A "swing" ride at a carnival consists of chairs that are swung in a circle by 19.8 m cables attached to a vertical rotating pole, as the drawing shows. Suppose the total mass of a chair and its occupant is 144 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 use Newton's second law of motion, which states that the net force acting on an object is equal to the mass of the object multiplied by its acceleration.
(a) Let's start by considering the forces acting on the chair. There are two forces: the tension in the cable pulling the chair outward and the force of gravity pulling the chair downward. Since the chair is moving in a circle, there is also a centripetal force acting towards the center of the circle.
The force of gravity can be calculated using the equation:
Force of gravity = mass * gravitational acceleration
where the gravitational acceleration is approximately 9.8 m/s^2.
Force of gravity = 144 kg * 9.8 m/s^2 = 1411.2 N
Since the chair is moving in a circle, the centripetal force is provided by the tension in the cable. Therefore, the tension in the cable is equal to the centripetal force.
Tension = centripetal force
Next, we need to calculate the centripetal force using the following equation:
Centripetal force = mass * (velocity)^2 / radius
In this case, the radius is given by the length of the cable, which is 19.8 m.
Centripetal force = 144 kg * (velocity)^2 / 19.8 m
Since the tension in the cable is equal to the centripetal force, we can equate the two equations:
Tension = 144 kg * (velocity)^2 / 19.8 m
To solve for the tension, we need to know the velocity of the chair.
(b) To find the speed of the chair, we can consider the forces acting on the chair in the vertical direction.
The force of gravity acting downward and the tension in the cable acting upward must add up to the total mass of the chair and its occupant multiplied by the acceleration due to gravity.
Force of gravity - Tension = mass * gravitational acceleration
Using the values we have:
1411.2 N - Tension = 144 kg * 9.8 m/s^2
Now, we can substitute the value of tension from the equation we derived in part (a) and solve for velocity.
144 kg * (velocity)^2 / 19.8 m = 1411.2 N - Tension
Once we have the tension, we can substitute it back into this equation and solve for velocity.