IB Physics Mass and Energy

A tutorial sheet on mass and energy. This is a topic of continual confusion. Notes for guidance are taken from the article by Keith Atkin in Physics Education 35, 5, Sep 2000, page 319.

1. Is matter the same as mass? No, matter is a qualitative term meaning substance, mass is a defined physical quantity which is a measure of inertia.
2. E = mc² states that mass (inertia) and energy are equivalent not interconvertible.
3. In the Cockroft-Walton reaction a proton interacts with a lithium nucleus forming two helium nuclei with a gain in kinetic energy. Is there a loss in mass? No, mass and energy are equivalent. Both energy and mass are conserved in the process, there is no destruction of mass or creation of energy. There is merely a redistribution of a fixed amount of energy (and its equivalent mass).

IB Physics Diffraction

This tutorial sheet is about diffraction, a concept that is not widely understood.

1. State the definition of diffraction.
2. Why does diffraction occur when light passes through a single slit?
3. Explain qualitatively the pattern that we see when light passes through a single slit if i. the slit width is much greater than the wavelength ii. the slit width is much less than the wavelength.
4. Divide a single slit of width b into two sections. By considering a ray from the top of each section, show that destructive interference occurs when b sin𝜃 = 𝜆. Do this with four sections in the slit and show that destructive interference occurs when b sin𝜃 = 2𝜆. This can be generalized to b sin𝜃 = n𝜆, n=1,2,..
5. When light is now incident normally on a diffraction grating what does the equation d sin𝜃 = n𝜆, n=0,1,2,3.. predict?
6. The incident monochromatic light is not normal to the grating. Can we use the equation d sin𝜃 = n𝜆 in this situation?
7. In question 6 the angle of incidence of the light on the grating is ϕ. Show that the angle 𝜃 to the normal to the grating where intensity maxima occur are given by d ( sin(𝜃 - ϕ) - sinϕ ) = n𝜆.
8. Describe the difference between primary maxima and secondary maxima.
9. Describe the interference pattern when white light is incident normally on a diffraction grating.

IB Physics TOK What did Planck Quantize?

A tutorial sheet on the process of science in the development of quantum phenomena. See Physics Education, Nov 2000 p 381-386, A brief history of quantum phenomena by Gren Ireson.

1. In 1900 Max Planck quantized the energy of the oscillators in the radiating body rather than the radiation from the body.
2. Planck presented an argument that consisted essentially phenomenological curve fitting, relying on classical idea of entropy at long wavelengths and an ad hoc conhecture due to Wien for short wavelengths, but which fitted the data perfectly at all wavelengths. However he was unable to offer any theoretical justification for the results.
3. Planck regarded the walls of the cavity as harmonic oscillators which could absorb and emit energy only in discrete amounts, E, which is related to the frequency, f, of the absorbed or emitted radiation by E = hf, where h is Planck's constant.
4. In 1905 Albert Einstein took the notion of quantization further by suggesting that electromagnetic radiation exists in the form of packets of energy that we now call photons.
5. Application of the photon model supplied Einstein with the means to solve the photoelectric effect. Classical electromagnetic theory predicted that the energy available in light is proportional to the intensity and independent of the frequency, but experimental evidence pointed to the opposite result.

IB Physics Flipping Rod

A tutorial sheet on the problem of a uniform rod moving in a vertical plane.

A uniform rod of length 2a and mass m lies horizontal on a smooth horizontal surface. One end of the rod is given a vertical velocity V upwards.

1. Is energy conserved as the rod moves in a vertical plane?
2. Is linear momentum conserved as the rod moves in a vertical plane?
3. Is angular momentum conserved as the rod moves in a vertical plane?
4. The rod turns right over without losing contact with the table. Show that 6ag < V² < 7ag.

IB Physics Entropy Decrease

A tutorial sheet on the entropy change in a rust reaction is given below. It is taken from Entropy and Rust by Daniel F Styer, Am J. Phys., 78 (11), November 2010, p 1077.

The rust reaction is 4Fe +3O₂ →2Fe₂O₃. The entropy (at room temperature, 298.15 K and at a pressure of 10⁵ Pa) of 1 mole of Fe is 27.280 JK^-1, of 1 mole of O₂ is 205.147 J K^-1 and of one mole of Fe₂O₃ is 87.404 J K^-1.

1. Find the entropy change when 1 mole of Fe rusts.
2. Describe the entropy change in the surroundings when 1 mole of Fe rusts.

IB Physics Point Charges at the Corners of a Triangle

This tutorial sheet investigates a problem involving two positive and a negative charge placed in a plane. It is taken from Electricity and Magnetism, The Berkeley Physics Course Volume 2, Second Edition by Edward M Purcell.

1. Two positive ions and one negative ion are placed at the vertices of an equilateral triangle. Where can a fourth charge be placed so that the force on it will be zero? Is there more than one such place? (page 35 q 1.12)

IB Physics Rod Swinging in a Vertical Plane

This tutorial sheet is about a uniform rod swinging in a vertical plane.

