Topic 6 – Further Mechanics

Q1. 

The diagram shows the carriage of a rollercoaster about to enter a vertical loop of diameter 17.0 m. The carriage is initially at rest at a height Δh above the bottom of the loop.

During one particular ride, the speed of a car at the bottom of the loop was 22.5 m s−1.

(i)  Calculate the acceleration of the passenger at the bottom of the loop as a multiple of g, the acceleration due to gravity.

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Acceleration = ...........................................................

(ii)  The maximum safe acceleration recommended for passengers is 4g. Most loop-the-loop rollercoasters do not have a circular loop. Instead, the radius of curvature of the loop varies.

Explain why making the radius of the loop vary in this way enables the acceleration at the bottom of the loop to be less than 4g.

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(Total for question = 10 marks)

Q2. 

Motors usually have a rotating component which can do work W.

(a)  A motor lifts a load in a time t. The axle of the motor has a radius r and exerts a force F.

The power produced by a motor can be calculated by using the following word equation.

Power = moment of the force exerted by the rotating axle × angular velocity

Derive this equation, starting with power (4)

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(b)  An electrostatic motor was first demonstrated by Benjamin Franklin in 1750.

The diagram shows a simplified version of part of this motor.This consists of a rod, with an oppositely charged sphere at either end, which rotates around a fixed pivot. Two stationary charged spheres apply a force on the spheres at either end of the rod.

(i)  In the diagram below, electric field lines have been drawn around one pair of these spheres.Add to the diagram to show

· the directions of the field lines· the lines of equipotential.

(3)

(ii)  The distance between the centres of each charged sphere in this pair is 5.0 cm.Show that the force between this pair of charged spheres is about 0.04 N.charge on each sphere = 0.10 μC

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(c)  The table shows the typical power and the corresponding angular velocity required for three different appliances.

Deduce which of these appliances, in principle, could use the electrostatic motor in (b).You should use the word equation in (a) and assume that the length of the rod in the electrostatic motor is 8.0 cm.Assume that the electrostatic motor would deliver a constant force throughout one complete rotation.

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(Total for question = 13 marks) 

Q3. 

A stationary ball is released from a height of 2.0 m onto a hard surface.

The simplified velocity-time graph shows the motion of the ball as it falls and bounces back to its maximum height.

(a)  Calculate the maximum height reached by the ball after bouncing.

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Maximum height = ...........................................................

(b)  Calculate the decrease in kinetic energy of the ball as it bounces.

mass of ball = 60 g(2)

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Decrease in kinetic energy = ...........................................................

(c)  Calculate the resultant force on the ball when it is in contact with the ground.

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Resultant force = ...........................................................

(d)  The ball is replaced with one that is softer. It is released from a height of 2.0 m onto the same surface as before. A velocity-time graph is drawn to show the motion of the new ball.

Describe the similarities and differences between the two graphs.(3)

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(Total for question = 10 marks) 

Q4. 

A centrifuge is a machine which rotates.

The United States' space agency, NASA, uses a centrifuge to test whether equipment will operate when experiencing large forces. The equipment to be tested is attached to the end of the frame of the centrifuge, which rotates around a vertical axis at its centre.

(i)  Show that the angular velocity of the centrifuge is about 5 rad s−1.(2)

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(ii)  Explain how the centrifuge applies large forces to the equipment under test.(2)

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(iii)  The NASA website says the centrifuge can be used to test whether the equipment can withstand accelerations of up to about 25g.

Deduce whether this claim is correct.(2)

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(Total for question = 6 marks) 

Q5. 

The photograph is of a roundabout in a children's playground.

A child of mass 20 kg sits on the roundabout without holding the bars.

The distance from the centre of the roundabout to the centre of gravity of the child is 0.80 m. The maximum frictional force between the roundabout and the child is 0.35 × the weight of the child.

(a)  Calculate the minimum time taken for one revolution of the roundabout if the child is not to slide off.

(4)

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Minimum time = ...........................................................

(b)  State and explain how this time would change if a child of larger mass sat at the same place on the roundabout.

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(Total for question = 6 marks) 

Q6. 

Hump back bridges are sometimes found in country areas. The road surface is curved, describing an arc of a circle. If cars cross the bridge at too high a speed there is a danger they will lose contact with the road surface.

The responsibility for setting speed limits on many roads on which hump back bridges are found rests with the local traffic authority. One local traffic authority has suggested that a speed limit of 25 mph is appropriate for a hump back bridge with radius of curvature 10 m.

Assess the suitability of a speed limit of 25 mph for this bridge.

10 mph = 4.5 m s−1

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(Total for question = 5 marks) 

Q7. 

* The photograph shows a probe moving in space.

Whilst moving, empty fuel tanks can be ejected by means of an explosion. This has the effect of increasing the speed of the probe.

Discuss whether conservation of momentum and conservation of energy apply in this situation and why the speed of the probe increases.

