positive zero error of 0.10 mm.

Indicate, in the table, the corrected readings for $L$ for each value of $W$.

Student A wants to determine the extension of the wire for each value of $W$. The vernier calliper can measure to the nearest 0.02 mm.

Using the data, calculate the extension of the wire when $W$ = 1.50 N, and give the absolute uncertainty.

Student B measures $L$ using a ruler and records their data in the following table.

The ruler can measure to the nearest 1.0 mm.

Calculate the fractional uncertainty in $L$ using a ruler when $W$ = 2.50 N. Give the final value with its fractional uncertainty.

Student B obtains the following repeated readings for $L$ for one value of $W$.

Student B asks Student A to help analyse the data.

Student A quotes the percentage uncertainty in $L$ as 33%.

Student B quotes the percentage uncertainty in $L$ as 10.2%.

Discuss the values that the students have quoted.

In an experiment to measure the acceleration due to gravity $g$, a student carries out a procedure using a simple pendulum.

The student suggests that the period $T$ of oscillation is related to the length $L$ of the pendulum by the equation

$T=2\mathrm{\pi }\sqrt{\frac{L}{g}}$

The student obtains the following repeated readings for $T$.

Determine the mean value of $T$ and its percentage uncertainty.

For a different value of $L$, the student measures the time for the pendulum to complete 20 oscillations to be (18.4 ± 0.1) s.

The percentage uncertainty in $L$ is determined to be 1.8%.

Calculate the percentage uncertainty in $g$.

The student plots a graph of the variation of $T^2$ with $L$.

Explain how the graph indicates the presence of systematic and random errors in the data.

The period $T$ of oscillation for a mass $m$ hanging on a spring performing simple harmonic motion is given by

$T=2\mathrm{\pi }\sqrt{\frac{m}{k}}$

where $k$ is the spring constant of the spring.

The fractional uncertainty in $T$ is $\alpha$ and the fractional uncertainty in $m$ is $\beta$.

Determine the fractional uncertainty in $k$ in terms of $\alpha$ and $\beta$.

The diagram shows the side and plan views of a microwave transmitter M_T and a receiver M_R arranged on a line marked on the bench.

The circuit connected to M_T and the ammeter connected to M_R are only shown in the plan view.

The distance $y$ between M_T and M_R is recorded.

M_T is switched on and the output from M_T is adjusted so a reading is produced on the ammeter.

M is kept parallel to the marked line and moved slowly away. The perpendicular distance $x$ between the marked line and M is recorded.

Describe one method to reduce systematic errors in the measurement of $x$.

At the first minimum position, a student labels the minimum n = 1 and records the value of $x$. The next minimum position is labelled n = 2 and the new value of $x$ is recorded. Several positions of maxima and minima are produced.

A relationship between $x$ and $y$ against $n$ is shown on the graph. The wavelength $\lambda$ is the gradient of the graph.

Determine the maximum and minimum possible values of $\lambda$.

Calculate:

(i) $\lambda$

[2]

(ii) the percentage uncertainty in $\lambda$.

[2]

A small cannon is designed to fire projectiles at an angle of 22° to the horizontal with an initial velocity of v. 

Calculate the vertical component of velocity if v = 10 m s^–1.

State the direction of the horizontal component of velocity. 

State and explain why the horizontal component of velocity stays constant in the absence of air resistance.

There is a point along the projectile's trajectory at which the vertical component of its velocity decreases to zero.

(i) State the location of this point.

[1]

(ii) Explain why the vertical component of the projectile's velocity decreases to zero at this point.

[2]

An object of weight $W$ is at rest on a slope inclined at an angle $\theta$ above the horizontal.

On the diagram, draw the components of the weight $W$ along the axes shown. 

Determine an expression for the magnitude of the component of weight acting parallel to the slope. 

Draw and label a vector arrow to represent the normal reaction force $R$ acting on the object.

Identify the third force acting on the object and describe its direction with respect to the slope. 

The diagram shows a skier travelling at constant speed down a slope inclined at 35° to the horizontal.

The weight of the skier is $W=850\mathrm{N}$. Two other forces, $P$ and $Q$, act on the skier parallel and perpendicular to the slope, respectively. Assume the friction between the skis and the slope is negligible.

(i) Identify the forces $P$ and $Q$.

[2]

(ii) Draw and label a vector diagram to represent the three forces acting on the skier.

[2]

Calculate the magnitude of force $Q$.

Calculate the magnitude of force $P$.

