HSC Physics Millikan's Oil Drop Experiment

A tutorial sheet of questions on Millikan’s oil drop experiment is given below.

1. Show that

This equation is taken from Millikan’s second paper on the oil drop experiment.

IB Physics Rate of Change of Momentum

A tutorial sheet on the rate of change of momentum follows. What does it mean?

1. A rocket moves through space. At a certain instant its mass is m and its speed relative to the Earth is u. At this instant it is ejecting mass backwards at a rate m' at a speed v relative to the Earth. What is the resultant force acting on the rocket at this instant? Assume u and v are much less than c.
2. In question 1 the rocket is ejecting mass backwards at a speed w relative to the rocket. Find the force acting on the rocket at this instant. Assume u and w are much less than c.
3. In which question, 1 or 2, does the rocket have greater speed after a given time if w = v?
4. In question 1 is the acceleration of the rocket relative to the Earth constant?
5. Find the total momentum of the fuel-rocket system at time t.
6. In question 1 the rocket is initially at rest. If m = 25.0 kg, v = 100.0 m s^-1 and m' = -0.5 kg s ^-1, show that at t = 10.0 s, u = 25.0 m s^-1 and s = 115.7 m, where s is the distance travelled by the rocket as it burns fuel.
7. In question 2 the rocket is initially at rest. If m = 25.0 kg, w = 100.0 m s^-1 and m' = -0.5 kg s ^-1, show that at t = 10.0 s, u = 28.125 m s^-1 and s = 125 m.

IB Physics Heat and Internal Energy

A tutorial sheet on the heat and internal energy follows.

1. An system gains an amount Q of heat energy. Does the internal energy increase by this amount?
2. An air conditioner extracts an amount Q of heat energy from a room. Is the work done by the air conditioner Q?
3. A heat engine has a heat inflow Q1 per cycle and a heat outflow of Q2 each cycle. What is the efficiency of this engine?
4. Must a heat engine always have a clockwise cycle on a p-V diagram?
5. Does a heat pump always have an anticlockwise cycle on a p-V diagram?

IB Physics "Show that" Collision Questions

A tutorial sheet of show that problems follows. Show all working and explain the Physics principles being used. Assume that all speeds are much less than the speed of light.

1. A mass m moves at a speed u. A mass M is at rest. A one dimensional elastic collision occurs. Show that the speed of M after the collision is 2mu/(m+M).

2. A paticle P of mass m moving at a speed u strikes a particle of mass nm that is at rest. If the collision is elastic show that the ratio of the final velocity of P to its initial velocity is (1-n)/(1+n).

3. A particle is at rest on a smooth horizontal surface. It is struck off centre by a particle of equal mass moving at a speed u. The collision is elastic. Show that the velocity vectors of the particles are perpendicular after the collision.

4. A particle of mass M is at rest on a smooth horizontal surface. It is struck by a particle of mass m moving at a speed u, the angle between the line of cetres of the particles and the initial velocity vector being 45° . The collision is elastic. Show that after collision the angle made by the velocity vector of m (measured in the laboratory reference frame) with its initial direction is given by tanθ = M/m

IB HL Physics Escape Speed

A tutorial sheet of questions on escape speed, from HL Topic 10, is given below.

1. Define gravitational potential.
2. Define gravitational potential energy.
3. Why is gravitational potential energy given a negative sign?
4. Does the escape speed from the surface of a planet depend on the angle at which the mass is projected?
5. Find the maximum mass of a planet that you can jump off. Make realistic assumptions. Assume that the planet does not recoil.
6. A person of mass m can jump at a speed u relative to the surface of a planet of mass M and radius R. Find the maximum mass of this planet if the person is to escape its gravitational field.

IB Physics Electric Field Lines

A charge is released from rest in an electric field. Is the path of the charge the same as that of the electric field line through its initial point?

IB Physics Electric and Magnetic Fields

Tutorial questions on problems with electric and magnetic fields present are given below.

