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 137 kg. (a) Determine the tension in the cable attached to the chair. (b) Find the speed of the chair.

a)  T = 1342.6 cos θ, b)  v = 13.93 √(sin θ tan θ)

Explanation:

We can solve this problem using Newton's second law

Let's fix a reference system with a horizontal axis and the other vertical, therefore the only force to decompose is the tension, in these problems the most common is to measure the angle with respect to the vertical. Let's use trigonometry to find the components of the dispute

      cos θ = / T

      T_{y} = T cos tea

     sin θ = Tₓ / T

     Tₓ = T sin θ

let's write Newton's second law

axis and vertical

      T cos θ - W = 0

       T = mg / cos θ

let's calculate

      T = 137  9.8 cos θ

       T = 1342.6 cos θ

unfortunately there is no drawing or indication of the angle

Axis x Horizontal

       T sin θ = m a

acceleration is centripetal

        a = v² / R

        T sin θ = m v² / R

        v² = (g / cos θ) R sin θ

        v = √ (gR tan θ)

let's use trigonometry to find radius of gyration

          sin θ = R / L

          R = L sin θ

         v = √ (g L  sin θ tan θ)

let's calculate

         v = √(9.8 19.8 sin θ tant θ)

         v = 13.93 √(sin θ tan θ)

they do not give the angle for which the calculation cannot be finished

the answer is false

explanation: