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?

IB HL Physics Field strength and potential

A tutorial sheet on the HL concepts of field strength and potential is given below.

1. Define the term electric potential.
2. Define the term electric field strength.
3. Is electric field strength equal to minus the change in potential divided by the change in displacement?
4. What is the relationship between electric field (E) and electric potential (V)?
5. True or false. If the electric potential at a point is zero then the electric field strength at this point must be zero.
6. True or false. If the electric field strength at a point is zero then the electric potential at this point must be zero.
7. True or false. The electric potential at the midpoint of the line joining charges +Q and +Q is a local minimum.
8. True or false. The electric potential at the midpoint of the line joining charges +Q and -Q is a global maximum.
9. Point charges +Q are placed at the vertices of an equilateral triangle of side L. Find the electric potential and field strength at the centre of the triangle.
10. Four point charges +Q are placed at the corners of a square of side L. A point charge -q is placed at the centre of the square. Describe the movement of the charge -q if it (a) pushed a small amount towards a corner charge, and b) pushed a small amount towards the midpoint of a side of the square.

IB Physics Resultant force

A tutorial sheet on resultant force is given below.

1. Define the word force.
2. Define resultant force.
3. A mass on the end of a rod moves in uniform circular motion in a vertical circle. Which of the following are constant in magnitude? (a) the velocity of the mass (b) the acceleration of the mass (c) the force exerted by the rod on the mass (d) the resultant force acting on the mass.
4. A mass is placed at rest on a vertical spring that has its upper end fixed. The mass is pulled down and released from rest. Draw a free-body diagram showing the forces acting on the mass when (a) it is held at rest and (b) after it is released from rest.
5. A smooth rod of length 2L rotates about a vertical axis through its centre in a horizontal plane at a constant angular speed. A particle can slide along the rod. Describe the subsequent movement of the particle if it is released from rest (relative to the rod) at (a) the centre of the rod (b) a distance L/2 from the centre of the rod.
6. In question 5 (b) what is the force that makes the particle slide along the rod? Is this force constant in magnitude?

IB SL Physics Charged oil drop in a uniform electric field

A tutorial sheet on a charged oil drop in a uniform electric field is given below.

1. An charged oil drop of mass 0.1 g is held at rest in a vertical uniform electric field between two oppositely charged parallel metal plates. The potential difference between the plates is 1000.0 V, the top plate is positive and the plates are 2.0 cm apart. Find the magnitude of the charge on the oil drop.
2. X-rays are directed on the air between the plates causing the charge on the drop to be halved. Find the new net force acting on the oil drop.
3. The drop falls from rest and reaches a terminal speed. Find the drag force acting on the oil drop.
4. Sketch graphs showing the (a) distance travelled by the oil drop (b) speed of the oil drop and (c) acceleration of the oil drop, as it falls from rest in part (3).
5. When the potential difference between two oppositely charged parallel metal plates is V1 an oil drop falls downwards with a terminal speed u1. When the potential difference between the plates is V2 the oil drop moves upwards with a terminal speed u2. Find the charge on the drop.

IB HL Physics Dielectrics and capacitors

A tutorial sheet with dielectric questions is given below.

1. What is a dielectric?
2. What does dielectric constant mean?
3. Plastic has a dielectric constant of 2.8. Find the permittivity of plastic.
4. A block of plastic is placed in a uniform electric field. Is the electric field strength in the plastic greater than, equal to or less than the electric field strength outside the plastic?
5. A block of dielectric fills the space between the parallel plates of a capacitor that is connected to a source of constant emf. Is the capacitance increased, stays the same or decreased when the dielectric is inserted?
6. A block of dielectric fills the space between the parallel plates of an isolated charged capacitor. Is the capacitance increased, stays the same or decreased when the dielectric is inserted?

IB Physics Average power or power?

A tutorial sheet of questions on power is given below.

