First, let's find the length of the vertical component of the cable. The length of the vertical component can be found by:
Length = 12 m * cos(56°)
Next, we'll find the horizontal component of the cable's tension. At the top of the swing, the centripetal force acting on the system is equal to the horizontal component of tension. So,
Fc = T * sin(θ)
mv^2 / r = T * sin(θ)
where m = 190 kg, r = 12 m, and θ = 56°
(a) To find the tension in the cable, first consider the vertical component of the force. The tension in the cable is equal to the gravitational force acting on the chair and its occupant:
T * cos(θ) = mg
T = mg / cos(θ)
T = 190 kg * 9.81 m/s^2 / cos(56°)
T = 1680 N
(b) To find the speed of the chair, we will use the centripetal force equation we found earlier:
mv^2 / r = T * sin(θ)
Solve for v,
v^2 = T * sin(θ) * r / m
v = sqrt((1680 N * sin(56°) * 12 m) / 190 kg)
v = 9.21 m/s
The tension in the cable is 1680 N, and the speed of the chair is 9.21 m/s.
A "swing" ride at a carnival consists of chairs that are swung in a circle by 12.0 m cables attached to a vertical rotating pole, as the drawing shows. (θ = 56.0°) Suppose the total mass of a chair and its occupant is 190 kg.
(a) Determine the tension in the cable attached to the chair.
(b) Find the speed of the chair.
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