IB Physics Relativistic train

A tutorial sheet on special relativity, a topic that most HL students find difficult. The tutorial questions are taken from the book University Physics by Harris Benson, Revised Edition, John Wiley and Sons, p 832. It is important to identify the proper time and proper length in a question. For questions where this choice is not obvious, apply the Lorentz transformations. Always find gamma first.

1. A train (frame S') of proper length 3.2 km moves at 0.6c relative to a platform (frame S). At t = t'= 0, two light pulses are emitted in opposite directions from the centre of the train. At what times do the pulses reach the ends of the train in (a) frame S' (b) frame S?
2. When it is 10 ⁸ m away in the Earth's frame (S), a rocket (frame S') travelling at 0.8c towards the Earth emits a flash. On receipt, it is immediately transmitted back to the rocket. How long does it take the flash return to the rocket according to the Earth frame?
3. A detector (frame S') moves away from the origin of frame S at a speed V along the +x axis. When it is at a distance x = L from the origin of S, a flash is emitted at the origin. How long does it take the flash to reach the detector according to observers (a) in S, (b) in S'?
4. A train of rest length 800 m approaches a platform of length 1 km at 0.6c. The front reaches the left end of the platform at t = 0 in the platform frame. At what time does the rear of the train reach the right end of the platform in the platform frame?

IB Physics Two climber problem

A tutorial sheet on the famous two climber problem ( Physics for Scientists and Engineers, Tipler, 3rd edition, p 133, q 44) follows. Why is this question famous? (Does Jay fall vertically downwards after going off the edge?)

1. Two climbers are on an icy (frictionless) slope that makes 40.0° with the horizontal are tied together by a 30.0 m rope. One climber (Paul, of mass 52.0 kg) is hanging over the top of the slope and the other climber (Jay, of mass 74.0 kg) has dropped his ice pick and is on the slope. At time t=0 the speed of each is zero. (a) Find the tension in the rope as Paul falls and his speed just before he hits the ground if initially he is 20.0 m above the ground. (b) If Paul unties his rope after hitting the ground, find Jay's speed as he hits the ground. Initially Paul is hanging 5.0 m below the top of the slope.
2. Assume that Paul does not untie his rope and that the coefficients of static and dynamic friction of Paul with the horizontal ground are 0.3 and 0.2 respectively. Find Jay's speed as he hits the ground.

IB Physics. Is constructive interference always a maximum value?

A tutorial sheet on interference follows.

1. At a point on the screen constructive interference occurs in a Young's double slit experiment. Do crests always meet crests at this point?
2. At another point on the screen destructive interference occurs. Does a crest always coincide with a trough at this point?
3. A graph is drawn of the change in pressure with distance as a sound wave passes. Is this graph the same as the displacement of the particles from their equilibrium position?
4. When a standing wave in an open pipe is drawn we have antodes at the open ends. What are we plotting if there are antinodes at the open end?
5. The first harmonic standing wave forms in an open pipe. Draw a graph showing the change in pressure along the length of the pipe. At what point in the pipe does the pressure change have its maximum value?

IB Physics Missing Orders in an Interference Pattern

This tutorial sheet examines the missing orders in the double slit interference pattern.

1. In a double slit interference pattern with coherent monochromatic light with d = 1.5b, are all interference maxima present? Explain.
2. In a double slit interference pattern with coherent monochromatic light with d = 3b, are all interference maxima present? Explain.
3. State the condition for the missing orders in a double slit interference pattern.
4. A double slit interference pattern contains missing orders of maximum interference. Where has the energy gone? Explain.

IB Physics Entropy

A tutorial discussion sheet on entropy is given below.

1. What does the word entropy mean?
2. State the second law of thermodynamics in terms of entropy.
3. Can entropy ever decrease? Explain.
4. Can entropy ever stay the same? Explain.
5. Does entropy always increase? Explain.

IB Physics. Electron Moving in a Magnetic Field

A tutorial sheet for the motion of a charge in a magnetic field is given below.

An electron enters a region of uniform magnetic field of flux density 1.2 × 10^-3 T with an initial velocity of 3.0× 10⁷ m s^-1 at an angle of 30.0° to the magnetic field.

1. Describe the path of the electron in the magnetic field.
2. Find the time taken by the electron to make one complete revolution about the magnetic field lines.
3. Find the cross sectional radius of the path of the electron.
4. Find the work done by the magnetic field on the electron in 10 ms.
5. Suppose that the magnetic field increases uniformly in time. Is the work done on the electron zero? Does the speed of the electron change?
6. Suppose that the magnetic field is proportional to z and the electron is injected at 30.0° to the z axis. Is the work done on the electron zero?