The diagram shows a straight section of a river where the water is flowing from left to right at a speed of 2.0 m s^-1.

A person crosses the river in a motorboat which moves at a constant speed of 5.0 m s^-1 relative to the water and perpendicular to the current.

Determine, relative to the river bank, the magnitude and direction of the motorboat’s velocity.

The distance between the motorboat's initial starting point and point X is 32 m. After crossing the river, the motorboat reaches point Y.

Calculate the distance between X and Y.

When the motorboat is crossing the river, the motor produces a constant forward force.

Explain why the motorboat moves at a constant speed.

The force $F$ of gravitational attraction between two objects with masses $m_1$ and $m_2$ separated by a distance $r$ is given by

$F=G\frac{m_1{m}_2}{r^2}$

where $G$ is the universal gravitational constant.

Show that the fundamental SI units of $G$ are m³ kg⁻¹ s⁻².

For a planet of mass $M$ and radius $R$, the escape velocity $v$ from the planet's surface is given by

$v=\sqrt{\frac{2GM}{R}}$

Show that this formula is dimensionally consistent.

An electron microscope is used to analyse the arrangement of atoms and their nuclei in a new design for a special sheet of silver foil. The foil is a new material being added to various components in a military medical aircraft. 

The sheet of silver foil has a thickness of 0.992 µm. A silver atom has a radius of 172 pm. 

Estimate the number of layers of atoms in a sheet of silver foil.

The barn (b) is a derived unit used to describe the cross-sectional area of a nucleus.

The cross-sectional area of a silver nucleus is approximately equal to 1 b. 1 b is equivalent to 100 fm².

Determine how many orders of magnitude greater the cross-sectional area of a silver atom is than the cross-sectional area of a silver nucleus.

Einstein's equation describes the interchangeable nature of energy and matter, and is given by

$E=m{c}^2$

where $E$ is energy, $m$ is mass, and $c$ is the speed of light.

The mass of a silver atom is 107.87 u.

Show that the mass of silver in 1 cm² of foil is equivalent to an energy of 4 × 10¹⁰ J.

X–rays emitted by a pulsar travel at the speed of light and are detected on Earth with a wavelength of 3.5 nm.

Calculate the frequency of the X–rays in PHz. Give your answer to an appropriate number of significant figures.

X–rays from the nearest pulsar, PSR J0108-1431, take 13.4 Gs to travel to Earth. 

Calculate the number of oscillations completed by the X–rays as they travel from the pulsar to the Earth. Give your answer to an appropriate number of significant figures.

Show that the distance from Earth to PSR J0108-1431 is around 420 light-years.

The relationship between the period $T$ and length $l$ of a simple pendulum is: 

$T=2\mathrm{\pi }\sqrt{\frac{l}{g}}$

where $g$ is the acceleration of free fall.

Determine the fractional increase in $T$ if $l$ is increased by 6%.

The time period T of a pendulum is also related to the amplitude of oscillations θ. Measurements are taken and a graph is obtained showing the variation of $\frac{T}{T_0}$ with angular amplitude θ, where T₀ is the period for small amplitude oscillations:

Use the information from the graph to

(i) deduce the condition for the time period T to be considered independent of angular amplitude θ.

[2]

(ii) determine the maximum value of θ for which T is independent of θ.

[2]

Typically, using a simple pendulum to determine the acceleration of free fall g involves measuring the periodic time T and the pendulum length l. 

State and explain which piece of measuring equipment is likely to have the largest impact on the accuracy of the value determined for g. 

An experiment is designed to explore the relationship between the temperature of a ball T and the maximum height to which it bounces h. 

The ball is submerged in a beaker of water until thermal equilibrium is reached. The ball is then dropped from a constant height and the height of the first bounce is measured. This is repeated for different temperatures. The results are shown in the graph, which shows the variation of the mean maximum height h_mean with temperature T:

Compare and contrast the uncertainties in the values of h_mean and T. 

The experimenter hypothesises, from their results, that h_mean is proportional to T². 

Suggest how the experimenter could use two points from the graph to validate this hypothesis. 

State and explain whether two points from the graph can confirm the experimenter's hypothesis.

It is known that the energy per unit time P radiated by an object with surface area A at absolute temperature T is given by

$P=e\sigma A{T}^4$

where e is the emissivity of the object and σ is the Stefan-Boltzmann constant.

