Topic 6 [166 marks]

1. A motorcyclist is cornering on a curved race track.Which combination of changes of banking angle θ and coefficient of friction μbetween the tyres and road allows the motorcyclist to travel around the corner atgreater speed?

[1 mark]

2. Satellite X orbits a planet with orbital radius R. Satellite Y orbits the sameplanet with orbital radius 2R. Satellites X and Y have the same mass.

What is the ratio ?

A.

B. C. 2D. 4

centripetal acceleration of Xcentripetal acceleration of Y

14

12

3. Object P moves vertically with simple harmonic motion (shm). Object Qmoves in a vertical circle with a uniform speed. P and Q have the sametime period T. When P is at the top of its motion, Q is at the bottom of its motion.

What is the interval between successive times when the acceleration of P is equaland opposite to the acceleration of Q?A.

B.

C. D. T

T4

T2

3T4

[1 mark]

[1 mark]

4. A particle of mass 0.02 kg moves in a horizontal circle of diameter 1 mwith an angular velocity of 3 rad s . What is the magnitude and direction of the force responsible for this motion?

π -1

5. Which graph shows the relationship between gravitational force Fbetween two point masses and their separation r?

[1 mark]

[1 mark]

6a.

The Moon has no atmosphere and orbits the Earth. The diagram shows the Moonwith rays of light from the Sun that are incident at 90° to the axis of rotation of theMoon.

A black body is on the Moon’s surface at point A. Show that themaximum temperature that this body can reach is 400 K. Assume thatthe Earth and the Moon are the same distance from the Sun.

6b. Another black body is on the Moon’s surface at point B.Outline, without calculation, why the aximum temperature of the black body atpoint B is less than at point A.

[2 marks]

[2 marks]

6c. The albedo of the Earth’s atmosphere is 0.28. Outline why the maximumtemperature of a black body on the Earth when the Sun is overhead isless than that at point A on the Moon.

6d. Outline why a force acts on the Moon.

6e. Outline why this force does no work on the Moon.

[1 mark]

[1 mark]

[1 mark]

7a.

A proton moves along a circular path in a region of a uniform magnetic field. Themagnetic field is directed into the plane of the page.

Label with arrows on the diagram the magnetic force F on the proton.

7b. Label with arrows on the velocity vector v of the proton.

7c. The speed of the proton is 2.16 × 10 m s and the magnetic fieldstrength is 0.042 T. For this proton, determine, in m, the radius of thecircular path. Give your answer to an appropriate number of significant figures.

6 -1

[1 mark]

[1 mark]

[3 marks]

8a.

The moon Phobos moves around the planet Mars in a circular orbit.

Outline the origin of the force that acts on Phobos.

8b. Outline why this force does no work on Phobos.

8c. The orbital period T of a moon orbiting a planet of mass M is given by

where R is the average distance between the centre of the planet and the centreof the moon.Show that

= kMR3

T 2

k = G

4π2

[1 mark]

[1 mark]

[3 marks]

8d. The following data for the Mars–Phobos system and the Earth–Moonsystem are available:Mass of Earth = 5.97 × 10 kgThe Earth–Moon distance is 41 times the Mars–Phobos distance.The orbital period of the Moon is 86 times the orbital period of Phobos.Calculate, in kg, the mass of Mars.

24

[2 marks]

8e. The graph shows the variation of the gravitational potential between theEarth and Moon with distance from the centre of the Earth. The distancefrom the Earth is expressed as a fraction of the total distance between the centreof the Earth and the centre of the Moon.

Determine, using the graph, the mass of the Moon.

[3 marks]

9a.

A proton moves along a circular path in a region of a uniform magnetic field. Themagnetic field is directed into the plane of the page.

Label with arrows on the diagram the magnetic force F on the proton.

9b. Label with arrows on the diagram the velocity vector v of the proton.

6 -1

[1 mark]

[1 mark]

9c.

The speed of the proton is 2.16 × 10 m s and the magnetic field strength is0.042 T.

For this proton, determine, in m, the radius of the circular path. Giveyour answer to an appropriate number of significant figures.

6 -1

9d. For this proton, calculate, in s, the time for one full revolution.

[3 marks]

[2 marks]

10. A particle of mass m and charge of magnitude q enters a region ofuniform magnetic field B that is directed into the page. The particlefollows a circular path of radius R. What are the sign of the charge of the particleand the speed of the particle?

11. Two isolated point particles of mass 4M and 9M are separated by adistance 1 m. A point particle of mass M is placed a distance from theparticle of mass 9M. The net gravitational force on M is zero.