IB Physics. Coulomb’s law. Four Point Charges

Four equal point charges are placed on a straight line of length 2a. Two of the charges are fixed at the ends of the line. Find the positions on the line where the other two charges are placed if they are to be at rest. (From D. J. Griffiths and Y. Li, Am. J. Phys., 64, 6, June 1996, p 708) Answer is at x = ± 0.36148a from the centre of the line.

IB Physics. The Jumping Hoop

This tutorial sheet is about the jumping hoop demonstration. A smooth vertical hoop of mass M has two beads of mass m slotted on it, both initially at rest side by side at the highest point of the hoop. The masses are released and slide down along the hoop. Find the smallest ratio of m/M if the hoop is to leave the ground during the subsequent motion. (Problem given in Isaac Physics, L. Jardine-Wright, Periphyseos Press, 2015, p24)

1. What is the first step in solving a mechanics problem? Draw a diagram showing the forces acting and the coordinates used.
2. Is total energy conserved as the beads slide on the hoop?
3. Is the linear momentum of the system conserved as the beads slide on the hoop?
4. Is the angular momentum of the system conserved as the beads slide?
5. What condition determines if the hoop leaves the surface?
6. Answer. Ratio = 3/2.

IB Physics. Compton Effect

A tutorial sheet of questions on the Compton effect follows.

1. Is momentum conserved in the scattering of an X-ray photon by a free electron?
2. Is kinetic energy conserved in the scattering of an X-ray photon by a free electron?
3. A photon of energy 20.0 keV is scattered through an angle of 90.0° when it is scattered by a free electron. Find the kinetic energy of the scattered electron.

IB Physics. Electric Potential and Field

A tutorial sheet of questions on electric potential and field follows.

1. Explain why the electric potential is negative in sign around a negative charge.
2. A spherical shell of radius R carries a charge Q. What is the electric potential at the centre of the sphere?
3. A spherical shell of radius R carries a charge Q. What is the electric field at the centre of the sphere?
4. A sphere of radius R contains charge of uniform density 𝜌. Find the electric potential at the centre of the sphere.
5. A sphere of radius R contains charge of uniform density 𝜌. Find the electric field at the centre of the sphere.
6. A sphere of radius R contains charge of uniform density 𝜌. The sphere contains a spherical cavity of radius R/4 at a distance R/2 from the centre of the sphere. Find the electric potential at the centre of the cavity.

IB Physics. Electricity

A tutorial sheet on the IB Physics topic B3 electricity is given below.

1. In a wire N electrons per minute move to the right past a point every second. Find an expression for the current.
2. The emf of a cell is E. The internal resistance of the cell is r. Find the work done by the cell in moving a charge q between its terminals.
3. The potential difference across a resistor is V when it is connected in series to a cell of zero internal resistance. Find the potential difference across this resistor when two equal resistances which are connected in parallel are placed in series with the first resistor.

HSC Physics. Momentum and Kinetic Energy

A tutorial sheet on momentum and kinetic energy is given below.

1. Two moving objects collide and join together. Is the total kinetic energy conserved in this collision?
2. A pellet of mass 100 g moves at 20 m s^-1 to the east. A clay block of mass 300 g moves at 5 m s^-1 to the west. A collision occurs and the objects join together. Find the kinetic energy of the combined mass after the collision.
3. The kinetic energy of an object is K. Find the new kinetic energy of this object if its mass is doubled and its momentum is halved.
4. A stationary nucleus has a mass M. It emits a small particle of mass m at a speed v. Find the ratio of the kinetic energy of the small particle to that of the remaining nucleus.

IB Physics and Maths. Simple harmonic motion

A tutorial sheet on simple harmonic motion using IB Physics and Maths skills is given below.

1. The period of oscillation of a particle moving in simple harmonic motion is T. For what fraction of this time are the velocity and acceleration vectors in the same direction?
2. The period of oscillation of a particle moving in simple harmonic motion is T. For what fraction of this time are the displacement from the equilibrium position and the velocity vector in the same direction?
3. The energy of a particle moving in simple harmonic motion is E. Find the energy of the particle if its mass is doubled and the frequency of oscillation is tripled.
4. A particle moves in simple harmonic motion of period 6.0 s and amplitude 12.0 cm. At a certain time the particle is moving away from the equilibrium position at 3.0 cm s^-1. Find the time taken by the particle to come to rest.

IB Physics and Maths. Straight Line Motion

A tutorial sheet of problems on motion in a straight line, solved using IB Physics and Mathematics skills, is given below.

