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. (è = 60.0°) Suppose the total mass of a chair and its occupant is 222 kg
To solve this problem, we can use the principles of circular motion and centripetal force.
The centripetal force acting on the chair is provided by the tension in the cable. This force can be found using the equation:
F_c = m * a_c
Where F_c is the centripetal force, m is the mass of the chair and occupant, and a_c is the centripetal acceleration.
The centripetal acceleration can be calculated using the equation:
a_c = (v^2) / r
Where v is the linear velocity and r is the radius of the circular path.
Now, let's calculate the centripetal force.
Given:
Mass of chair and occupant (m) = 222 kg
Radius of the circle (r) = 12.0 m
Angle (θ) = 60.0°
First, we need to find the linear velocity (v) using the angle (θ). The linear velocity can be calculated using the formula:
v = 2πr * (θ / 360°)
Plugging in the values:
v = 2π(12.0) * (60.0 / 360)
v ≈ 12.57 m/s
Now, substitute the values of mass (m) and linear velocity (v) into the equation for centripetal force:
F_c = m * a_c
F_c = 222 kg * (v^2) / r
F_c = 222 kg * (12.57 m/s)^2 / 12.0 m
F_c ≈ 2942 N
Therefore, the tension in the cable, or the centripetal force acting on the chair, is approximately 2942 Newtons.