While riding a Ferris wheel, the rider determines that the Ferris wheel makes 1.5 revolutions per minute.
a) knowing that the diameter of the Ferris wheel is 100 ft, determine the angular speed (in rad/s) of the Ferris wheel.
b) Determine the linear speed (in ft/s) of the rider on the Ferris wheel. (Hint: Use the equation for arc length L=θr, where L is the arc length, θ is the angle of the arc, and r is the radius. Realize that the equation for circumference of a circle is the equation for arc length with an arc angle of 2π).
c) If the ride is 5 minutes long, how many linear feet does the rider travel during the ride?
d) There are a total of 12 cars on the Ferris wheel. When exiting the ride, the rider of the interest has to wait for 2 other cars to empty until they exit the ride. How many extra linear feet are traveled during exit time? (Hint: Use the equation for arc length).
You need to make an effort.
(a) 1.5 revolutions is 3 pi radians, and that takes 60 seconds. Compute 3 pi/60 as for the angular speed. Call it w.
(b) w*D/2 is the rider's speed.
(c) Multiply the speed from (b) by 300 seconds
(d) 1/6 radian * (D/2)
18
a) To determine the angular speed of the Ferris wheel in radians per second, we need to convert the number of revolutions per minute to radians per second.
1 revolution = 2π radians
1 minute = 60 seconds
So, the angular speed (ω) can be calculated as:
ω = (1.5 revolutions/minute) * (2π radians/revolution) * (1 minute/60 seconds)
= (1.5 * 2π) / 60
≈ 0.157 radians/second
Therefore, the angular speed of the Ferris wheel is approximately 0.157 radians per second.
b) To determine the linear speed of the rider on the Ferris wheel, we can use the equation for arc length:
L = θr
Since the angle of the arc for one revolution of the Ferris wheel is 2π, we can calculate the linear speed (v) using the formula:
v = θr * (1/T)
where T is the time taken for one revolution. In this case, T = 60 seconds/1.5 revolutions = 40 seconds.
v = (2π) * (100/2) * (1/40)
= (π * 100) / 40
= (π * 5) / 2
≈ 7.85 ft/s
Therefore, the linear speed of the rider on the Ferris wheel is approximately 7.85 ft/s.
c) To calculate the linear distance traveled by the rider during a 5-minute ride, we can use the equation:
distance = speed * time
The speed of the rider is calculated as approximately 7.85 ft/s, and the time is given as 5 minutes.
distance = 7.85 ft/s * 5 minutes * (60 seconds/1 minute)
= 7.85 ft/s * 5 minutes * 60 seconds/minute
= 7.85 ft/s * 300 seconds
≈ 2355 feet
Therefore, the rider would travel approximately 2355 feet during the 5-minute ride.
d) During the exit time, the rider has to wait for 2 other cars to empty. Since there are 12 cars on the Ferris wheel, this means the rider spends 2/12 of the time exiting.
The total time for the ride is 5 minutes. Therefore, the exit time can be calculated as:
Exit time = (2/12) * 5 minutes
= (2/12) * 5 * 60 seconds/1 minute
= (2/12) * 5 * 60 seconds
= 600 seconds/12
= 50 seconds
To calculate the extra linear distance traveled during the exit time, we use the linear speed of the rider (approximately 7.85 ft/s) and the time spent during the exit.
Extra distance = speed * exit time
= 7.85 ft/s * 50 seconds
= 392.5 ft
Therefore, the rider would travel an extra distance of approximately 392.5 feet during the exit time.