A swing ride at a carnival consists of chairs that are swung in a circle by 11.0-m cables attached to a vertical rotating pole, as the drawing shows. Suppose the total mass of a chair and its occupant is 164 kg, the angle θ = 58°. Find the speed of the chair.
To find the speed of the chair, we can use the concept of centripetal acceleration. Centripetal acceleration is given by the equation:
a = (v^2) / r
where:
a = centripetal acceleration
v = velocity of the chair
r = radius of the circular motion
In this case, the radius of the circular motion is given as the length of the cable, which is 11.0 m.
To find the velocity, we need to calculate the centripetal acceleration. The centripetal acceleration can be calculated using the formula:
a = g * tan(θ)
where:
a = centripetal acceleration
g = acceleration due to gravity (approximated as 9.8 m/s^2)
θ = angle of inclination
In this case, θ is given as 58°.
So, let's calculate the centripetal acceleration:
a = (9.8 m/s^2) * tan(58°)
Now, substituting the value of a in the centripetal acceleration equation:
(9.8 m/s^2) * tan(58°) = (v^2) / (11.0 m)
Now, we can solve for v. Let's rearrange the equation:
v^2 = (9.8 m/s^2) * tan(58°) * (11.0 m)
v^2 = (9.8 m/s^2) * (0.0271) * (11.0 m)
v^2 = 3.07574 m^2/s^2
Taking the square root of both sides, we get:
v = √(3.07574 m^2/s^2)
v ≈ 1.75 m/s
Therefore, the speed of the chair is approximately 1.75 m/s.