1. Two charged particles moving to the right at the same speed enter a uniform electric field. Both particles curve upwards in identical paths. Both particles must have the same (a) mass, (b) charge, (c) charge to mass ratio, (d) initial momentum
2. Did J. J. Thomson use the method of "balancing" the electric and magnetic forces on a beam of cathode rays in his original experiment?

IB Physics Wheatstone Bridge

Below is a tutorial sheet on resistor combinations.

1. Five identical resistors form a bridge configuration. A current I enters the group. Find the current in the bridge resistor.

2. Resistances of 2Ω, 3Ω, 4Ω and 5Ω are placed clockwise around the sides of a square loop (in order from top left). Find the value of the vertical bridge resistor if no current flows through it when the configuration is connected to a 12 V battery.

3. more to come

IB Physics Climate Model

The following questions are taken from the book The Physics of Atmospheres by the late Sir John Houghton (3rd edition, 2007, Cambridge University Press).

To make a crude estimate of the surface temperature of a planet we can equate the solar radiation it absorbs with the infrared radiation it emits using the equation

In this equation the power of the radiation emitted by a planet of radius a (assumed to be emitting as a perfect black body) is equal to the average power absorbed by the planet having an albedo A at a distance R from the Sun (R is measured in terms of Earth distances from the Sun, for the Earth, R = 1 ). F is the solar constant.

1. In the equation when the planet absorbs the solar radiation we use an area of 𝜋a² but when the planet emits radiation we use an area of 4𝜋a². Why is this? (The rays from the Sun are effectively parallel when they reach the planet and we use the area perpendicular to these rays which is the area of a circle, the cross sectional area that the rays "see". When the planet emits radiation it does so perpendicular (radially) outwards to its surface and so we use the surface area of a sphere)
2. Given that F = 1370 W m^-2 and for the Earth R = 1, A = 0.30, find the effective surface temperature of the Earth. (256 K). This result is lower than average temperature of 288 K. Why?
3. Find the effective surface temperature for Venus given that R = 0.72 and A = 0.77. (227 K)
4. Find the effective surface temperature for Mars given that R = 1.52 and A = 0.15. (216 K)
5. Find the effective surface temperature for Jupiter using this model given that R = 5.20 and A = 0.58. The approximate measured surface temperature is 130 K. The difference is due to the energy generated inside Jupiter. (98 K)

IB Physics Circle problems

A tutorial sheet of questions involving circles follows.

1. A car moves at a constant speed V around a circular path. Find the magnitude of the change in velocity of the car after it turns through an angle of 𝜽 degrees.
2. Three equal point charges +Q are placed at the corners of an equilateral triangle. Find the magnitude of the force acting on one of the charges.

3. In question 2 the charges move in a circular path about the centre of the circle. Find the speed of each charge if they are a distance d apart. Is the circular path stable?

4. Three equal masses are placed at rest a distance d apart at the corners of an equilateral triangle. Find the time taken for the masses to collide after they are released.

5. Planets of equal mass m orbit a star of mass M each at a distance a from the centre of the star but at opposite sides of a diameter of the common circular path. Does Kepler's law of periods apply in this situation? How is it modified?

IB Physics Waves

A tutorial sheet on wave concepts follows.

1. A wave of wavelength 1.0 m travels to the right along a long string at 2.0 m s ^-1 . Another wave of wavelength 1.0 m travels along the same string at 2.0 m s ^-1 to the left. Initially, destructive interference has occurred. Does destructive interference always occur on this string?
2. In the previous question draw the shape of the string one-quarter of a second later.
3. more to come

HSC Physics Doppler effect

A tutorial sheet on the Doppler effect for sound is given below.

1. The speed of sound in air is 343 m s^-1. A train moves towards a stationary observer at 100 m s^-1. What is the speed of the sound waves made by the train relative to the observer?
2. In question 1 the train has a horn that emits sound waves of frequency 200 Hz. Find the frequency of these sound waves that is measured by a stationary observer in front of the train.
3. In question 2 find the wavelength of the sound waves in front of the train.
4. The sound waves reflect back towards the train from a vertical mountain in front of the train. Find the frequency of the sound waves arriving back at the train.
5. A second train moves at 50 m/s towards the first train. Find the frequency of the sound waves that reflect off the second train and arrive back at the first train.
6. A stationary observer is at a perpendicular distance of 50 m from the straight train line. Assuming that the first train moves at a constant speed, find the maximum and minimum frequencies detected by the observer as the train approaches and recedes.