1. Define instantaneous power.
2. Define average power.
3. Is average power always one-half the maximum power ?
4. Which is the correct equation, 𝘗 = Fv or 𝘗 = Fv cos𝜃 ?
5. The resultant force acting on an object is given by F = a v + b, where v is the speed of the object and a and b are constants. The "area" under the graph of F versus v between v = 0 and v = V is equal to (a) the power when v = V, (b) the average power from v = 0 to v = V, (c) the total power from v = 0 to v = V (d) none of the above
6. The engine of a car of mass m has a constant power output P. A drag force kv acts on the car as it is driven along a horizontal road, where v is the speed of the car. If the car starts from rest the speed of the car (a) increases at a constant rate, (b) increases at an increasing rate, (c) increases at a decreasing rate, (d) remains zero due to drag force.
7. In question 6 the speed of the car (a) approaches a maximum value, (b) reaches a maximum value, (c) continues to increase, (d) approaches a maximum value that is proportional to k.

IB HL Physics Multiple slit interference

A tutorial sheet of questions on multiple slit interference is given below.

1. Laser light of wavelength 𝜆 passes through a double slit of separation d and slit width b. Describe the intensity pattern that is observed on a screen at a distance D from the double slit.
2. Sketch a graph showing the intensity on the screen versus distance along the screen.
3. Three parallel narrow slits are now used. Describe the interference pattern on the screen if the wavelength, slit spacing, slit width and distance to the screen are unchanged.
4. A triple slit interference pattern is compared to a double slit interence pattern pattern using the same wavelength of light, slit spacing, slit width and distance from the slits. In the double slit pattern (a) the maxima are closer together and dimmer, (b) the maxima are closer together and sharper, (c) the maxima are the same distance apart and dimmer, (d) the maxima are the same distance apart and sharper
5. Distinguish between primary maxima and secondary maxima. How many secondary maxima form between the primary maxima when 4 slits are used?

IB HL Physics Magnetic flux

A tutorial sheet on magnetic flux is given below.

1. Define magnetic flux density.
2. Define magnetic flux.
3. Define one tesla.
4. Define one weber.
5. Can the magnetic flux through a coil be increased by increasing the number of turns in the coil?
6. The equation giving the magnetic flux through an area is ϕ = BAcosθ. How is the angle θ defined?
7. Define magnetic flux linkage.
8. Several properties of the magnetic flux are given below. Which are correct and how could these be improved? a) magnetic flux is the amount of magnetic field crossing an area, b) magnetic flux is the number of magnetic field lines crossing an area, c) magnetic flux through a plane area is the product of the area and the sum of all of the perpendicular B components crossing it, d) the magnetic flux through any closed surface is always zero.

IB Physics Why is the magnetic force perpendicular to the current?

Why does the magnetic force on a current push perpendicular to the conductor? The magnetic field direction around a straight wire carrying a current in a uniform external magnetic field is shown below. The current is flowing out of the page and the uniform magnetic field direction is to the right. The diagram shows the resultant magnetic field found by adding the magnetic field created by the current and the uniform magnetic field. Magnetic field lines can be thought of as exerting a tension along their length and a pressure perpendicular to their length, like elastic bands. What does the diagram suggest about the direction of the magnetic force acting on the current? Upwards, in agreement with the right hand palm rule.

IB Physics Emissivity

A tutorial sheet of questions on emissivity is given below.

1. What is the emissivity of a perfect black body?
2. Is emissivity the same as albedo?
3. A metal sphere of radius 2.50 cm is placed in a vacuum and has a surface temperature 523 °C. The intensity of the radiation from the sphere at a distance of 12.0 cm from its centre is 2.37 kW m^-2. Find the emissivity of the sphere.
4. Does emissivity have an effect on the peak wavelength in the radiation spectrum of an object?
5. Determine the peak wavelength in the radiation spectrum of the sphere in question 3.
6. Would a human eye "see" the sphere in question 3?
7. Infra red radiation has a wavelength between 750 nm and 1.0 mm. Visible light has wavelength between 380 nm and 750 nm. By referring to a Planck radiation curve, determine the ratio of the total energy emitted by the sphere as infra red radiation to that emitted as visible light.

IB Physics emf

The term emf is a misunderstood concept. Listed below are a number of possible definitions. Which are correct? What must be added/deleted to make them correct?

