Trigonometry problem

Question

London eye.

The wheels 32 passenger capsules are attatched to the external circumfrence of the wheel and are rotated by electric motors. Each excursion lasts the lenght of time it takes to complete one revolution.
The observation wheel has diameter 120m and the centre oof each passenger capsule is 4m from the external circumfrence of the wheel. The centre of a passenger capsule is 6m above ground level at the point that passengers enter the capsule.

The wheel does not usually stop to allow passengers on, as the rate of movement is slow enough to allow visitors to walk on and off moving capsules. The wheel can complete 2.5 revolutions per hour if it does not stop.

 

(a) What is the maximum height that the centre of a passenger capsule reaches above ground level and how long does it take a capsule to reach this height if its wheel hasnt stopped?

 

(b) Find the angle of rotation of a passenger capsule from the point of entry to the capsule, to after 5 minutes

 

(c)

(i) Write down the terms of t, the angle of rotation of a passenger capsule at any given time, where t is time in minutes after a person enters the capsule, if it hasnt stopped

 

(ii) Hence, by drawing a sketch or otherwise, find an expression for height h in metres of the centre of the capsule above ground level, in terms of t

 

(iii) The wheel does not normally stop to allow passengers to embark and disembark and, therefore, the speed of the wheel is constant. Suggest a reason why this is desirable

Answer 1

a

Max height = distance between ground and wheel + wheel diameter + distance between wheel and capsule

Max height = (6+4) + 120 + 4 = 134

Time taken to make 0.5 revolutions = Distance / Speed

Time taken to make 0.5 revolutions = 0.5 revolutions / 2.5 revolutions per hour = 12 minutes

 

b

Number of revolutions after 5 minutes = Speed * Time

Number of revolutions after 5 minutes = 2.5 revolutions per hour * (1/12) hour = (5/24) revolutions

Angle of rotation after 5 minutes = Number of revolutions * 360 degrees

Angle of rotation after 5 minutes = (5/24) revolutions * 360 degrees = 75 degrees

 

ci

Number of revolutions after t minutes = 2.5 revolutions per hour * (t/60) hour = (t/24) revolutions

Angle of rotation after t minutes = (t/24) revolutions * 360 degrees) = 15t degrees

 

cii

Note that t takes values between 0 and 24 minutes.

When t = 0, h = 6

When t = 12, h = 134 (see part a)

When t = 24, h = 6

Between t = 0 and t = 12: h = (32/3)t + 6

Between t = 12 and t = 24: h = (-32/3)t + 262

Combining both equations: h = 134 - |(-32/3)t + 128|

 

ciii

A constant speed is more energy efficient. If the wheel has to decelerate for passengers, there is wasted energy from applying brakes. Similarly, if the wheel has to accelerate after slowing down, there is wasted energy from motor friction.