What is ? A. m

B. m

C. m

D. m

x

x

413

25

35

913

[1 mark]

[1 mark]

12. The mass at the end of a pendulum is made to move in a horizontal circleof radius r at constant speed. The magnitude of the net force on the massis F.

What is the direction of F and the work done by F during half a revolution?

13. A mass m attached to a string of length R moves in a vertical circle with aconstant speed. The tension in the string at the top of the circle is T. Whatis the kinetic energy of the mass at the top of the circle?

A.

B.

C.

D.

R(T+mg)2

R(T−mg)2

Rmg

2

R(2T+mg)2

[1 mark]

[1 mark]

14a.

Ion-thrust engines can power spacecraft. In this type of engine, ions are created ina chamber and expelled from the spacecraft. The spacecraft is in outer spacewhen the propulsion system is turned on. The spacecraft starts from rest.

The mass of ions ejected each second is 6.6 × 10 kg and the speed of each ion is5.2 × 10  m s . The initial total mass of the spacecraft and its fuel is 740 kg.Assume that the ions travel away from the spacecraft parallel to its direction ofmotion.

Determine the initial acceleration of the spacecraft.

–6 4 –1

14b.

An initial mass of 60 kg of fuel is in the spacecraft for a journey to a planet. Half ofthe fuel will be required to slow down the spacecraft before arrival at thedestination planet.

Estimate the maximum speed of the spacecraft.

[2 marks]

[2 marks]

14c. Outline why scientists sometimes use estimates in making calculations.

14d.

In practice, the ions leave the spacecraft at a range of angles as shown.

Outline why the ions are likely to spread out.

[1 mark]

[2 marks]

14e. Explain what effect, if any, this spreading of the ions has on theacceleration of the spacecraft.

14f.

On arrival at the planet, the spacecraft goes into orbit as it comes into thegravitational field of the planet.

Outline what is meant by the gravitational field strength at a point.

14g. Newton’s law of gravitation applies to point masses. Suggest why the lawcan be applied to a satellite orbiting a spherical planet of uniform

density.

[2 marks]

[2 marks]

[1 mark]

15a.

There is a proposal to place a satellite in orbit around planet Mars.

Outline what is meant by gravitational field strength at a point.

15b. Newton’s law of gravitation applies to point masses. Suggest why thelaw can be applied to a satellite orbiting Mars.

[2 marks]

[2 marks]

15c.

The satellite is to have an orbital time T equal to the length of a day on Mars. Itcan be shown thatT = kRwhere R is the orbital radius of the satellite and k is a constant.

Mars has a mass of 6.4 × 10  kg. Show that, for Mars, k is about 9 ×10 s m .

2 3

23–13  2  –3

4

[3 marks]

15d. The time taken for Mars to revolve on its axis is 8.9 × 10 s. Calculate,in m s , the orbital speed of the satellite.

4–1

15e.

The ratio = 1.5.

Show that the intensity of solar radiation at the orbit of Mars is about600 W m .

distance of Mars from the Sundistance of Earth from the Sun

–2

[2 marks]

[2 marks]

15f. Determine, in K, the mean surface temperature of Mars. Assume thatMars acts as a black body.

15g. The atmosphere of Mars is composed mainly of carbon dioxide and hasa pressure less than 1 % of that on the Earth. Outline why the mean

temperature of Earth is strongly affected by gases in its atmosphere but that ofMars is not.

[2 marks]

[3 marks]

16. An object of mass m at the end of a string of length r moves in a verticalcircle at a constant angular speed ω.What is the tension in the string when the object is at the bottom of the circle?A. m(ω r + g)B. m(ω r – g)C. mg(ω r + 1)D. mg(ω r – 1)

2

2

2

2

17. Newton’s law of gravitationA. is equivalent to Newton’s second law of motion.B. explains the origin of gravitation.C. is used to make predictions.D. is not valid in a vacuum.

18. A mass at the end of a string is swung in a horizontal circle at increasingspeed until the string breaks.

The subsequent path taken by the mass is aA. line along a radius of the circle.B. horizontal circle.C. curve in a horizontal plane.D. curve in a vertical plane.

19. An object of mass m moves in a horizontal circle of radius r with aconstant speed v. What is the rate at which work is done by thecentripetal force?

A.

B.

C. D. zero

mv3

r

mv3

2πr

mv3

4πr

[1 mark]

[1 mark]

[1 mark]

[1 mark]

20a.

An electron moves in circular motion in a uniform magnetic field.

The velocity of the electron at point P is 6.8 × 10 m s in the direction shown.The magnitude of the magnetic field is 8.5 T.

