IB Physics Motion in a Vertical Circle

A sheet of vertical circle problems is given below.

1. A mass m is tied to a string of length L and hangs vertically at rest. The mass is given an initial horizontal speed u. What is the initial acceleration of the mass?
2. A mass on a string is released from rest swings as a simple pendulum. What is the speed of the mass when its vertical drop is ∆h?
3. A mass m is placed on the end of a light rigid rod of length r and made to move in a vertical circle at a constant speed v. Is the magnitude of the acceleration of the mass a constant value of v²/r? (no, this is the radial component of the acceleration only)
4. In the previous question draw a free-body diagram if the radius to the mass makes an angle 𝜽 with the vertical when 𝜽 < 90°. Show that the force exerted on the mass by the rod has a component mgsin𝜽 perpendicular to the rod and a component mv²/r - mgcos𝜽 parallel to the rod.
5. In the previous question at what angle to the vertical is the force exerted by the rod maximum? (180°)

IB Physics Spring Problems

A sheet testing Physics concepts involving a mass on a spring is given below.

1. At a certain instant a mass on a spring has zero velocity. Must it have zero acceleration?
2. An object on a spring has zero acceleration at a certain instant. Must it have zero velocity?
3. A mass on a spring is at rest. Is there no force acting on the mass?
4. Is the tension in a spring equal to the resultant force acting on the mass?
5. State Hooke's law (be careful).
6. A mass moves in SHM on a spring. Is the speed of the mass at the instant it passes through the equilibrium position the same as the average speed during one oscillation?
7. A mass M is placed on a spring of constant k that hangs vertically. If the mass is released from rest does the mass fall a distance Mg/k before coming to rest?
8. A block of mass M rests on a smooth horizontal surface. A spring of force constant k connects the block to a wall on the left. On the right a constant horizontal force F pulls on the block. Is the extension of the spring F/k when the block comes to rest?
9. A mass M is on a smooth horzontal table. Two identical springs, one on the left and one on the right, are connected to the mass. The other end of each spring is held at rest. If the mass is displaced a small amount in the line of the springs and released, what is the period of the oscillations? Each spring has a spring constant k.
10. A spring has a spring constant k. The spring is cut in half. What is the spring constant of each section?
11. A 2.0 kg mass slides from rest down an inclined plane of angle of elevation 30°. After travelling 4.0 m the mass encounters a spring of spring constant 100 Nm^-1. Find the maximum compression of the spring if (a) the incline is smooth, (b) the coefficient of friction between the mass and the plane is 0.20. ((a) 0.989 m, not 0.885 m (b)0.783 m)
12. In question 11 (b) determine the compression of the spring when the mass eventually comes to rest. (8.6 cm, from Tipler Physics 2nd edition page 220)

IB Physics Pulling Apart a Hydrogen Atom

1. When a hydrogen atom is ionised does the mass decrease, stay the same or increase?

2. When an electron “jumps down” to its ground state does the mass of the hydrogen atom decrease, stay the same or increase?

3. When an electron “jumps up” from its ground state to its first excited state does the mass of the hydrogen atom decrease, stay the same or increase?

4. Is the mass of a hydrogen atom equal to the mass of a proton plus the mass of an electron?

5. Imagine that a proton and an electron link to form a hydrogen atom. Is energy released or absorbed in the process? In what form is the energy change?

6. Is the proton in the hydrogen atom the same as a free proton?

7. In the Bohr theory of the hydrogen atom the speed of the electron in the ground state is v. What is the speed of the electron in the first excited state?

8. In the Bohr theory of the hydrogen atom the energy of the ground state is E. What is the energy of the first excited state?

9. How can the hydrogen atom have a negative energy?

10. The ground state energy of the hydrogen atom is -13.6 eV. A potential difference is applied to some hydrogen gas in a discharge tube. Will an applied voltage of 3.4 V cause the gas to emit light?

