www.mathspanda.com

Banked tracksStarter1. (Review of last lesson) A light inextensible string of

length m is threaded through a smooth ring, , and carries a particle at each end. Particle of mass kg is at rest at a distance of m below the ring. The other particle, , of mass kg is rotating in a horizontal circle whose centre is .(a) Express in terms of .(b) Find the angular velocity of .

NotesThe second type of situation where the forces are not acting horizontally is banked tracks.

The vehicle is modelled as a particle which has acceleration in a horizontal direction towards the centre of the circle.

When a car moves at the design speed it has no tendency to slip sideways on the track so there is no friction up or down the track.

When car travels faster than the design speed the tendency is for the car to slide outwards so friction acts down the banked track.

When car travels slower than the design speed,the tendency is for the car to slide inwards so friction acts up the banked track.

Lowest speed — minimum needed to prevent object from slipping inwards (towards the centre)

Highest speed — a higher speed would see the object slip outwards (away from the centre)

Solving problems involving banked tracksUse a similar method to conical pendulums.

Use or or depending on the situation.

Use radially (i.e. horizontally).Resolve vertically, .

N.B. Do not resolve or use parallel or perpendicular to the banked plane.

0.8 RA m

0.3B M

AM m

B

ar = rω2 ar = vω ar =v2

rF = ma

R( ↑ )

F = ma

Page of 1 4

AB

0.3

R

mg

Rar

Centre of circle α

rF

mg

Rar

Centre of circle α

r F

www.mathspanda.comE.g. 1 A car is rounding a bend of radius m which is banked at an angle of to the

horizontal. At what speed must the car travel to ensure it has no tendency to slip sideways (i.e. design speed)?

Working:

Using :

Dividing:

The car must travel at m/s (3 s.f.) for there to be no friction.

N.B. The design speed of the car does not depend on the mass of the car.

E.g. 2 A car is travelling around a circular bend on a road banked at an angle of to the horizontal. The car may be modelled as a particle moving in a horizontal circle of radius

m. When the car is moving at a constant speed of m/s there is no frictional force up or down the slope. Find the angle , giving your answer to 1d.p.

Banked train trackFor a banked train track the outer rail is higher than the inner one. The difference with roads is that there is no significant friction force. Sideways movement is prevented by the lateral thrusting force provided by the rails.

E.g. 3 A railway line is to be laid round a circular arc of radius m. Trains are expected to travel at km/h around the bend. Find:(a) the force exerted by the outer edge on the flanges of the wheel if the track is level

and the mass of the train is tonnes(b) the height, in mm, at which the outer rail should be raised so that there is no

pressure on the wheel flanges, given that the rails are m apart. Give your answer in centimetres.

Working: (a) km/h = m/s,

Using :

N kNThe force on the outer edge of the flanges of the wheel is N

100 10∘

r = 100

ar =v2

rar =

v2

100F = ma( → ) : R sin 10 = m

v2

100R( ↑ ) : R cos 10 = mg

sin 10cos 10

=v2

100gv2 = 100g tan 10v ≈ 13.1

13.1

α

120 20α

50045

35

1.5

v = 45 12.5 r = 500

ar =v2

rar =

12.52

500=

516

F = ma( → ) : F = 35000 ×5

16F = 10937.5 ≈ 11

10937.5

Page of 2 4

mg

Rar

10∘

r = 100

www.mathspanda.com

(b) As before ,

and

Dividing:

Using trigonometry:

metresThe height the outer rail should be raised is cm

E.g. 4 A train of mass kg travels at km/h round a bend of radius m. (a) If the track is level, calculate the lateral thrust on the outer rail. (b) At what height should the outer rail be raised to eliminate lateral thrust at this speed

if the distance between the rails is m. Give your answers to 2 s.f.

Video: Banked tracks

Solutions to Starter and E.g.s

Exercisep96 4C Qu 4, 6, 9

Further questions on banked tracks1. A car is moving at a constant speed of m/s round a curve which is part of a horizontal

circle of radius m. The acceleration of the car is m/s2.(a) Calculate the value of .(b) The car has a mass of kg. Calculate the magnitude of the horizontal force

acting on the car perpendicular to its direction of motion and suggest the physical cause of this force.

2. A curved section of race track, of radius m, is banked at .(a) Find the design speed.(b) Given that the car is of mass kg and travels at m/s, find the magnitude

and direction of the lateral frictional force exerted by the track.

3. A road banked at goes round a bend of radius, m. Vehicles can travel at m/s around the bend without tending to side-slip. Find the value of .

v = 12.5 r = 500ar =

516

F = ma( → ) : R sin α = 35000 ×516

R( ↑ ) : R cos α = 35000gtan α =

516g

α ≈ 1.826∘

sin α =h

1.5h = 1.5 sin 1.826∘

h = 0.04784.78

80000 40 1200

1.4

v50 3.38

v900

120 40∘

840 36

10∘ r 11r

Page of 3 4

αh

1.5 m

www.mathspanda.comAnswers to further questions1. (a) (b) N, friction

2. (a) m/s(b) Since m/s design speed, car tries to slide out, so friction acts down the

banked track.

3. The radius is about m

SummaryWhen a car moves at the design speed it has no tendency to slip sideways on the track so there is no friction up or down the track.

Solving problems involving banked tracks:

Use or or depending on the situation.

Use radially (i.e. horizontally).Resolve vertically, .

N.B. Do not resolve or use parallel or perpendicular to the banked plane.

For train tracks, sideways movement is prevented by the lateral thrusting force provided by the rails.

v = 13 900 × 3.38 = 3042

31.436 >

70

ar = rω2 ar = vω ar =v2

rF = ma

R( ↑ )

F = ma

Page of 4 4