In an amusement park ride, cars are suspended from L = 4.22 m cables attached to rotating arms at a distance of D = 6.25 m from the axis of rotation. The cables swing out at an angle of θ = 49° when the ride is operating. What is the angular speed of rotation?

Do they want it in radians per second or revolutions per minute?

V^2/(R*g) = tan 49 = 1.150

R = L sin49 = 3.04 m

V^2 = 34.3 m^2/s^2
V = 5.86 m/s

The angular speed in radians/sec is V/R

To find the angular speed of rotation, we can use the concept of simple harmonic motion and the relationship between angular speed and the angle at which the cables swing out.

First, let's understand the terms involved in the problem:
- L: Length of the cables = 4.22 m
- D: Distance from the axis of rotation to the attachment point of the cables = 6.25 m
- θ: Angle at which the cables swing out = 49°

The motion of the cars on the ride is similar to a pendulum, where the cars are swinging back and forth. This motion is called simple harmonic motion. In simple harmonic motion, the angle at which the pendulum swings out is related to the period and frequency of the motion.

However, in this problem, we are interested in finding the angular speed of rotation, which is the rate at which the ride is rotating, and it is not directly related to the period or frequency.

To find the angular speed, we need to consider the relationship between the angle at which the cables swing out and the centripetal force acting on the cars.

In this case, the gravitational force acting on the cars provides the necessary centripetal force, which can be expressed as:

mg = T * sin(θ)

Where:
- m: Mass of the cars (not given)
- g: Acceleration due to gravity (9.8 m/s²)
- T: Tension in the cables

The tension in the cables can be further decomposed into horizontal and vertical components:

T * cos(θ) = mg
T * sin(θ) = mv² / D

Since we are interested in finding the angular speed of rotation and not the velocity, we can use the relationship between linear velocity and angular velocity:

v = ω * R

Where:
- v: Linear velocity
- ω: Angular velocity
- R: Distance from the axis of rotation to the cars

Substituting the above equations, we can find the angular speed:

mg / T * cos(θ) = mv² / D
mg * D * cos(θ) = T * m * (ω * R)²
g * D * cos(θ) = ω² * R

Rearranging the equation, we get:

ω = √(g * D * cos(θ) / R)

Now, let's substitute the values given in the problem:
- D = 6.25 m
- θ = 49° (convert to radians by multiplying by π/180)
- R = D (since the cables are attached at a distance of D from the axis of rotation)
- g = 9.8 m/s²

Plugging in the values, we can find the angular speed of rotation:

ω = √(9.8 m/s² * 6.25 m * cos(49°) / 6.25 m)

Calculating the above expression should give us the final value of the angular speed of rotation.