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

vapour or gas assuming atmospheriv pressure

air exerts forces on falling objects near earth’s surface.

explanation:

the term 'free falls' refer to object that are falling towards the earth, in a situation in which gravity is the only force acting on the object.

for an object in free fall, the acceleration is constant (g, the gravitational acceleration), and so the velocity of the object constantly increases.

for objects near earth's surface, there is always another force acting on the object: the resistance due to the air. this force is opposed to the direction of fall of the object, and it is proportional to the velocity of the object, so as the object falls down, this force increases up to a point when it balances the force of gravity, and the object continues its motion at constant velocity (called terminal velocity). since in this case gravity is not the only force acting on the object, we are not in a situation of free fall.