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(Total for question = 6 marks) 

Q8. 

A coil of wire is placed around the lower end of an iron rod. The coil is supplied with an alternating current.

A thick aluminium ring is placed around the iron rod above the coil. The coil remains in the position shown.

The current is switched off and the aluminium ring comes to rest on top of the coil. The supply to the coil is changed and a direct current (dc) is switched on. An upwards force F acts on the ring for 0.05 s accelerating it to a final speed, v. The ring then moves freely through a height of 30 cm.

Mean diameter of ring = 4.8 cm Mass of ring = 0.019 kg Magnetic field strength = 0.032 T

(i)  Use conservation of energy to calculate the speed v of the ring after 0.05 s.(2)

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v = ...........................................................

(ii)  Use the idea of impulse to calculate the magnitude of the mean force F acting on the ring and

hence the mean current I in the ring.(6)

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F = ...........................................................

I = ...........................................................

 

(Total for question = 8 marks)

Q9. 

The diagram shows a battery-powered clock on a wall. It has a minute hand and an hour hand.

(a)  Calculate the average angular velocity of the minute hand.

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Angular velocity = ...........................................................

(b)  The diagram shows the position of the minute hand when the time is 1 : 44 and when the time is 1 : 46.

Show that the work done against the force of gravity to move the minute hand from 1 : 44 to 1 : 46 is about 1 mJ.mass of minute hand = 14 glength of minute hand = 8.0 cm

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(c)  The clock uses a 1.5 V cell and draws a maximum current of 8.0 μA.

After a time, the maximum power of the cell has reduced to 65% of its initial value making the clock run slowly.Calculate the time taken for the minute hand to move from the 1 : 44 position to the 1 : 46 position.

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Time taken = ...........................................................

 

(Total for question = 10 marks) 

Q10. 

An electron beam tube can be used to demonstrate the deflection of electrons in a uniform magnetic field. The tube contains a very low pressure gas so that electron paths can be seen.

Electrons are emitted from the electron gun travelling vertically upwards into a region of uniform horizontal magnetic flux density.

Explain why the electrons follow a circular path.

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(Total for question = 3 marks)

Q11. 

In 2015 the Messenger spacecraft crashed into the surface of the planet Mercury after four years in orbit observing the surface of Mercury.

Messenger's orbit was highly elliptical, varying between 200 km and 15 000 km above the surface of Mercury. Messenger completed one full orbit every 12 hours.

mass of Messenger spacecraft = 565 kg mass of planet Mercury = 3.30 × 1023 kg radius of planet Mercury = 2430 km

It has been suggested that the same orbital period of about 12 hours could have been achieved if Messenger was in a circular orbit 7690 km above the surface of Mercury.

(i)  Determine whether this suggestion is correct.

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(ii)  The elliptical orbit chosen had advantages over this circular orbit.

Explain one advantage.(2)

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(Total for question = 6 marks) 

Q12. 

A 'Gauss gun' can be made from five ball bearings of equal mass and two magnets, as shown.

Pairs of ball bearings are placed to the right of two strong magnets. A single ball bearing is released from the left, as shown. The ball bearing is attracted to, and collides with, the first magnet. This and all subsequent collisions can be assumed to be elastic.

Explain what happens to make the last ball bearing on the right subsequently move off with a large velocity.

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(Total for question = 3 marks) 

Q13. 

Mass spectrometry is a technique used to separate ions based on their charge to mass ratio.

The atoms in a sample are ionised and then accelerated and formed into a fine beam. This beam is passed into a region of uniform magnetic field and the ions are deflected by different amounts according to their mass.

Analysis of mass spectrometer data shows that chlorine exists in nature as two isotopes, chlorine-35 and chlorine-37.

After passing through the velocity selector the ion beam enters a region of uniform magnetic flux density 0.35 T with the ions travelling at right angles to the field direction.

(i)  Explain why the ions travel in a circular path.

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(ii)  Calculate the radius of the circular path.

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Radius = ...........................................................

(iii)  The diagram shows the path of the chlorine-35 ions in the field. Chlorine-37 ions enter the magnetic field with the same velocity.

1.   Add another line to the diagram to show the path of these chlorine-37 ions.(1)

2.   Explain any differences in the paths.(2)

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(Total for question = 7 marks) 

Q14. 

A cyclotron is a particle accelerator which can be used to accelerate protons. The cyclotron consists of two semicircular electrodes called 'dees'. An alternating potential difference is applied across the gap between the dees. A uniform magnetic field is applied at right angles to the plane of the dees.

(i)  Complete the diagram to show the path of the protons.

(1)(ii)  State the direction of the magnetic field needed in order to produce the path you have sketched.

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(iii)  Explain how the kinetic energy of the protons is increased as they follow the path you have shown.

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(iv)  Show that the magnetic flux density B of the applied magnetic field is given by

where f is the frequency of the alternating potential difference, m is the mass of the proton and e is the charge on the proton.

