Alright, let's break down these questions one by one. I'll explain the calculations for each part.
a) What is the speed of the gondola when the Ferris wheel is running at full speed?
First, we need to find the distance the gondola travels in one revolution. This is equal to the circumference of the circle, which is given by the formula:
`Circumference = 2πr`
Where:
- r = radius of the Ferris wheel = 50.0 m
So, Circumference = 2π * 50 = 100π meters.
The gondola completes one revolution every 60 seconds, so its speed (v) is given by the distance traveled divided by the time taken.
Speed = Distance / Time
v = Circumference / time
v = 100π m / 60 s = (5/3)π ≈ 5.236 m/s.
b) What is the centripetal acceleration of the gondola when the Ferris wheel is running at full speed?
Centripetal acceleration (a) is given by the formula:
`a = v² / r`
Where:
- v = speed of the gondola = (5/3)π m/s
- r = radius of the Ferris wheel = 50.0 m
So, a = [((5/3)π)²] / 50 ≈ 0.549 m/s².
c) If a gondola has a mass of 120 kg, what is the centripetal force acting on the gondola?
Centripetal force (F) is given by the formula:
`F = ma`
Where:
- m = mass of the gondola = 120 kg
- a = centripetal acceleration = 0.549 m/s²
So, F = 120 kg * 0.549 m/s² = 65.88 N.
Remember, these calculations are simplified and ignore factors like gravitational effects, friction, and the fact that the gondola's speed may not be constant throughout its revolution.
Here is the simplest way to solve this:
1. Calculate the circumference of the Ferris wheel using the radius.
2. Determine the speed of the gondola by dividing the circumference by the time for one revolution.
3. Calculate the centripetal acceleration using the speed and radius.
4. Determine the centripetal force by multiplying the mass of the gondola by the centripetal acceleration.