In an experiment to determine the emissivity e of a circular surface of diameter d, the following measurements are taken: 

• P = (3.0 ± 0.2) W

• d = (6.0 ± 0.1) cm

• T = (500 ± 1) K

Determine the value of the emissivity e of the surface and its uncertainty. Give your answer to an appropriate degree of precision. 

The power dissipated in a resistor can be investigated using a simple electrical circuit. The current in a fixed resistor, marked as 47 kΩ ± 5%, is measured to be (2.3 ± 0.1) A. 

Determine the power dissipated in this resistor with its associated uncertainty. Give your answer to an appropriate degree of precision. 

A student investigates the relationship between two variables T and B. Their results are plotted in the graph shown: 

Comment on the absolute and fractional uncertainty for a pair of data points.

The student suggests that the relationship between T and B is of the form:

$T=a\sqrt{B}+c$

where a and c are constants. To test this suggested relationship, the following graph is drawn:

Describe a method that would determine the value of c and its uncertainty. 

Comment on the student's suggestion from (b). 

A load W is supported by two strings kept in tension by equal masses m hung from their free ends, with each string passing over a smooth pulley. 

Draw a free body force diagram for the load W, expressing tensional forces in terms of each mass m.  

The mass of the load W is M.

Determine an expression for

(i) m in terms of W, g and the angle to the vertical θ

[2]

(ii) M in terms of m and the angle to the vertical θ.

[1]

A crane hook is held in equilibrium by three forces of magnitude 16.5 kN, T₁ and T₂.

Construct a diagram, including an appropriate scale, to determine the magnitude of T₁ and T₂. 

A crate rests on an inclined plane. 

Describe the effect on X, Y and Z if the angle of inclination increases. 

A plane flying across the Lake District sets off from base camp to Lake Windermere, 28 km away, in a direction of 20.0° north of east. 

After dropping off supplies it flies to Lake Coniston, which is 19 km at 30.0° west of north from Lake Windermere. 

By constructing a scale drawing, determine the distance from Lake Coniston to base camp. 

The plane now flies due north with a speed v. It moves through air that is stationary relative to it. 

Suddenly, the plane enters a region where the wind is blowing with a speed u from a direction of θ anticlockwise from south.

Determine an expression for the time taken t for the plane to fly a distance D due north of its current position in this windy region. 

In still air, the plane travels 180 km every 30 minutes. In the windy region described in part (c), the aircraft takes an extra 4 minutes to travel the same distance, when the wind blows at an angle 53° anticlockwise from south. 

Assuming the orientation of the plane does not change, calculate the speed of the wind u in km h^–1. 

The wind now blows due south with the same speed as in part (c). The plane continues to travel at the same speed in this windy region. 

The pilot wishes to cross the sky along the straight line AB. In order to do so, they must turn the plane at an angle φ clockwise from north. 

Construct a scale drawing to determine φ. 

Two taut, light ropes keep a pole vertically upright by applying two tension forces, one of magnitude 200 N and one of magnitude T. 

Construct a scale diagram to determine the weight of the pole W and the magnitude of T. 

A canoeist can paddle at a speed of 3.8 m s^–1 in still water. But, she encounters an opposing current, moving at a speed of 1.5 m s^–1 at 30° to her original direction of travel. 

Construct a scale diagram to determine the magnitude of the canoeist's resultant velocity. 

The boat shown is being towed at a constant velocity by a towing rope, which exerts a tension force F_T = 2500 N. There are two resistive forces indicated – the force of the water on the keel F_K and the force of the water on the rudder, F_R. 

By calculation, or by constructing a diagram, determine the magnitude of F_R. 

You may wish to use the result: 

$\tan \theta =\frac{\sin \theta }{\cos \theta }$

Another boat wishes to cross a river. The river flows from west to east at a constant velocity of 35 cm s^–1 and the boat leaves the south bank, due north, at 1.5 m s^–1. 

Construct a scale diagram to determine the resultant velocity of the boat. 

A ladder rests against a vertical wall as shown. 

Explain how the diagram shows that there is no coefficient of static friction between the ladder and the wall. 

Draw a vector arrow on the diagram to show the direction of the resultant force from the ground exerted on the ladder. Label this vector G.

G acts at an angle of 62° to the ground. 

Show that the coefficient of static friction between the ladder and the ground at the point of slipping is 0.53. 

You may wish to use the result: 

$\tan \theta =\frac{\sin \theta }{\cos \theta }$

The ladder weighs 125 N. 

Calculate the magnitude of vector G.