State the direction of the magnetic field.

5 –1

20b. Calculate, in N, the magnitude of the magnetic force acting on theelectron.

20c. Explain why the electron moves at constant speed.

[1 mark]

[1 mark]

[1 mark]

20d. Explain why the electron moves on a circular path.

21a.

A small ball of mass m is moving in a horizontal circle on the inside surface of africtionless hemispherical bowl.

The normal reaction force N makes an angle θ to the horizontal.

State the direction of the resultant force on the ball.

[2 marks]

[1 mark]

21b. On the diagram, construct an arrow of the correct length to representthe weight of the ball.

21c. Show that the magnitude of the net force F on the ball is given by thefollowing equation.

F =mg

tan θ

[2 marks]

[3 marks]

21d. The radius of the bowl is 8.0 m and θ = 22°. Determine the speed ofthe ball.

21e. Outline whether this ball can move on a horizontal circular path ofradius equal to the radius of the bowl.

[4 marks]

[2 marks]

21f. A second identical ball is placed at the bottom of the bowl and the firstball is displaced so that its height from the horizontal is equal to 8.0 m.

The first ball is released and eventually strikes the second ball. The two ballsremain in contact. Determine, in m, the maximum height reached by the twoballs.

22a. State the direction of the resultant force on the ball.

[3 marks]

[1 mark]

22b. On the diagram, construct an arrow of the correct length to representthe weight of the ball.

22c. Show that the magnitude of the net force F on the ball is given by thefollowing equation.

F =mg

tan θ

[2 marks]

[3 marks]

22d. The radius of the bowl is 8.0 m and θ = 22°. Determine the speed ofthe ball.

22e. Outline whether this ball can move on a horizontal circular path ofradius equal to the radius of the bowl.

[4 marks]

[2 marks]

22f.

The ball is now displaced through a small distance x from the bottom of the bowland is then released from rest.

The magnitude of the force on the ball towards the equilibrium position is given by

where R is the radius of the bowl.

Outline why the ball will perform simple harmonic oscillations aboutthe equilibrium position.

mgx

R

22g. Show that the period of oscillation of the ball is about 6 s.

[1 mark]

[2 marks]

22h. The amplitude of oscillation is 0.12 m. On the axes, draw a graph toshow the variation with time t of the velocity v of the ball during one

period.

23a.

A planet has radius R. At a distance h above the surface of the planet thegravitational field strength is g and the gravitational potential is V.

State what is meant by gravitational field strength.

23b. Show that V = –g(R + h).

[3 marks]

[1 mark]

[2 marks]

23c. Draw a graph, on the axes, to show the variation of the gravitationalpotential V of the planet with height h above the surface of the planet.

23d. A planet has a radius of 3.1 × 10 m. At a point P a distance 2.4 × 10 mabove the surface of the planet the gravitational field strength is 2.2 N

kg . Calculate the gravitational potential at point P, include an appropriate unitfor your answer.

6 7

–1

[2 marks]

[1 mark]

23e. The diagram shows the path of an asteroid as it moves past the planet.

When the asteroid was far away from the planet it had negligible speed. Estimatethe speed of the asteroid at point P as defined in (b).

23f. The mass of the asteroid is 6.2 × 10 kg. Calculate the gravitationalforce experienced by the planet when the asteroid is at point P.

12

[3 marks]

[2 marks]

24a. Bohr modified the Rutherford model by introducing the condition mvr =n . Outline the reason for this modification.h

2π

24b. Show that the speed v of an electron in the hydrogen atom is related tothe radius r of the orbit by the expression

where k is the Coulomb constant.

v = √ ke2

mer

[3 marks]

[1 mark]

24c. Using the answer in (b) and (c)(i), deduce that the radius r of theelectron’s orbit in the ground state of hydrogen is given by the following

expression.

r =h2

4π2kmee2

24d. Calculate the electron’s orbital radius in (c)(ii).

106 106 –

[2 marks]

[1 mark]

24e.

Rhodium-106 ( ) decays into palladium-106 ( ) by beta minus (β )decay. The diagram shows some of the nuclear energy levels of rhodium-106 andpalladium-106. The arrow represents the β decay.

Explain what may be deduced about the energy of the electron in the βdecay.

10645Rh 106

46Pd –

–

–

24f. Suggest why the β decay is followed by the emission of a gamma rayphoton.

–

24g. Calculate the wavelength of the gamma ray photon in (d)(ii).

[3 marks]

[1 mark]

[2 marks]

25. A mass attached to a string rotates in a gravitational field with a constantperiod in a vertical plane.

How do the tension in the string and the kinetic energy of the mass compare at Pand Q?

