Physics

posted by Lilly

A bicycle is turned upside down while its owner repairs a flat tire. A friend spins the other wheel and observes that drops of water fly off tangentially from point A. She measures the heights reached by drops moving vertically (see figure). A drop that breaks loose from the tire on one turn rises vertically 0.44 m above the tangent point. A drop that breaks loose on the next turn rises 0.58 m above the tangent point. The radius of the wheel is 0.40 m. Neglecting air friction and using only the observed heights and the radius of the wheel, find the wheel's angular acceleration (assuming it to be constant).

1. MathMate

The key is to find the tangential velocities at each of the turns, v1 and v2.

The initial velocity v for an object thrown upwards will reach a height of h is related by the formula:
h=v²/2g
where g is acceleration due to gravity.

Conversely, given h, we can find v from:
v=√(2gh)

So the tangential velocities v1 and v2 can be determined from h1 and h2 (work in metres).

Since the tangential velocities v1 and v2 are known, the average tangential velocity is
va=(v1+v2)/2

The time it took to make the turn was
t=2πr/va

From angular velocity

We can determine angular acceleration, a
=(v2-v1)/r/t

Solve for a. I get about 1.4 rad./s.
Check my calculations.

2. Lilly

Thanks!

3. MathMate

Angular acceleration is defined as (ω2-ω1)/time, since ω2 is bigger, evidenced by the fact that the droplet shoots higher in the second turn than the first, the value calculated should be positive.
If you got a negative answer according to the equations, it's time to give a thorough check.

4. Lilly

No, I thought the answer was positive too, but when I entered it in, a message popped up and said "Your anwer has the wrong sign". That's why I asked.

5. MathMate

Unless the heights of the drops have been inverted, I would be tempted to say that there is an error in the answer.

Double check the order of the heights of the droplets, i.e. 0.44 m followed by 0.58 m on the next turn. If that's the case, you could report your findings to your teachers so you don't get penalized.

After all, the people who put the answers on the computer are human, so mistakes are possible.

6. Ed

I have a question similar to this but I don't know what you did in each step. Could you plug in all the numbers step by step.

7. MathMate

I suggest you post a new question and it will get a personalized answer.

Otherwise, you can post what you get and if you don't get the right answer, we'll help you find the problem.

8. Blanche

You would use linear kinematics first to find the initial velocities for drop 1 and 2. Then you would find the angular velocity (initial and final) by using the equation v=wr and rearranging to be w=v/r.
Then you would substitute all the values into one of your angular kinematic equations and solve for the angular acceleration.

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