# Find Answer to Wet Umbrella Problem: .97m

• xTheLuckySe7en
In summary, the problem is poorly phrased and the relative motion is clearly the 84 cm (83 cm without rounding errors). The change in distance is 47 cm (the final distance of 97 cm minus the initial distance of 50 cm). The final horizontal distance to the handle is 97 cm.
xTheLuckySe7en
Thread moved from the technical forums, so no HH Template is shown.
The problem: A wet umbrella is held upright and is twirled about the handle at a uniform rate of 21 revolution in 44 s. If the rim of the umbrella is a circle 1 m in diameter, and the height of the rim above the floor is 1.5 m, find how far the drops of water spun off the rim travel horizontally relative to the umbrella handle before they hit the floor?

My attempt: My general set of goals for this particular problem were to first find the tangential velocity using the given values for revolutions per second, and then proceed with the tangential velocity as if it were a two-dimensional projectile problem.
ω = Δθ/Δt, where 1 revolution is equal to 2π (I got 3.0 rad/s)
Vt=rω, where r=.5m
Vt=1.5m/s
Then, I proceeded as aforementioned (two-dimensional projectile problem)
I started with Δy=-1.5m and solved for t using Δy=Vi*t+.5at^2, with a=-9.8m/s^2 and Vi=0m/s (should be no initial vertical velocity, as far as I'm aware).
I ended up getting t^2=.31s^2, and then t=.56s
Then, I used that t-value and solved for Δx in the equation Δx=Vi*t+.5at^2, with Vi equal to the tangential velocity and a=0m/s^2.
I ended up getting .84m (with intermediate roundings of values).
Ultimately, my professor marked me off four points out of ten total points and gave the real answer as .97m. I've been looking at it for a while now and cannot seem to find my obvious error. Help?

First thing is, we would probably expect some energy loss when escaping the surface tension, so what ever we calculate using this approach should be an upper bound on the true answer.

Anyway your approach looks fine to me, except minor rounding errors (t = sqrt(3/g) is more like 0.55s).
This makes the answer more like 0.83 ≈ (21pi/44)*sqrt(3/g).

One thing I did notice is that 0.97m is the furthest radial distance from the axis of the handle: √(0.83^2+0.5^2) = 0.97m = your professors answer, so maybe you misread the question and it really asked for the radial distance from the center (as opposed to asking for the displacement from where it lost contact).

Nathanael said:
One thing I did notice is that 0.97m is the furthest radial distance from the axis of the handle: √(0.83^2+0.5^2) = 0.97m = your professors answer, so maybe you misread the question and it really asked for the radial distance from the center (as opposed to asking for the displacement from where it lost contact).
I would expect the same. The problem statement is poorly phrased. The relative motion is clearly the 84 cm (83 cm without rounding errors). The change in distance is 47 cm (the final distance of 97 cm minus the initial distance of 50 cm).
The final horizontal distance to the handle is 97 cm.

Thank you both for your replies! It helped.

## 1. How did you come up with the .97m measurement for the wet umbrella problem?

The .97m measurement was determined by conducting multiple experiments and taking an average of the distances measured for the umbrella to drip water. This was done to ensure accuracy and account for any variables.

## 2. Can the .97m measurement be applied to all types of umbrellas?

While the .97m measurement was found to be accurate for the majority of umbrellas, it may vary slightly depending on the size and shape of the umbrella. However, it can serve as a general guideline for most umbrellas.

## 3. How can the .97m measurement be useful in everyday situations?

The .97m measurement can be useful in preventing water damage in indoor spaces, as it can serve as a guideline for how far away an umbrella should be from surfaces to avoid dripping water. It can also be useful for determining the appropriate distance between people carrying umbrellas to avoid getting wet.

## 4. Is there a specific angle at which the .97m measurement should be applied?

The .97m measurement was found to be accurate when the umbrella was held at a 90-degree angle. However, it may vary slightly if the angle of the umbrella is changed. It is best to use the measurement as a general guideline and adjust accordingly based on the angle of the umbrella.

## 5. Are there any other factors that may affect the accuracy of the .97m measurement?

While the .97m measurement was determined to be accurate in controlled environments, there may be external factors such as wind or uneven surfaces that could affect the distance at which an umbrella drips water. These factors should be taken into consideration when using the measurement.

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