HSC Physics Induced emf in a generator

A tutorial sheet on the induced emf in a spinning coil is given below.

1. A coil spins at a constant rate in a uniform magnetic field. At what posiion is the magnetic flux through it a maximum value?
2. A coil spins at a constant rate in a uniform magnetic field. At what position is the induced emf in the coil a maximum value?
3. When a coil spins in a uniform magnetic field the induced current produces its own magnetic field. Does this magnetic field affect the current flowing in the coil?
4. A coil spins at a constant rate in a uniform magnetic field. Is the mechanical power supplied to the coil constant?
5. A coil spins in a radial magnetic field of constant magnitude. Is an emf induced in the coil?

IB Physics When is a real gas most like an ideal gas?

A common question is the departure of the behaviour of a real gas from ideal gas behaviour. A tutorial sheet on this concept follows.

1. Can an ideal gas be liquefied?
2. Do the particles of an ideal gas collide with each other?
3. The thermodynamic temperature of an ideal gas is increased. How does this affect the collision frequency of the particles?
4. At which temperature, high or low, are the particles of an ideal gas more likely to be in the vicinity of each other?
5. At which pressure, high or low, are the particles of an ideal gas more likely to be in the near vicinity of each other?
6. For an ideal gas pV/nT has a constant value for all values of p, V and T. Sketch a graph showing pV/nT versus p for oxygen gas.

IB Physics Phase of a wave

An important term applied to an oscillating particle and a wave is phase. Below is a tutorial sheet on phase.

1. Two points on a progressive wave are in phase if they have the same displacement and velocity. True or false?
2. Two points on a progressive wave are in phase if the distance between the points is a whole number of wavelengths. T or F?
3. P and Q are two points on a progressive wave. The gradients of the tangents to the wave at P and Q have the same value. Are P and Q in phase?
4. What is the phase difference between two points on a progressive wave of wavelength 8.0 cm that are 3.0 cm apart?
5. The displacement from the equilibrium caused by a progressive wave is given by the equation y = 2sin(2x - t). What is the phase difference when t = 1 between points on the wave at x = 3 and x = 1?
6. X and Y are two points on a standing wave. X is at a distance 𝜆/4 from a node and Y a distance 𝜆/3 from this same node, both points being to the right of the node. Are X and Y in phase?
7. R and S are two points on a standing wave. The gradients of the tangents to the wave at R and S are different. Could R and S be in phase?
8. What is the phase difference between two points on a standing wave of wavelength 6.0 cm that are 2.0 cm apart?
9. The displacement from the equilibrium caused by a standing wave is given by the equation y = 2sin(3x)cos(2t). What is the phase difference when t = 1 between points on the wave at x = 4 and x = 1?

IB Physics What causes the resistance of a wire?

What causes the electrical resistance of a conductor? Some explanations from textbooks are given below:

Tipler and Mosca, Physics for Scientists and Engineers page 840 (6th edition)

When an electric field is applied, the field exerts a force -eE on each free electron giving it a change in velocity in the direction opposite the field. However any additional kinetic energy acquired is quickly dissipated by collisions with the lattice ions in the wire.

Serway and Jewett, Physics for Scientists and Engineers with Modern Physics page 779 (eighth edition)

The excess energy acquired by the electrons in the electric field is transferred to the atoms of the conductor when the electrons and atoms collide. The energy transferred to the atoms increases their vibrational energy which causes the temperature of the conductor to increase.