See Resnick and Halliday, Basic Concepts in Relativity and Early Quantum Theory , second edition, page 113

IB HL Physics Capacitors

1. Where does the charge “go” when a capacitor discharges?

2. Where does the energy “go” when a capacitor discharges?

3. Is there an electric field between its plates when a capacitor discharges?

4. Is there a magnetic field between its plates when a capacitor discharges?

5. Is I = dq/dt correct when a capacitor discharges?

6. Two charged capacitors are placed in series. Does the equation on page 11 of the Physics Data Booklet apply in this case?

IB HL Physics Misunderstood Concepts 2

Below are some HL questions on concepts that are not well understood.

1. A metallic surface is irradiated with infrared radiation and photoelectrons are emitted from the surface. This radiation is replaced by ultraviolet light of the same intensity. How does this affect the kinetic energy of the emitted photoelectrons and their rate of release?
2. A mass m is tied to a string and revolves in a circular path of radius R in a vertical plane. What is the least total energy of the mass if it is to maintain its circular motion?
3. A train has a whistle of frequency f. The train moves to the east at a constant speed v. Another train also has a whistle of the same frequency and this train moves to the west at a constant speed v. What is the frequency of the sound heard on the train moving to the west?
4. The diameter of a nuclide of mass number 2A is 3r. What is the radius of a nuclide of mass number 3A?
5. A mass m has a total energy E when it moves in simple harmonic motion of amplitude A on a spring. What is the total energy of the motion if a mass m is added and the amplitude is increased by 50%?
6. A wind farm consisting of 65 turbines, each of power 15 MW, would generate the same amount of energy as a typical nuclear plant of 975 MW. True or false? (reference, Physics World, March 2011)
7. Which is closest to the efficiency of commercial solar panels made of silicon that are used on rooftops, 20%, 50%, 75%, 100%? (reference, Physics World, April 2021)

IB Physics Relativity

A common examination response is that a rapidly moving object is observed to be contracted in length as it passes an observer. Is this correct? No. The observed overall length of a far away object is the same and it is seen to be rotated with the back of the object visible to the observer and is seen as red shifted. The below extract from Electromagnetic Fields and Waves by Lorrain, Corson and Lorrain (3rd edition, 1988) p254 summarises this.

"Suppose one looked through a telescope at a far-away cube moving at a velocity v≅c perpendicular to the line of sight. Then the face normal to the line of sight would appear to be foreshortened in the direction of motion by the factor 1/ɣ. Also one would see the trailing face for the following reason. At a given instant, the eye senses the photons that arrive at that instant. Photons originating from distant parts of the object have left earlier than others and the object has moved in the meantime. The net effect is that the cube would appear to be rotated through an angle arctan(v/c). If the cube were not far away, then it would appear distorted in peculiar ways, depending on its distance and velocity. This effect, which never has been observed, was discovered by James Terrell in 1959, 54 years after the publication of Einstein's first paper on relativity."

IB SL Physics Poorly Answered Questions 1

Below are 6 questions encountered in IB SL Physics that are poorly answered. Can you do them?

1. A motor is used to lift a load by a rope passing over a pulley. The output power of the battery supplying the motor is 60 W. If in a time interval of 2.0 min the load gains 3.0 kJ of energy and friction in the pulley releases 0.5 kJ of energy, determine the efficiency of the motor. Neglect air resistance.
2. A stationary closed vertical cylinder of mass M, cross sectional area A and length L contains N particles of an ideal gas at a kelvin temperature T and pressure P. The cylinder sits on an digital balance. Determine an expression for the reading on the balance.
3. A particle is moving in SHM on a spring with total energy E on a smooth horizontal table. The amplitude of the motion is x₀. Determine, in terms of E only, the work done by the spring when the extension of the spring changes from 3x₀/4 to x₀/2.
4. Sketch a graph showing the speed of a particle moving in SHM in terms of the displacement from the equilibrium point.
5. A buoy floating in the ocean moves in simple harmonic motion of period 20 s. Which of the following energy resources can cause this oscillation? tidal motion, wind energy, solar energy.
6. One ampere is now defined using which of the following natural constants? e, h, c, k_B, f_Cs, N_A, K_cd .

IB HL Physics Poorly Answered Questions 1

Below are 11 questions encountered in IB HL Physics that are poorly answered. Can you do them?

