Lab 9: Hurricane Speed and Angle

The Hurricane is a common amusement park ride created by the Allan Herschell Company in the 1940s. It consists of a central support pillar with levers extending out to rides, which will spin and, using the behaviors of centripetal acceleration, cause the rides to left up. The angle that the ride lifts, as it turns out, depends on the speed by which it turns. The real version also uses pneumatic cylinders to oscillate the height, but we are modelling this without the hydraulics.

Since we don't have enough rulers to measure the Hurricane itself, we use our own rotating apparatus: a tripod tied to a motor and a stick, and a string tied to a mass on the other end. It sounds dangerous, but it's quite impressive in person.

The motor can be controlled to spin at a certain volt, which implies a constant speed if unimpeded. As it spins, the mass, a rubber stopper connected by a string at one end of the horizontal bar, rises at an angle. The first step to setup any experiment is to measure all the relevant angles and hypothesize on the relationship, so we could have an idea on what to expect.

Now we draw a model with the relevant dimensions:

Using the model, we draw a free body diagram:

And we're set to calculate for the relationship between angular velocity ω and the angle θ. Because there is centripetal acceleration on the horizontal access, we say that (where r is radius):

F_x = m a = m r ω²

Resolving the x and y components of the forces, we have:

x: T sinθ = m r ω²

y: T cosθ = m g

Dividing them, we get:

tanθ = r ω² / g

A word about the radius r: Since the width of the top horizontal bar is the radius of the system with the mass at rest, and the mass only swings outward, we could say that the radius is the radius at rest r₀ plus the length that forms a triangle between the string at rest and the string that is swung out at a certain angle θ. Thus...

r = r₀ + L sinθ
tanθ = ω² (r₀ + L sinθ) / g
g tanθ = ω² (r₀ + L sinθ)
g tanθ / (r₀ + L sinθ) = ω²
ω = sqrt[g tanθ / (r₀ + L sinθ)]

With the theoretical relationship armed in hand, we are ready to start experimenting to see if it confirms our analysis. To find the speed of the mass, we use a basic timer to time 10 revolutions, to improve accuracy.

As the motor spins, we expect that the string will form an angle outward, but this also raises the height of the mass. If we know the difference in height of the mass compared to the position at rest, then it is possible to solve the horizontal component. To measure the height, we place a piece of folded paper clamped to a metal bar under the apparatus.

We slowly raise the height until the mass just hits it as it swings around.

Studiously measuring the height:

And we do this 8 times...

Time (s)	Rev	Height (cm)	Velocity (ω)	Angle (θ)
39.68	10	50.0	1.5835	9.0125
32.88	10	62.8	1.9109	24.6816
29.53	10	76.2	2.1277	34.3175
26.98	10	95.5	2.3288	45.0257
25.14	10	106.6	2.4993	50.3369
22.05	10	114.1	2.8495	53.7027
19.40	10	142.5	3.2388	65.3757
18.50	10	148.7	3.3963	67.7657

 

We get the velocity ω by dividing 2π radians per revolution by T, the period, or seconds per revolution. The angle θ is found by subtracting the measured height from the overall height h of the apparatus, then dividing it by the length L of the string, and taking an inverse cosine of the result.

A similar set of data derived in class, from the experimental angle θ, then using the analytic relationship calculated earlier to reach the velocity ω:

We graph the analytically derived velocity ω_theory against the experimental velocity ω_expt to see if there's a correlation:

The regression line coefficient is 0.96483. The slope is 1.0135, meaning that we are under 2% error. That isn't too bad considering all the measurements we had to take, each of which could introduce error. What's interesting is that we find that velocity depends wholly on radius, angle, and gravity, and not on mass. But perhaps that isn't too surprising given that the pendulum period equation also excludes mass, presumably for the same reason.