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(v)  In a particular cyclotron B is 1.2 mT. Calculate the frequency f of the alternating potential difference.

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f = ...........................................................

 

Q15. 

A teacher is demonstrating the principle of conservation of momentum using a flat glass surface and air pucks. Lightweight tubing supplies compressed air to the pucks which is forced out from the bottom of the pucks. This means that the pucks move with very little friction across the glass surface.

(a)  Explain, using ideas about molecular movement, how the puck is able to hover a small distance above the glass surface.

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*(b)  Applying Newton's 2nd and 3rd laws of motion to the collision between two pucks leads to the conclusion that momentum is conserved.

Justify this statement.(6)

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(c)  The teacher uses two identical pucks to investigate collisions. In one collision, one puck moves with a velocity of 4.0 m s−1 and collides with a second puck that is stationary. After the collision, the first puck has a velocity v at an angle of 30° to its original direction, and the second puck moves off with a velocity of 2.0 m s−1 at an angle of 60° to the original direction.

(i)  Show that the magnitude of the velocity v of the first puck after the collision is about 3.5 m s−1.(3)

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(ii)  Use the data to determine if the collision is elastic or inelastic.(3)

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(Total for question = 16 marks) 

Q16. 

The diagram shows two black holes, A and B, orbiting each other.

Assume that the centre of mass C of the system is the centre of a circular orbit for each black hole as shown in the diagram.

Black hole A is in an orbit of radius 2.9 × 1010 m and black hole B is in an orbit of radius 3.6 × 1010 m. Both orbit with the same period, so the total distance between them is 6.5 × 1010 m.

(a)  Calculate the force between the black holes.

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Force = ...........................................................

(b)  By considering the orbit of one black hole about C, determine the period of the orbit.

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Period = ...........................................................

 

(Total for question = 5 marks) 

Q17. 

A centrifuge is a machine which rotates.

A particle in a centrifuge moves in a circle of radius r, centre O, with a constant speed v.

The diagram represents two positions of the particle.

Derive the equation for centripetal acceleration by considering the velocity at these two positions.

Your answer should include a vector diagram.

(5)

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(Total for question = 5 marks) 

Q18. 

The discovery of the Higgs particle was an important contribution to our understanding of particle physics.

(a)  Describe the standard model for subatomic particles. You should identify the fundamental particles and the composition of the particles we can observe.

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(b)  The mass of the Higgs particle is 2.2 × 10−25 kg.

Calculate this mass in GeV/c2.(3)

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Mass = ........................................................... GeV/c2

(c)  The Higgs particle was discovered using the Large Hadron Collider (LHC) in 2012. Two beams of very high energy protons, moving in opposite directions, were made to collide.

(i)  Explain the need for such high energy collisions.

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(ii)  The beams of protons are contained within a ring of superconducting magnets.

Calculate the momentum of a proton in a beam.(3)

magnetic field strength = 8.3 T circumference of the ring = 27 km

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Momentum = ...........................................................

(iii)  State the total momentum of the products of the collision between the two beams of protons.

(1)Total momentum = ...........................................................

(d)  A student used the equation to predict the energy of a proton in the beam, using the momentum calculated in (c)(ii), but found the energy was far higher than 7 TeV.

Explain why.

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(Total for question = 17 marks) 

Q19. 

The diagram shows the carriage of a rollercoaster about to enter a vertical loop of diameter 17.0 m. The carriage is initially at rest at a height Δh above the bottom of the loop.

(i)  So that a passenger remains in contact with their seat at the top of the ride, show that the minimum speed of the car at the top of the loop is 9.1 m s−1.

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(ii)  Calculate the minimum value of Δh that will enable the passenger to remain in contact with their seat at the top of the loop.

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Δh = ...........................................................

 

(Total for question = 4 marks)

Q20. 

Answer the question with a cross in the box you think is correct ( ). If you change your mind about an answer, put a line through the box ( ) and then mark your new answer with a cross ().

Which of the following are the base units for impulse?

   A    kg m s−1

   B    kg m s−2

   C    N m

   D    N s

 

(Total for question = 1 mark) 

Q21. 

A pendulum consists of a bob of mass m and a string of length x.

The diagram shows the pendulum swinging through the arc of a circle. At the bottom of its swing the tension in the string is T and the velocity of the bob is v.

Which of the following is correct for the bob at the bottom of the swing?

   A    T = − mg

   B    T = + mg

   C    T = mg −

   D    T =

 

(Total for question = 1 mark) 

Q22. 

A bullet is fired into a block of wood. Select the line of the table that applies to this situation.

 

(Total for question = 1 mark)

Q23. 

A mass is attached to a light thread which is fixed at X. The mass is moving at constant speed in a horizontal circle, centre O.

Which of the following is a correct free-body force diagram for this mass?

(1)

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   B   

   C   

   D   

 

(Total for question = 1 mark)