Unlike a roller coaster, the seats in a Ferris wheel swivel so that the rider is always seated upright. An 80-ft-diameter Ferris wheel rotates once every 24 s.
A. What is the apparent weight of a 60 kg passenger at the lowest point of the circle?
b. What is the apparent weight of a 60 kg passenger at the highest point of the circle?
A. m g + m v^2/R
B. m g - m v^2/R
where
m = 60 kg
g = 9.81 m/s^2 or so
R = 40 feet = 12.2 meters
v = 2 pi R / 24 = 3.19 m/s
A. Well, at the lowest point of the circle, the rider would probably feel perfectly grounded, just like a potato firmly stuck in the ground. The apparent weight would be the same as their actual weight, which is 60 kg. The only difference is that they're not being used to make french fries.
B. Now, at the highest point, things get a little more giggly. You see, gravity likes to play tricks when you're up in the air. The passenger would feel a little lighter, as if their weight is being momentarily shaved off by a cosmic hairdresser. The apparent weight would be slightly less than 60 kg, but don't worry, they won't float away like a balloon. They're still firmly rooted in their seat. Well, as firm as a seat on a spinning wheel can be!
To find the apparent weight of a passenger at the lowest and highest points of the Ferris wheel, we can use the concept of centripetal force.
A. Apparent Weight at the Lowest Point:
At the lowest point, the passenger experiences both the gravitational force and the centripetal force. The apparent weight is the net force acting on the passenger.
1. Calculate the gravitational force acting on the passenger at the lowest point:
Gravitational Force = mass × acceleration due to gravity
Gravitational Force = 60 kg × 9.8 m/s^2 (acceleration due to gravity)
Gravitational Force = 588 N
2. Determine the centripetal force acting on the passenger at the lowest point:
Centripetal Force = mass × centripetal acceleration
Centripetal acceleration = (angular velocity)^2 × radius
Angular velocity = (2π radians) / time period (T)
Time period (T) = 24 s
Radius = (80 ft / 2) × 0.3048 m/ft (converting to meters)
Centripetal acceleration = ((2π / 24 s)^2) × (80 ft / 2 × 0.3048 m/ft)
3. Calculate the centripetal force:
Centripetal Force = 60 kg × centripetal acceleration
4. Calculate the apparent weight:
Apparent Weight = Gravitational Force - Centripetal Force
B. Apparent Weight at the Highest Point:
At the highest point, the passenger experiences both the gravitational force and the centripetal force. The apparent weight is the net force acting on the passenger.
Repeat steps 1-4 using the radius and centripetal acceleration at the highest point. The radius will remain the same, but the centripetal acceleration will be different as the angular velocity changes.
Note: The radius is constant as the Ferris wheel rotates, but the centripetal acceleration varies depending on the position on the wheel (lowest or highest point).
To find the apparent weight of a passenger at different points on a rotating Ferris wheel, we need to consider the forces acting on the passenger.
A. At the lowest point of the circle, the passenger experiences two forces: the gravitational force (weight) and the centripetal force (resulting from circular motion).
To calculate the apparent weight, we can subtract the centripetal force from the gravitational force. The centripetal force can be determined using the formula:
Fc = m * ac,
where Fc is the centripetal force, m is the mass of the passenger, and ac is the centripetal acceleration.
The centripetal acceleration can be found using the formula:
ac = v^2 / r,
where v is the linear velocity and r is the radius (half the diameter) of the Ferris wheel.
First, let's find the linear velocity:
v = 2 * π * r / T,
where T is the period (time for one complete revolution) of the Ferris wheel.
Given:
- Diameter of the Ferris wheel = 80 ft
- Mass of the passenger = 60 kg
- Period of rotation = 24 s
1. Calculate the radius (r):
r = diameter / 2 = 80 ft / 2 = 40 ft.
2. Calculate the linear velocity (v):
v = 2 * π * r / T
= 2 * 3.14 * 40 ft / 24 s
≈ 8.377 ft/s.
3. Calculate the centripetal acceleration (ac):
ac = v^2 / r
= (8.377 ft/s)^2 / 40 ft
≈ 1.74 ft/s^2.
4. Calculate the centripetal force (Fc):
Fc = m * ac
= 60 kg * 1.74 ft/s^2
≈ 104.4 N.
5. Calculate the gravitational force (weight):
The weight can be calculated using the formula:
Fg = m * g,
where Fg is the gravitational force, m is the mass of the passenger, and g is the acceleration due to gravity (approximately 9.8 m/s^2).
Fg = m * g
= 60 kg * 9.8 m/s^2
≈ 588 N.
6. Finally, calculate the apparent weight:
Apparent weight = Fg - Fc
≈ 588 N - 104.4 N
≈ 483.6 N.
Therefore, the apparent weight of a 60 kg passenger at the lowest point of the circle is approximately 483.6 N.
B. To find the apparent weight of the passenger at the highest point of the circle, we follow a similar process.
At the highest point, the passenger still experiences two forces: the gravitational force (weight) and the centripetal force. However, the direction of these forces is different.
When the Ferris wheel is at the highest point, the passenger is moving in a circular motion, but they are experiencing an upward acceleration, which reduces their apparent weight.
The steps to find the apparent weight at the highest point are the same as in part A, but the centripetal acceleration (ac) will be negative since it opposes the gravitational force.
1. The radius (r) remains the same at 40 ft.
2. Calculate the linear velocity (v) using the same formula as in part A.
3. Calculate the centripetal acceleration (ac) using the formula ac = v^2 / r. However, in this case, ac will be negative.
4. Calculate the centripetal force (Fc) using the same formula as in part A.
5. Calculate the gravitational force (weight) using the same formula as in part A.
6. Finally, calculate the apparent weight by subtracting the centripetal force (negative value) from the gravitational force.
Follow the calculations similar to part A to determine the apparent weight of the passenger at the highest point of the circle.