26. A satellite X of mass m orbits the Earth with a period T. What will be theorbital period of satellite Y of mass 2m occupying the same orbit as X?A. B. TC. D. 2T

T2

√2T

[1 mark]

[1 mark]

27a.

A satellite powered by solar cells directed towards the Sun is in a polar orbit aboutthe Earth.

The satellite is orbiting the Earth at a distance of 6600 km from the centre of theEarth.

Determine the orbital period for the satellite.Mass of Earth = 6.0 x 10 kg24

[3 marks]

27b.

The satellite carries an experiment that measures the peak wavelength emittedby different objects. The Sun emits radiation that has a peak wavelength λ of 509nm. The peak wavelength λ of the radiation emitted by the Earth is 10.1 μm.

Determine the mean temperature of the Earth.

SE

27c. Suggest how the difference between λ and λ helps to account forthe greenhouse effect.

S E

[2 marks]

[3 marks]

27d. Not all scientists agree that global warming is caused by the activities ofman.

Outline how scientists try to ensure agreement on a scientific issue.

28. Two pulses are travelling towards each other.

What is a possible pulse shape when the pulses overlap?

[1 mark]

[1 mark]

29. A horizontal disc rotates uniformly at a constant angular velocity about acentral axis normal to the plane of the disc.

Point X is a distance 2L from the centre of the disc. Point Y is a distance L from thecentre of the disc. Point Y has a linear speed v and a centripetal acceleration a.What is the linear speed and centripetal acceleration of point X?

30. An object of constant mass is tied to the end of a rope of length l andmade to move in a horizontal circle. The speed of the object is increaseduntil the rope breaks at speed v. The length of the rope is then changed. At whatother combination of rope length and speed will the rope break?

[1 mark]

[1 mark]

31. Two satellites of mass m and 2m orbit a planet at the same orbit radius. IfF is the force exerted on the satellite of mass m by the planet and a is thecentripetal acceleration of this satellite, what is the force and acceleration of thesatellite with mass 2m?

32. The gravitational field strength at the surface of Earth is g. Another planethas double the radius of Earth and the same density as Earth. What is thegravitational field strength at the surface of this planet?A.

B. C. 2gD. 4g

g

2g

4

[1 mark]

[1 mark]

33. A small ball of weight W is attached to a string and moves in a verticalcircle of radius R.

What is the smallest kinetic energy of the ball at position X for the ball to maintainthe circular motion with radius R?A. B. W RC. 2 W RD.

W R2

5W R2

34. The centre of the Earth is separated from the centre of the Moon by adistance D. Point P lies on a line joining the centre of the Earth and thecentre of the Moon, a distance X from the centre of the Earth. The gravitationalfield strength at P is zero.

What is the ratio ?

A.

B.

C.

D.

mass of the Moonmass of the Earth

(D−X)2

X 2

(D−X)X

X 2

(D−X)2

X

D−X

[1 mark]

[1 mark]

35a.

The diagram below shows part of a downhill ski course which starts at point A, 50m above level ground. Point B is 20 m above level ground.

A skier of mass 65 kg starts from rest at point A and during the ski course some ofthe gravitational potential energy transferred to kinetic energy.

From A to B, 24 % of the gravitational potential energy transferred tokinetic energy. Show that the velocity at B is 12 m s .–1

[2 marks]

35b. Some of the gravitational potential energy transferred into internalenergy of the skis, slightly increasing their temperature. Distinguish

between internal energy and temperature.

35c. The dot on the following diagram represents the skier as she passespoint B.

Draw and label the vertical forces acting on the skier.

35d. The hill at point B has a circular shape with a radius of 20 m. Determinewhether the skier will lose contact with the ground at point B.

–1

[2 marks]

[2 marks]

[3 marks]

35e. The skier reaches point C with a speed of 8.2 m s . She stops after adistance of 24 m at point D.

Determine the coefficient of dynamic friction between the base of the skis and thesnow. Assume that the frictional force is constant and that air resistance can beneglected.

–1

35f.

At the side of the course flexible safety nets are used. Another skier of mass 76 kgfalls normally into the safety net with speed 9.6 m s .

Calculate the impulse required from the net to stop the skier and statean appropriate unit for your answer.

–1

[3 marks]

[2 marks]

Printed for Jyvaskylan Lyseonlukio

© International Baccalaureate Organization 2020 International Baccalaureate® - Baccalauréat International® - Bachillerato Internacional®

35g. Explain, with reference to change in momentum, why a flexible safetynet is less likely to harm the skier than a rigid barrier.

[2 marks]