A "swing" ride at a carnival consists of chairs that are swung in a circle by 16.6 m cables attached to a vertical rotating pole, as the drawing shows. Suppose the total mass of a chair and its occupant is 239 kg. (a) Determine the tension in the cable attached to the chair. (b) Find the speed of the chair.
To determine the tension in the cable attached to the chair, we can start by analyzing the forces acting on the chair.
(a) Tension in the cable:
When the chair is at the lowest point of its swing, two forces are acting on it: the tension in the cable and the force due to gravity. At this point, the net force is equal to the centripetal force.
The centripetal force is given by the equation: Fc = (m*v^2) / R
Where:
- Fc is the centripetal force
- m is the mass of the chair and occupant
- v is the speed of the chair
- R is the radius of the circular path
At the lowest point, the radius of the circular path is the length of the cable, which is 16.6 m. The mass of the chair and occupant is given as 239 kg.
Now, we need to determine the speed of the chair.
(b) Speed of the chair:
The speed of the chair is related to the period of rotation (T) by the equation:
v = (2πR) / T
Where:
- v is the speed
- R is the radius
- T is the period of rotation
Let's calculate the period of rotation using the following formula:
T = 2π * √(R / g)
Where:
- T is the period of rotation
- R is the radius
- g is the acceleration due to gravity (approximately 9.8 m/s^2)
After calculating the period of rotation, we can substitute it back into the speed equation to find the speed of the chair.
I will now calculate the values for you.