1. The current in the primary coil of an ideal transformer having 200 turns in the primary coil and 120 in the secondary coil is a sine function of period 2.0 s and phase angle 𝜋/2. Draw the current-time graph for the secondary coil from t=0 to t=2.0 s.
2. Photoelectrons are being emitted from a metal surface. Light of the same intensity but higher frequency now shines on the metal. Does the rate of photoelectron emission stay the same, decrease or increase?
3. The initial activity of a radioactive substance is known. The half-life is required to be found. Can the half-life be found if the mass remaining at a later time is measured?
4. A coil spins at a constant rate of 2.0 rads^-1 in a uniform magnetic field. The average electrical power due to ohmic heating in the coil is 40 W. Draw a graph showing the electrical power versus time for one complete rotation of the coil. Initially the plane of the coil is perpendicular to the magnetic field lines.
5. A stationary vertical cylinder of cross sectional area A and length L contains N particles of an ideal gas at a kelvin temperature T and pressure P. The mass of the gas is M. What is the force exerted by the gas on the base of the cylinder?
6. Two stars are observed through a telescope. The images are just resolved when a yellow filter is placed over the telescope. Are separate images seen when the yellow filter is replaced by a red filter? Why?
7. Two parallel wires of lengths 1.0 m and 2.0 m carry currents of 2.0 A and 1.0 A in opposite directions in a vacuum respectively. If the magnitude of the electromagnetic force acting on the shorter wire is 2.0 mN determine the magnitude and direction of the electromagnetic force acting on the longer wire.
8. The air column in a pipe is vibrating in its second harmonic. Are the nodes always at the centre of a compression?
9. A single electron atom has 4 energy levels. In terms of energy values, the lower two energy levels are close together and the top two levels are very close together. Draw the emission spectrum of this atom labelling the longest wavelength transition.
10. A weight W is suspended from the roof by two strings of equal length. When one of the strings is suddenly cut the initial tension in the remaining string (a) always increases (b) always decreases (c) always stays the same (d) can stay the same
11. One kilogram is now defined in terms of which of the following natural constants? e, c, h, k_B, f_Cs, N_A, K_Cd

IB Physics Where do Magnetic Fields Come From?

A tutorial sheet of questions, from SL, HL and the Relativity topic is given below.

S is a reference frame at rest relative to the laboratory. S' is another reference frame moving at a constant non-zero velocity relative to S.

1. An electron is at rest in S. Is there an electric field in S? Is there a magnetic field in S?
2. An electron is at rest in S'. Is there an electric field in S'? Is there a magnetic field in S'?
3. An electron is at rest in S. Is there an electric field in S'? Is there a magnetic field in S'?
4. A uniform magnetic field B acts along the X axis in S. There is no electric field in S. S' moves parallel to the X axis of S. Is there an electric field in S'? Is there a magnetic field in S'?
5. A uniform magnetic field B acts along the X axis in S. There is no electric field in S. S' moves parallel to the Y axis of S. Is there an electric field in S'? Is there a magnetic field in S'?
6. A uniform electric field E acts along the X axis in S. There is no magnetic field in S. S' moves parallel to the X axis of S. Is there an electric field in S'? Is there a magnetic field in S'?
7. A uniform electric field E acts along the X axis in S. There is no magnetic field in S. S' moves parallel to the Y axis of S. Is there an electric field in S'? Is there a magnetic field in S'?
8. Two electrons each move at a constant velocity v along parallel paths. In S' the electrons are always on the Y axis at a distance d apart. (a) what are the forces between the electrons in S? (b) what are the forces between the electrons in S'? (c) is the magnitude of the force between the electrons the same in each reference frame?

What is an electron?

Particle or wave? Here is a quote from a textbook describing what an electron “is”.