A uniform rod of length 2L mass M can swing freely in a vertical plane about a horizontal axis through one end. The rod is released from rest when it is horizontal.

1. What are the acceleration components of the end of the rod when it makes an angle 𝜃 with the vertical?
2. Find the magnitude of the force exerted by the point of support of the rod when it makes an angle 𝜃 with the vertical.
3. Draw a graph showing the angular speed of the rod in terms of the angle that the rod makes with the vertical.
4. The rod is now released from the horizontal position and is given an initial angular speed. Find the least initial angular speed that will allow the rod to complete a full revolution.

IB Physics Mass on a Swinging Rod

A tutorial sheet on the problem of a mass on the end of a uniform rod swinging in a vertical plane. A question answered correctly by only a small percentage of the world.

A uniform rod of mass 2m and length L has a particle of mass m attached to its lower end. The rod is free to swing in a vertical plane about a horizontal axis through its upper end. The rod is released from rest at the horizontal.

1. Draw a free-body diagram showing the forces acting on the (i) particle, (ii) rod. The rod exerts an inwards force Y acting along the rod and a perpendicular force X on the particle along the tangent to the circular path. This question involves a rod and not a string. For a string X=0.
2. Use Newton's second law in terms of forces to write down the two equations of motion of the particle. The equations are written in terms of components parallel to the rod and perpendicular to the rod.
3. Use Newton's second law for rotational motion to write down the equation of motion for the system in terms of the net torque acting on it.
4. Solve the three equations simultaneously to find the force components exerted by the rod on the particle.
5. Show that X = mgsin𝜃/5 and Y = 17mgcos𝜃/5, where 𝜃 is the angle made by the rod with the vertical.
6. The force exerted by the rod on the particle is directed towards the centre of the circle only when the rod is vertical (𝜃=0°).

IB Physics Point Charges at the Corners of a Square

This sheet practices finding the electric potential of a system of point charges.

1. Define electric potential at a point in an electric field.
2. A point charge -q is placed at (-d/2,0) and a point charge +q is placed at (d/2,0). Find the electric potential at the point P that has coordinates (2d,0).
3. In question 2 the point P is moved to a very large distance x from the origin along the x-axis. Show that the potential at P is approximately d k q/x².
4. In question 2 find the electrostatic potential energy of the two charge system. What does this mean?
5. Point charges of +q are placed at the corners of a square of side of length d. Find the electric potential at the centre of the square.
6. In question 5 sketch a graph showing the variation in electric potential along the y-axis. The origin is at the centre of the square and the y-axis is parallel to the side of the square.
7. Show that the potential at the point (x,y) near the origin is approximately V = 8kq/d(1 + ( x² + y² )/d² + ..).
8. In question 5 a charge +q is placed at the origin and given a small displacement along the y-axis. Describe the subsequent motion of this charge and calculate its period. The charges at the four corners of the square are fixed in position.

IB Physics HL Mechanics

A sheet of questions on rolling objects, a topic in HL mechanics. These are explain questions requiring elaboration.

1. A sphere rolls from rest down an inclined plane. A block of the same mass slides the same distance down a smooth inclined plane of the same angle to the horizontal. Which object reaches the bottom first? Explain.
2. Why is friction necessary for rolling? Explain.
3. Can we use the equation F_f = µ F_N for rolling objects? Explain.
4. Does friction force do work as an object rolls? Explain.

IB Physics. What is conserved in the Compton effect?

The HL IB Physics course gives an equation for use in the Compton effect. What are the Physics principles expressed in this equation? These are general questions requiring elaboration in the answers.

1. Is momentum conserved in the Compton effect?
2. Is mass conserved in the Compton effect?
3. Is kinetic energy conserved in the Compton effect?
4. Is energy conserved in the Compton effect?
5. In the Compton effect the photon is scattered and in the photoelectric effect the photon is absorbed. Why is there a difference?

IB Physics Two blocks on a horizontal table

A tutorial sheet on two blocks on a table follows.

1. A 2.0 kg block sits on a 4.0 kg block that is resting on a frictionless table. The coefficients of static and dynamic friction between the blocks are 0.3 and 0.2 respectively. What is the maximum horizontal force that can be applied to the 4.0 kg block if the 2.0 kg block is not to slide on the 4.0 kg block? (Tipler Physics 3rd edition, p 133, q 47. )

2. A 60 kg block slides along the top of a 100 kg block with an acceleration of 3.0 m s ^-2 when a horizontal force of 320 N is applied. The 100 kg block sits on a smooth horizontal surface, but there is friction between the two blocks. Find (a) The coefficient of kinetic friction between the blocks. (b) The acceleration of the 100 kg block during the time that the 60 kg block maintains contact. (loc. cit, q 42)

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 follows. It is taken from Physics for Scientists and Engineers, Tipler, 3rd edition, p 133, q 44.

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. In question 1 does the path followed by Jay after the rope is untied affect his speed at the ground?
3. Now 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 antinodes 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.