1. The voltage of a battery.
2. The potential difference of a battery.
3. The potential difference of the battery when no current is flowing through it.
4. The work done per unit positive charge.
5. The power per unit current.
6. The work done in moving a charge between two points.
7. The work done in moving a +1 C charge at a constant speed along a path in an electric field.
8. The integral of the electric field along the path taken.
9. The line integral of the electric field component parallel to the path taken along the path taken.
10. The work done in moving a +1 C charge at a constant speed around a closed path in an electric field.
11. The emf of a cell is the work done per unit positive charge in going around the circuit.

IB Physics Galileo's Pendulum

A tutorial sheet on Galileo’s pendulum, a pendulum where the string strikes a peg as the pendulum swings, is given below. The length of the string is L and the peg is a distance d below the point of support of the string.

1. In the diagram the mass is released from rest on the end of a light inextensible string with the string initially horizontal. Determine the speed of the mass at the lowest point in its swing.
2. In question 1, d = 0.5L. Find the maximum height reached by the mass above the peg after the string strikes the peg.
3. In question 1 show that d must be at least 0.6L if the mass is to swing completely around the peg.
4. The mass is placed on a string of length 1.0 m and released with the string making an angle of 60° with the vertical. If d = 0.7 m, find the angle made by the string with the horizontal when the mass reaches its highest point. (41.8°)
5. If d = 2L/3, find the least angle to the vertical at which the string can be released if the mass is to swing completely around the peg. (80.41°)
6. The string is replaced by an elastic spring of force constant k. The mass m is released from rest with the spring horizontal and unextended. Is the speed of the mass at the bottom of the swing greater than, equal to or less than, the speed of the mass when the string is inelastic?

Force on a Current

A tutorial sheet of questions on the force on a current is given below

1. Define current.
2. Define the SI unit for current.

3. What are the charge carriers in a copper wire?

4. When a current carrying wire is in an external magnetic field do the moving charge carriers in the wire experience a force?

5. How is the force on the moving charge carriers transferred to the wire? (charge carriers experience a force moving them to one side of the wire creating a charge separation that exerts a force on the postive ions in the lattice of the wire pulling the wire sideways)
6. Is the direction of the magnetic force on the wire the same as the magnetic force on the charge carriers?
7. A uniform magnetic field is directed into the page. A current in a wire flows to the right. What is the direction of the force on the charge carriers in the current?
8. A uniform electric field acts into the page. A current in a wire flows to the right. What is the direction of the force on the charge carriers?
9. A positively charged particle moves to the right. The charge enters a uniform electric field directed into the page. Describe the force on the charge.
10. In a Physics experiment a constant current flows in a long straight wire. Does the magnetic field produced by the current exert a force on the wire? Give a reason.

Doppler Effect for Sound

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

1. State a definition of the Doppler effect for sound waves.
2. A train moves to the east at a speed v. The speed of sound in air is v_s. The train emits sound waves. Find the speed of the sound waves (a) in front of the train, (b) behind the train
3. A fire truck has a siren of frequency f. The truck moves at a constant velocity v relative to the ground. The speed of sound in air is v_s. What is the frequency of the sound waves heard by a stationary observer as the truck approaches them?
4. In question 2 does the frequency of the sound heard by the observer increase, stay the same or decrease as the fire truck approaches?
5. A train has a whistle of frequency f that emits sound waves that travel through air at a speed v_s. The train moves at a speed u to the east. A second train moves to the east at a speed v, where u is less than v. Is the frequency of the sound waves heard on the second train f(v - u)/v_s?

6. A train has a whistle of frequency f that emits sound waves which travel through air at a speed v_s. The train moves at a speed u towards a tunnel in a vertical mountain. Is the frequency of the reflected sound waves heard on the train 2fu/v_s?

7. A physics student drops a vibrating 440 Hz tuning fork down an elevator shaft in a tall building. When the student hears a frequency of 400 Hz, how far has the tuning fork fallen? Take the speed of sound in air as 340 m s^-1. (Tippler, Physics, second edition, p 483)

IB HL Physics Poorly Answered Questions 3

A third tutorial sheet of questions on poorly answered HL concepts is given below.

