The London Eye is a large ferris wheel. Each sealed and air-conditioned passenger capsule holds about 25 passengers. The diameter of the wheel is 135 m, and the wheel takes about half an hour to complete one revolution.
a) Determine the exact angle, in radians, that a passenger will travel in 5 min.
b) How far does a passenger travel in 5 min.?
c) How long would it take a passenger to travel 2 radians?
d) What is the angular velocity of a passenger, in radians per second?
e) What is the angular velocity of a passenger, in degrees per second?
b) 22.5 m
To solve these questions, we need to analyze the given information and use the formulas for angular displacement, arc length, angular velocity, and conversion between radians and degrees.
a) To determine the exact angle in radians that a passenger will travel in 5 minutes, we need to calculate the fraction of a full revolution they complete in that time. We know that the wheel takes half an hour to complete one revolution, which means it takes 30 minutes to complete 2π radians (a full revolution in radians). Therefore, in 5 minutes, a passenger will travel:
(5 min / 30 min) * 2π radians = (1/6) * 2π radians = π/3 radians
b) To find how far a passenger travels in 5 minutes, we need to calculate the length of the arc they cover. The circumference of the wheel can be determined using the formula:
Circumference = π * diameter = π * 135 m
Now, to calculate the length of the arc traveled by a passenger in 5 minutes, we can use:
Arc length = (angle in radians) * (radius)
Since the passenger travels an angle of π/3 radians and the radius is half the diameter (67.5 m), we get:
Arc length = (π/3) * 67.5 m
c) To determine how long it would take a passenger to travel 2 radians, we can set up a proportion. If π/3 radians are covered in 5 minutes (from part a), then the time needed to cover 2 radians would be:
(5 min) * (2 radians / (π/3) radians) = (5 min) * (2/ (π/3)) min = 30/π min
d) The angular velocity of a passenger is defined as the rate at which the angle changes per unit of time. In this case, the time unit is in minutes. So to find the angular velocity in radians per minute, we can divide the angle in radians by the time in minutes:
Angular velocity = (angle in radians) / (time in minutes)
= π/3 radians / 5 min
e) To convert the angular velocity from radians per minute to degrees per second, we need to apply the following conversions:
1 radian = 180/π degrees
1 minute = 60 seconds
Therefore, the angular velocity in degrees per second would be:
Angular velocity = (π/3 radians / 5 min) * (180/π degrees / 1 radian) * (1 min / 60 seconds)
= 36 degrees / (3 * 60) seconds
= 0.2 degrees per second
In summary:
a) The exact angle in radians that a passenger will travel in 5 minutes is π/3 radians.
b) A passenger travels an arc length of (π/3) * 67.5 meters in 5 minutes.
c) It would take a passenger 30/π minutes to travel 2 radians.
d) The angular velocity of a passenger is π/15 radians per minute.
e) The angular velocity of a passenger is 0.2 degrees per second.