1. The acceleration of a particle is given by a = - k x², where k is a constant. Is the motion simple harmonic motion? Explain.
2. In question 1, k = 0.01 m^-1 s^-2 and the mass of the particle is 100.0 g. Using a graphical method, find the work done on the particle when it moves from x = ∛2 m to x = 0.
3. Find the speed of the particle at the origin if it is at rest when x = ∛2 m.
4. Show that the velocity of the particle is given by, -√( ( 2 - x³)/150 ).
5. Using a GDC show that the time taken to reach the origin is 15.3 s.

IB HL Physics. Spacetime diagrams

A tutorial sheet on HL topic A.4 special relativity is given below.

1. On a spacetime diagram the axes are inclined at 45° to each other. What does this mean?
2. A spacetime diagram is drawn for observers in reference frames that are moving at a constant velocity of 0.8c relative to each other. When drawn on paper the unit length interval on the x and ct axes is 2.4 cm. Is the length interval on the x' and ct' axes less than, equal to or greater than this? Explain why and give its value.

IB HL Physics. P-V diagrams

A tutorial sheet on P-V diagrams from Topic B.4 Thermodynamics is given below.

1. During a cyclic process on a P-V diagram is the entropy change always zero?
2. Does the efficiency of a heat engine depend on how rapidly the process is carried out?
3. State the definition of an adiabatic process.
4. On a P-V diagram the initial state is (P₁,V₁) and the final is (P₂, V₂). If the ratio of specific heats is ɣ and the specific heat capacity at constant volume is c_v, find the entropy change between the two states.

IB Physics. Centripetal Force

A tutorial sheet on centripetal force follows.

1. The Earth rotates on its axis once every 23 h 56 m 4 s. Why do objects remain at rest on the Earth (relative to the Earth) ?
2. Imagine that the Earth's period of rotation is 2 h. Would an object on the Earth stay at rest on the Earth? Explain.
3. A planet of mass M has a radius R and has a period of rotation about its axis of T. Find the weight, relative to the surface of the planet, of a mass m at latitude θ on the planet.
4. A centrifuge is spinning at a rate ω. Do heavier particles accumulate near the centre or the outside? Explain.
5. A mass m can slide on a smooth horizontal rod. The mass is placed at the centre of the rod. The rod rotates with a constant angular speed ω about an axis perpendicular to one end. Does the mass stay at rest relative to the rod, move inwards or outwards? Explain.

IB Physics. Motion in a Vertical Circle

A sheet of harder questions on a mass moving on a string in a vertical circle is given below.

1. A mass m hangs at rest on the end of a light inextensible string of length L. The mass is given a horizontal velocity U and the maximum vertical height to which the mass rises is L. Find U.
2. In the previous question find the tension in the string and the magnitude of the acceleration of the mass at its highest point.
3. A mass m is on the end of a string of length L that has its upper end fixed. The mass is released from rest and swings in a circular arc. Find an expression for the tension in the string when the string makes an angle 𝜃 with the vertical.
4. A mass hangs at rest on the end of a light inextensible string of length L. The mass is given a horizontal speed U. In its subsequent motion the string becomes slack and the mass hits the point of supprt of the string. Find the initial speed U.

HSC Physics Atwood's Machine

Atwood’s machine, the name given to two blocks connected by a light inelastic string passing over a smooth pulley, is used in many mechanics experiments.

1. A single Atwood's machine is suspended from the roof of an elevator. Masses of of m₁ and m₂ are connected by a light inelastic string that passes over the smooth pulley. Find the acceleration of the elevator if m₁ and m₂ are at rest relative to the elevator.
2. A double Atwood’s machine consists of a fixed pulley with a string connected to a mass m₁ passing over it, with another Atwood’s machine connected to the other end of the string passing over the first pulley. On the second Atwood’s machine there are masses of m₂ and m₃ connected by a light inelastic string. We neglect the mass of each pulley.Show that the acceleration of the second pulley is given by a = g(4 m₂ m₃ - m₁(m₂+ m₃))/(4 m₂ m₃ + m₁(m₂ + m₃))

HSC Physics Elastic Collision

A sheet of harder mechanics questions appears below.

An elastic collision occurs between an alpha partice of mass m moving at a speed u and a stationary nucleus of mass M. After collision the alpha particle moves at speed v at an angle 𝜃 to its initial direction and the nucleus moves at speed V at an angle ϕ to the initial direction of the alpha particle.

1. Show that v = u sinϕ/sin(ϕ + 𝜃).
2. Show that V = m u sin𝜃/( M sin(ϕ + 𝜃)).
3. Show that m/M = sin(2ϕ + 𝜃)/sin𝜃.