1. A 6 Ω resistor is connected in series with a cell of emf 12 V of zero internal resistance, an ideal ammeter and an ideal voltmeter. Find the reading on each meter.
2. State the definition of emf.
3. An electron is in the fourth excited state of the hydrogen atom. How many different spectral lines could appear in the emission spectrum of this atom?
4. The strong nuclear interaction between two protons (a) can be equal to zero (b) is repulsive for distances less than 3 fm, (c) is attractive for distances less than 3 fm, (d) is always greater than the Coulomb interaction between the protons
5. A particular radioactive substance undergoes β^- decay. Does the daughter nuclide always have the same energy?
6. Two positive point charges are at a fixed distance apart. When the electric field due to the charges is zero the electric potential is (A) zero, (B) a maximum value, (C) a minimum value, (D) not defined
7. In quantum tunnelling (A) the particle tunnels under the obstacle and fully appears on the other side (B) the particle is described by a wave function that exponentially decays through the barrier and is detected with less energy on the other side (C) the particle is described by a wave function that exponentially decays through the barrier and is detected with the same energy on the other side (D) the particle passes through the barrier instantaneously (see Nature July 22, 2020)

IB SL Physics Poorly Answered Questions 4

A fourth tutorial sheet of questions that are not well answered in SL Physics is given below.

1. A resistor X has the shape of a cylinder of cross sectional area A and length L. The resistance of X is R. What is the resistance of a resistor Y made of the same material as X having the same volume as X but one-half the length of X? The resistors are at the same temperature.
2. An ideal gas particle of mass m moving at a speed v strikes the wall of the container at an angle of 𝜽 to the surface. What is the magnitude of the change in momentum of the particle due to the collision?
3. A cell of emf E and internal resistance r is connected to a variable resistance R. The potential difference across the cell is V and the current in the circuit is I. The gradient of the graph of V versus I is (a) R, (b) r, (c) -r, (d) -R
4. The emf of a cell is the work done in moving a unit positive charge through the cell. True or false?
5. A trolley of mass M is pulled along a smooth horizontal surface with an acceleration a by a string tied to a falling mass. The string passes over a smooth pulley at the end of the table. Express the mass of the falling object in terms of a, g and M.
6. The following nuclear reaction occurs, P + Q -> R + S. This reaction can be called nuclear fission if (a) energy is released, (b) P is a neutron, (c) R and S have a lower binding energy per nucleon than P + Q , (d) R and S have a higher binding energy per nucleon than P + Q
7. A block of mass M moves at a speed U to the east. After a time interval t the speed of the block is V to the west. The magnitude of the impulse given to the block during this time interval is (a) M(V - U)t (b) M(V - U) (c) M(V + U)t (d) M(V + U)
8. Which of the following equations is now used to define the kilogram? (a) F = mg, (b) F = ma, (c) hf = mc² (d) E = mc².

HSC Physics Moving Reference Frames

A tutorial sheet of questions on a moving object seen in a moving reference frame is given below.

1. A bus moves to the west at a constant velocity. A ball is released from rest relative to the bus. Describe the path of the ball (a) relative to the bus (b) relative to the ground.
2. A bus is moving to the east with a constant acceleration to the east. A ball is released from rest relative to the bus. Describe the path of the ball (a) relative to the bus (b) relative to the ground.
3. A bus is moving to the east with a constant acceleration to the west. A ball is released from rest relative to the bus. Describe the path of the ball (a) relative to the bus (b) relative to the ground.
4. An elevator is moving upwards at a constant velocity. A ball is thrown horizontally inside the elevator. Describe the path of the ball (a) relative to the elevator (b) relative to the ground.
5. An elevator is moving upwards with a constant acceleration upwards. A ball is thrown horizontally inside the elevator. Describe the path of the ball (a) relative to the elevator (b) relative to the ground.
6. A boy and a girl sit at opposite ends of the diameter on a rotating table. At a certain instant the boy is moving to the west and the girl to the east. The girl throws a ball that is caught by the boy. Describe the initial velocity components of the ball (a) relative to the table, (b) relative to the ground.
7. A simple pendulum has a period of oscillation T. The pendulum is supported over the centre of a rotating reference frame of period of revolution P. Find the period of oscillation of the pendulum in the rotating reference frame when (a) P = T, (b) P > T, (c) P < T