1. Giancoli, Physics 5th edition page 837. We might ask ourselves: "What is an electron?" The early experiments of J. J. Thomson indicated a glow in a tube that moved when a magnetic field was applied. The results of these and other experiments were best interpreted as being caused by tiny negatively charged particles which we now call electrons. No one, however has actually seen an electron directly. The drawings we sometimes make of electrons as tiny spheres with a negative charge on them are merely convenient pictures (now recognized to be inaccurate). Again we must rely on experimental results, some of which are best interpreted using a particle model and others the wave model. These models are mere pictures that we use to extrapolate from the macroscopic world to the tiny microscopic world of the atom. And there is no reason to expect that these models somehow reflect the reality of an electron. We thus use a wave or a particle model (whichever works best in the situation) so that we can talk about what is happening. But we shouldn't be led to believe that an electron is a wave or a particle. Instead, we could say that an electron is the set of its properties that we can measure. Bertrand Russell said it well when he wrote that an electron is a "logical construction".

IB Physics Challenging Mechanics Questions

A tutorial sheet of difficult mechanics questions is given below.

1. A ball rolls horizontally from a table of height 2.0 m. On collision with the smooth horizontal floor the kinetic energy of the ball is reduced by one-half. At what speed does the ball leave the table if it strikes a vertical wall 1.0 m from the table at a height of 1/3 m above the floor? (0.75 m/s, 1.20 m/s, 1.92 m/s)
2. Adapted from the Moscow Physics Problems (MPP) 1986-2005, 1.15. A projectile moves horizontally at 10.0 ms^-1 at a height above the ground of 30.0 m. When it is 25.0 m from a vertical wall it explodes and disintegrates into many fragments flying in all directions and all having an initial speed of 20.0 ms^-1 relative to the projectile. Find the area on the surface of the wall that will be hit by the debris. Assume that fragments hitting the ground do not bounce and ignore air resistance. ( 4673.91 m² , x axis along initial direction of motion, max height on wall z = 50.19 m, hits wall at ground level at y = 61.75 m, g = 9.81 m s^-2 )
3. In the previous question the wall is not present. Determine the area on the ground that can be struck by debris from the explosion. (1.8007 ha, on x axis debris lands 104.14 m ahead and 28.73 m behind explosion point)
4. In question 3 determine the time interval during which debris is hitting the ground. ( 4.08 s )
5. In question 3 the explosion occurs at a vertical height of 30.0 m over an inclined plane of angle of elevation 20°. Determine the area on the incline on which debris lands. ( 1.1801 ha, debris lands a maximum 70.92 m ahead and 46.38 m behind, on the incline measured from the perpendicular to the plane through the explosion point )

IB Physics Why does a Satellite not Fall to the Earth?

A tutorial sheet on satellite orbits is given below.

1. A satellite orbits the Earth in a circular path. Is the satellite "falling" towards the Earth?
2. As a satellite orbits the Earth in a circular path does the Earth "fall" away from the satellite?
3. Determine the orbital speed of a geostationary satellite.
4. How far does a geostationary satellite "fall" towards the Earth in one second? How far does the Earth "fall" away from the satellite orbit in one second?

IB Physics Double Slit Interference

A tutorial sheet on Young’s double slit experiment follows.

1. A Young's double slit experiment is carried out with the wavelength much greater than the slit separation. Describe the intensity pattern on a distant screen.
2. In a Young's double slit experiment the wavelength is much less thn the slit separation. Describe the intensity pattern on a distant screen.
3. In a double slit experiment with slits of width b separated by a distance d are all of the maxima on a distant screen of the same intensity?
4. A double slit experiment is carried out with b much less than d. Describe the intensity variation on a distant screen.

IB Physics Energy of Capacitors

A tutorial sheet of problems on the HL topic energy of capacitors is given below.

1. A capacitor gains a charge Q when it is connected by wires to a battery of emf V and zero internal resistance. Is the work done by the battery equal to the energy stored in the capacitor?
2. A charged capacitor of capacitance C is connected by wires to an uncharged capacitor of equal capacitance. Show that one half of the original energy is lost. Where does it go?
3. A capacitance C holds a charge Q and is connected in series with a resistance R. What is the heat energy released in R as the capacitor discharges?
4. A dielectric slab is placed between the plates of a parallel plate capacitor. Explain the change in energy of the capacitor if the slab is inserted keeping (i) the potential difference between the plates constant, (ii) the charge on the plates constant.
5. See Physics Education, vol 5, p 33, Sep 1998

IB Physics Half-life

A tutorial sheet on the terms decay constant, mean lifetime and half-life is given below.

