A swing ride at a carnival consists of chairs that are swung in a circle by 13.5-m cables attached to a vertical rotating pole. Suppose the total mass of a chair and its occupant is 170 kg, the angle θ = 53° with the vertical pole. Find the speed of the chair.
To find the speed of the chair, we need to consider the forces acting on the system. The only horizontal force acting on the chair is the tension in the cable, which causes it to move in a circular path.
First, let's find the tension in the cable. The gravitational force acting on the chair can be broken down into two components: one parallel to the cable (mg sin θ) and one perpendicular to the cable (mg cos θ).
The centripetal force required to keep the chair moving in a circle is equal to the tension in the cable. This can be calculated using the equation:
Tension = Mass × Centripetal Acceleration
The centripetal acceleration is given by:
Centripetal Acceleration = (Velocity^2) / Radius
In this case, the radius is the length of the cable, which is 13.5 m.
The centripetal acceleration can also be written as the angular acceleration (α) multiplied by the distance from the rotation axis (r), where r is equal to the length of the cable multiplied by sin θ.
Centripetal Acceleration = α × r
Now, we can equate the two expressions for centripetal acceleration:
(velocity^2) / 13.5 m = α × 13.5 m × sin θ
We can rearrange this equation to solve for the angular acceleration:
α = (velocity^2) / (13.5 m × sin θ)
Since the angular acceleration is related to the tension through the equation:
Tension = Mass × Angular Acceleration
We can substitute the expression for angular acceleration into this equation:
Tension = Mass × [(velocity^2) / (13.5 m × sin θ)]
Now we can solve for the tension:
Tension = 170 kg × [(velocity^2) / (13.5 m × sin 53°)]
Finally, we equate the tension to the horizontal component of the gravitational force (mg sin θ) to solve for the velocity:
mg sin θ = 170 kg × [(velocity^2) / (13.5 m × sin 53°)]
We can rearrange and solve for the velocity:
velocity = sqrt[(g × 170 kg × sin θ) / (13.5 m × sin 53°)]
where g is the acceleration due to gravity (9.8 m/s^2).
Plugging in the given values:
velocity = sqrt[(9.8 m/s^2 × 170 kg × sin 53°) / (13.5 m × sin 53°)]
Calculating this expression will give us the speed of the chair.