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?

Since you state that the wheel takes about half an hour to complete one rotation, how can you expect an exact angle in a)

a) in 30 min it rotates 2pi radians
in 1 min it rotates 2pi/30 radians
in 5 min it rotates 5(2pi/30) or pi/3 radians
b) find the circumference (C) of the wheel, that will tell you how far a person travels in one rotation.
Now, what fraction of 30 minutes is 5 minutes ?
c) set up a ration:
C/(2pi) = x/2 , solve for x

d) it moves at 2pi radians every 30 min
= 2pi/30 rad/min
= 2pi/(30*60) rad/sec
= pi/900 radians/sec

e) same thing, use 360º instead of 2pi

To solve these problems, we need to apply some basic formulas and conversions related to circular motion.

a) To determine the exact angle a passenger will travel in 5 minutes, we need to find the fraction of a full revolution that corresponds to 5 minutes. Since the wheel takes half an hour (30 minutes) to complete one revolution, we have:

Fraction of a revolution = (5 min) / (30 min)

b) To find the distance traveled by a passenger in 5 minutes, we need to consider the circumference of the circle formed by the London Eye. The formula for calculating the circumference is:

Circumference = π * diameter

c) To determine how long it would take a passenger to travel 2 radians, we need to find the time it takes for a full revolution and then calculate the fraction of that time required to travel 2 radians. Since a full revolution corresponds to 2π radians:

Fraction of time for 2 radians = (2 radians) / (2π radians) * (30 minutes)

d) The angular velocity of a passenger is the rate at which the angle changes with respect to time. It can be calculated using the formula:

Angular velocity = θ / t,

where θ represents the angle traveled and t represents the time taken.

e) To convert the angular velocity from radians per second to degrees per second, we need to multiply it by the conversion factor of 180/π.

Now, let's solve each part step by step:

a) Determine the exact angle, in radians, that a passenger will travel in 5 min:

Fraction of a revolution = (5 min) / (30 min) = 1/6

To find the angle in radians, we multiply the fraction by 2π (since a full revolution is 2π radians):

Angle in radians = (1/6) * 2π

b) How far does a passenger travel in 5 min.?

Circumference = π * diameter = π * 135 m

Distance traveled by a passenger = Fraction of a revolution * Circumference = (1/6) * (π * 135 m)

c) How long would it take a passenger to travel 2 radians?

Fraction of time for 2 radians = (2 radians) / (2π radians) * (30 minutes)

d) What is the angular velocity of a passenger, in radians per second?

Angular velocity = θ / t = (Angle in radians) / (t)

Since we already calculated the angle in radians in part a), we just need to divide it by the time taken for a full revolution (30 minutes).

e) What is the angular velocity of a passenger, in degrees per second?

To convert the angular velocity from radians per second to degrees per second, we multiply it by the conversion factor of 180/π.

a) To determine the exact angle in radians that a passenger will travel in 5 minutes, we need to know the time it takes the London Eye to complete one revolution.

Given that the wheel takes about half an hour to complete one revolution, we can calculate the time it takes for one revolution (T) in minutes:
T = 30 minutes

Next, we need to find the fraction of the revolution completed in 5 minutes. We can use the formula:

angle in radians = (2π * t) / T

where t is the time elapsed and T is the total time for one revolution.

Substituting the given values, the formula becomes:

angle in radians = (2π * 5) / 30

Using a calculator, the calculation is:
angle in radians = 1.047 rad

Therefore, a passenger will travel approximately 1.047 radians in 5 minutes.

b) To determine how far a passenger travels in 5 minutes, we need the circumference of the wheel.

The circumference of a circle (C) is given by the formula:
C = π * diameter

Substituting the given value,
C = π * 135 m

To find the distance traveled by a passenger in 5 minutes, we can use the formula:

distance = (angle in radians) * radius

Since the angle in radians is already calculated in part a) and the radius is half the diameter, the formula becomes:

distance = (1.047 rad) * (135/2) m

Using a calculator, the calculation is:
distance ≈ 89.49 m

Therefore, a passenger would travel approximately 89.49 meters in 5 minutes.

c) To determine how long it would take a passenger to travel 2 radians, we can use the formula:

time = (angle in radians) * (time for one revolution) / (2π)

Substituting the given values, the formula becomes:

time = 2 rad * 30 min / (2π)

Using a calculator, the calculation is:
time ≈ 28.65 minutes

Therefore, it would take approximately 28.65 minutes for a passenger to travel 2 radians.

d) Angular velocity is the rate at which an object rotates, usually measured in radians per second. To calculate the angular velocity of a passenger, we need the time it takes for one revolution.

Angular velocity (ω) is given by the formula:
ω = (2π) / T

Substituting the given value for T (total time for one revolution), the formula becomes:

ω = (2π) / 30

Using a calculator, the calculation is:
ω ≈ 0.209 rad/s

Therefore, the angular velocity of a passenger is approximately 0.209 radians per second.

e) To convert radians per second to degrees per second, we need to know that there are 180 degrees in one radian.

The conversion can be done using the formula:

angular velocity in degrees per second = (angular velocity in radians per second) * (180/π)

Substituting the given value for the angular velocity in radians per second, the formula becomes:

angular velocity in degrees per second = (0.209 rad/s) * (180/π)

Using a calculator, the calculation is:
angular velocity ≈ 11.97 degrees/second

Therefore, the angular velocity of a passenger is approximately 11.97 degrees per second.