1. Is the mean lifetime of a particle the same as the half-life of a sample? [No]
2. Show that mean lifetime of a nucleus = 1/λ, where λ is the decay constant.
3. Show that half-life of a sample = ln2/λ.
4. What is the definition of half-life of a sample?
5. Is the decay constant equal to the probability of a nucleus decaying?
6. The mean lifetime of a muon is 2.2 μs. Show that the half-life of a sample of muons is 1.52 μs.
7. A large number of dice are thrown and the die showing a 6 are removed on each throw. A graph is drawn showing the number of remaining dice versus throw number. Show that the decay constant is 1/6 and the half-life is 6ln2 throws.
8. Radioactive decay is a spontaneous, random process. Define each term.
9. Two samples of the same element have a different mass. How do the half-lifes of the samples compare?
10. Why is the half-life constant as a material decays?
11. A sample of half-life T1 decays into a material of half-life T2. How many daughter nuclei are present when N parent nuclei have decayed?
12. When a nucleus undergoes beta decay the daughter nucleus has the same number of nucleons as the parent nucleus but the atomic number increases or decreases by 1. True or false?
13. In the alpha decay of a nucleus the change in binding energy appears as the kinetic energy of the alpha particle and the daughter nucleus. Since the products have equal but opposite momenta the alpha particle has a much greater kinetic energy than the daughter nucleus. True or false?
14. Gamma decay occurs very often after alpha or beta decay which leaves the daughter nucleus in an excited state. The excess energy is released as a photon from the nucleus. The mass of the excited nucleus is greater than that of the ground state nucleus.True or false?
15. See P.Dunne et.al, Physics Education, Measurement of the mean lifetime of cosmic ray muons in the A level laboratory, 33, Number 5, 1998.

IB Physics Centrifugal Force?

A tutorial sheet on rotating reference frames is given below.

1. "For matter in a circular orbit at radius r around a mass M the centrifugal force and the gravitational attraction balance". Is this statement correct?
2. A tube containing liquid is spun about a vertical axis through one end. Where in the tube is the pressure greatest? Show that the pressure difference between the end of a straight tube of length L and its axis is ⍴⍵²L²/2, where ⍴ is the density of the fluid in the tube and ⍵ is the angular speed of the tube. The tube rotates about an axis through one end perpendicular to the tube.
3. A bucket of water spins at a constant rate about an axis through its centre. Why does the water build up at the edge of the bucket? (the fluid at the edge has greatest speed, creating a pressure gradient in the fluid between the centre and the edge pushing the fluid highest at the edges, there is no centrifugal force in an inertial reference frame)
4. Why does the acceleration due to gravity on the surface a planet decrease if the planet spins faster?
5. A space station has a large toroidal (doughnut) shape. When it spins are objects inside the space station weightless?
6. A bead is placed half-way along a smooth horizontal wire. The wire now spins about a vertical axis through one end. Why does the bead move outwards along the wire?
7. In question 3 there are now two fluids in the bucket, each of different density ⍴₁ and ⍴₂. What is the shape of the common surface when the bucket is spun at a constant angular speed ⍵?

IB Physics Atomic Mass

A tutorial sheet on atomic mass follows.

1. Define the term atomic mass.
2. What is the atomic mass of the C-12 atom?
3. The atomic mass of carbon is 12.011 u. Why is this not exactly 12 u?
4. Is 1 u the mass of one proton?
5. Is the mass of a nucleus equal to A u where A is the mass number of the nucleus?
6. Give two reasons why the mass in u of a nucleus is not equal to the mass number of the nucleus. (protons and neutrons have a slightly different mass, the nuclear binding energy varies between nuclei; for two nuclei having the same mass number the one with greater binding energy will have a smaller mass).