Serway, Physics for Scientists and Engineers with Modern Physics, page 784 (fourth edition)

According to quantum mechanics, electrons have wave like properties. If an array of atoms is regularly spaced, (that is periodic), the wave-like character of the electrons makes it possible for them to move freely through the conductor and a collision with an atom is unlikely. Electron waves are scattered only if the atomic arrangement is irregular (not periodic) as a result of, for example, structural defects or impurities. At high temperatures, the resistivity is dominated by scattering caused by collisions between the electrons and the atoms in the conductor, which are continually displaced as a result of thermal agitation. The thermal motion of the atoms causes the structure to be irregular (compared with an atomic array at rest), thereby reducing the electron’s mean free path.

Mathematical Tripos Dynamics and Relativity 2022

Below are my answers to the 2022 Mathematical Tripos Part 1A Dynamics and Relativity questions examined on 08 June 2022.

1. 3C (a) The vector equation of the plane is ͢r = ͢x0 + 𝜇 ͢E + λ ͢v . The motion is in a plane containing the initial point (x0) and is perpendicular to the magnetic field since the initial velocity has no component in the direction of B. The parameters 𝜇 and λ depend on m, q, t, E and B. These are identified in part (b). The scalar equation of the plane is ͢r · ͢B = ͢x0 · ͢B , since ͢E and ͢v are both perpendicular to ͢B.
2. 3C (b)...answer to come

IB Physics Nodes and antinodes

A tutorial sheet on nodes and antinodes is given below.

1. Is a node always at a rarefaction in a standing wave in a pipe containing air?
2. Is the pressure always a minimum at a node in the standing wave pattern in a pipe containing air?
3. In a sound wave are the displacement and pressure variations 90 degrees out of phase?
4. The open end of a pipe containing air is a displacement antinode and a pressure node. True or false?
5. When a tuning fork is struck and held above a vertical pipe containing water only odd harmonics can form as the water is drained from the pipe. True or false?
6. The length of the air column above the water level in a vertical pipe is 120 cm. Near the open end a loudspeaker can produce sound waves having a frequency ranging from 10 to 5000 Hz. (a) Find the lowest frequency that can produce resonance in the air column (b) How many different frequencies of the loudspeaker will produce resonance. Neglect the end correction. Take the speed of sound as 343 m/s.

IB HL Physics Simple harmonic motion

A tutorial sheet of SHM questions is given below.

1. Define simple harmonic motion (SHM).
2. True of false. SHM always occurs in a straight line.
3. True or false. The reference point in describing SHM is always the centre of the motion.
4. True or false. The period of damped SHM is constant during the motion.
5. Sketch a graph showing the kinetic energy of a mass versus time as it oscillates on a spring. Does it have 'sharp' points when it touches the t axis? (no)
6. A spring of force constant k hangs vertically. A mass M is placed on the end of the spring and released from rest. Determine the distance that the mass falls before it comes to rest.
7. A ball of mass M falls freely from rest. When it has reached a speed v it strikes a vertical spring of force constant k. Determine the amount that the spring is compressed when the ball comes to rest.
8. A particle moves in SHM. At a certain instant the displacement from the equilibrium position is 5.0 cm. Three seconds later the displacement is 2.0 cm and after two more seconds it is -3.0 cm. Find the largest possible period and the amplitude of the motion when this occurs. (12.60 s, 5.26 cm)
9. A particle is moving in SHM. At a certain instant its velocity is -4.0 cm/s. Five seconds later its displacement from the equilibrium position is 2.0 cm and after a further 3.0 s its displacement is -4.0 cm. Determine the largest period and its amplitude in this motion. (6.394 s, 10.77 cm)

IB Physics Speed

A tutorial sheet on speed is given below.

1. Define the word speed.
2. Is the speed of an object at a certain time equal to its rate of change of distance at this time?
3. Is the speed of an object at a certain instant always equal to the gradient of the distance-time graph at this instant?
4. Define the term average speed.
5. Is average speed equal to distance over time?
6. A car travels between two towns A and B. The car travels from A to B at a constant speed of 40 km h^-1. It immediately returns to A on the same road with a constant speed of 50 km h^-1. Is the average speed for the entire journey 45 km h^-1?