You have been hired by a theme-park operator to help with the design of their new ride. This ride consists of a 10 m tall pole, with a 3m diameter disk on top. From the edge of the disk hang four seats, on 8m long cables. The plan is that customers will sit in these seats, and then the disk will start to rotate. As it rotates faster, your boss tells you that centrifugal force will push the seats out, until the customers are doing circles around the pole. She wants the cables to be at an angle of 60° to the vertical when the ride is going at full speed, as she thinks that will be suitably thrilling. Knowing that you have a physics background, she asks you to calculate how many revolutions per minute the disk will have to spin at to make this happen, and how strong the cables will have to be. The seats weigh 30 kg and passengers can weigh up to 170 kg.

To tackle this problem, we need to analyze the forces acting on the seats when the ride is in motion. The goal is to determine the necessary rotational speed of the disk to achieve a 60-degree angle for the cables and to calculate the tension in the cables under these conditions. Let's break this down step by step.

Understanding the Geometry

First, let's visualize the setup. The disk is 10 meters tall with a 3-meter diameter, which means the radius of the disk is 1.5 meters. The cables are 8 meters long and will form a triangle with the vertical pole and the horizontal distance from the pole to the seat when the ride is in motion.

Calculating the Horizontal Distance

When the cables are at a 60-degree angle to the vertical, we can use trigonometry to find the horizontal distance (r) from the pole to the seat:

• The vertical component of the cable length (h) can be calculated as:
• h = L * cos(θ) = 8 m * cos(60°) = 8 m * 0.5 = 4 m
• The horizontal component (r) is then:
• r = L * sin(θ) = 8 m * sin(60°) = 8 m * (√3/2) ≈ 6.93 m

This means that when the ride is at full speed, the seats will be approximately 6.93 meters away from the pole horizontally.

Applying Centripetal Force

Next, we need to analyze the forces acting on the seats. The key force here is the centripetal force required to keep the seats moving in a circular path. The centripetal force (F_c) can be expressed as:

F_c = m * ω² * r

Where:

• m = total mass (seat + passenger) = 30 kg + 170 kg = 200 kg
• ω = angular velocity in radians per second
• r = horizontal distance from the pole = 6.93 m

Finding Angular Velocity

We also need to consider the vertical forces acting on the seat. The tension in the cable (T) must balance both the gravitational force and provide the centripetal force. The vertical component of the tension can be expressed as:

T * cos(θ) = mg

And the horizontal component provides the centripetal force:

T * sin(θ) = m * ω² * r

From the first equation, we can express T:

T = mg / cos(θ)

Substituting this into the second equation gives:

(mg / cos(θ)) * sin(θ) = m * ω² * r

Now, we can simplify this to find ω:

g * tan(θ) = ω² * r

Substituting the values:

• g ≈ 9.81 m/s²
• tan(60°) = √3 ≈ 1.732
• r ≈ 6.93 m

Now we can solve for ω:

ω² = (g * tan(θ)) / r

ω² = (9.81 m/s² * 1.732) / 6.93 m ≈ 2.51

ω ≈ √2.51 ≈ 1.58 rad/s

Converting to Revolutions Per Minute

To convert angular velocity from radians per second to revolutions per minute (RPM), we use the conversion factor:

RPM = (ω * 60) / (2π)

Substituting the value of ω:

RPM = (1.58 * 60) / (2 * 3.14) ≈ 15.1 RPM

Calculating Cable Tension

Now, let's calculate the tension in the cables using the expression we derived earlier:

T = mg / cos(θ)

Substituting the values:

T = (200 kg * 9.81 m/s²) / cos(60°) = (1962 N) / 0.5 = 3924 N

Summary of Findings

To achieve the desired thrilling experience for the ride:

• The disk must spin at approximately 15.1 RPM.
• The tension in the cables will need to be around 3924 N.

This analysis ensures that the ride operates safely while providing the excitement that your boss is looking for!