a carnival ride is designed with seats that carry two people weighing 910.0 N each. the seats itself weights 630 N.each seat is connected to central spindle by one chain and rotates on a radios of 6.07 meters. the ride is designed to rotate fast enough that the seats swing out horizontally at first and then the central spindle rotates so that the seats spin vertically. calculate the minimum velocity required to just keep the seats with the two people in them form falling out of rotation when they are spinning vertically?

Question

a carnival ride is designed with seats that carry two people weighing 910.0 N each. the seats itself weights 630 N.each seat is connected to central spindle by one chain and rotates on a radios of 6.07 meters. the ride is designed to rotate fast enough that the seats swing out horizontally at first and then the central spindle rotates so that the seats spin vertically. calculate the minimum velocity required to just keep the seats with the two people in them form falling out of rotation when they are spinning vertically?

please show work

Answer 1

the centripetal force at the top of the circle equals gravity

v^2 / r = g

v = √(g r)

Answer 2

To find the minimum velocity required to keep the seats with the two people from falling out of rotation when they are spinning vertically, we can analyze the forces acting on the seats.

Let's start by considering the gravitational force acting on each person when they are spinning vertically. The weight of each person is 910.0 N, and since there are two people, the total weight of the people in the seats is 2 * 910.0 N = 1820.0 N.

Now, when the seats are spinning vertically, the centrifugal force should provide the necessary inward force to balance the gravitational force. The centrifugal force is given by the equation:

Fc = m * v^2 / r

Where:
Fc is the centrifugal force
m is the mass of the people in the seats (1820.0 N / 9.8 m/s^2)
v is the velocity of the seats
r is the radius of rotation (6.07 m)

To prevent the people from falling out of rotation, the centrifugal force should be equal to the gravitational force. So we can set up the following equation:

Fc = 1820.0 N
m * v^2 / r = 1820.0 N
(1820.0 N) * r / m = v^2
v = sqrt((1820.0 N) * r / m)

Let's substitute the known values into the equation and solve for the minimum velocity:

v = sqrt((1820.0 N) * (6.07 m) / (2 * 9.8 m/s^2))

v = sqrt(1119.54 N*m / 19.6 kg)

v = sqrt(57.12 m^2/s^2)

v ≈ 7.56 m/s

Therefore, the minimum velocity required to just keep the seats with the two people from falling out of rotation when they are spinning vertically is approximately 7.56 m/s.