IB Physics Static Electricity

A tutorial sheet on electroscope questions is given below

1. A radioactive sample is placed inside a charged electroscope. Describe what happens to the leaves of the electroscope.(radiation ionises the air inside the electroscope creating ions which are attracted to the leaves reducing their net charge causing them to collapse gradually).
2. A rod is brought near the cap of a positively charged electroscope. The leaves of the electroscope collapse and then diverge as the rod comes closer to the cap. Must the rod have a net negative charge? (yes, in this case the size of the charge on the rod is much greater than the size of the charge on the electroscope).
3. An electroscope has a net positive charge. A rod is brought near the cap of the electroscope and the leaves continually diverge as the rod approaches. Must the rod have a net positive charge? (yes).
4. A rod is brought near the cap of a positively charged electroscope. The leaves of the electroscope diverge when the rod is near the cap. Must the rod have a net positive charge? (no).
5. What is the test for determining the relative sign of the charge on a charged electroscope and a rod? (observe the initial effect; if the leaves collapse they are oposite in sign,if they diverge they have the same sign).
6. A charged body at first attracts an uncharged object. True or false? (True. When a charged ebonite rod is brought near small pieces of paper a charge of unlike sign is induced on the near side of the paper attracting them to the rod. When contact is made with the rod charge of like sign is transferred to the paper repelling it from the rod).
7. An uncharged metal disc has a radius a. A point charge q is placed on the axis of the disk at a distance d from its centre. What is the magnitude of the electric force acting on the charge q?
8. A metal disc of radius a has a charge Q. What work was done in charging this disc?

IB Physics Harder Questions

1. Two parallel wires each of length 2.00 m and mass 50.0 g are supported from the same point by light insulating strings of length 1.00 m and carry equal currents of 100.0 A in opposite directions. Determine the angle between the strings. (5.18°)
2. Two point charges of +6.00 μC and mass 50.0 g are suspended from a point by two light insulating threads of length 37.0 cm. Find the angle made by each string with the vertical when the charges are in equilibrium. (58.8°)
3. A 1200 kg car rounds a curve of radius 70 m banked at an angle of 12°. If the car is travelling at a constant speed of 90kmh ^-1 , determine the magnitude of the friction force between the tyres and the road. (8035N)
4. A bicycle and rider of total mass 75.0 kg can coast down a 4.00° hill at a constant speed of 10.0 kmh^-1. At maximum exertion the cyclist can descend the hill at a speed of 30.0 kmh^-1. Using the same power, at what speed can the cyclist climb this hill? Assume that the drag force on the cyclist is proportional to the square of the speed of the cyclist. (27.7 kmh^-1)
5. A ball is tied to a string of length L the other end of the string being fixed. The string is held horizontal and the ball is released from rest. A peg is located a distance 0.8L directly below the point of attachment of the string. Find the speed of the ball when it reaches the top of its circular path about the peg.
6. Tarzan's problem. A rope of length 4.00 m hangs from a tree branch at the edge of a cliff. Tarzan runs at 10.0 ms^-1 and grabs the rope at a height of 2.00 m. What is the maximum width of the valley that he can jump across? (13.6m, lets go of rope at 37.76° to the vertical). See The Physics Teacher, Jan 2014, page 6 for references.
7. Tarzan tries again. Tarzan needs to jump across a valley of width 15.0 m. If the rope length and grab height are the same as in the previous problem, what is the least speed at which he must run if he is to reach the other side? (10.6 ms^-1, lets go at 38.46° to the vertical)

Mathematical Tripos Dynamics and Relativity 2020

Below are my answers to the 2020 MT Part 1A Dynamics and Relativity questions.

1. 4C (i) 2b²/(p(b² - p²)) (ii) A = (L²/b² - 2k/R), B = 2k, C = - L²
2. 11C, ⍵ = aq(1-e^{-λt/(a2(1+q))})/λ , q = 93/70
3. 12C (a) sin(𝜽/2) = E/√( 4E² - M²c⁴ ) (b) f(r) = √( (1 - r²)/(1 + 3r²) ), ct = c𝛕/2√( 3 + 1/r² ), x = c𝛕/2√( 1/r² - 1 )