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).
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
= v/r radians/sec
We can determine angular acceleration, a
=(v2-v1)/r/t
Solve for a. I get about 1.4 rad./s.
Check my calculations.
I also got about 1.4 rad/s, but is the answer negative? -1.4 rad/s?
Thanks!
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.
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.
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.
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.
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.
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.
To find the wheel's angular acceleration, we can use the following formula:
h = rω^2 - (1/2)gω^2
where
h is the height reached by the drop,
r is the radius of the wheel,
ω is the angular velocity of the wheel, and
g is the acceleration due to gravity.
We are given the heights reached by the drops of water, h1 = 0.44 m and h2 = 0.58 m. The radius of the wheel, r, is given as 0.40 m and we can assume the acceleration due to gravity, g, as 9.8 m/s^2.
For the drop that reaches a height of 0.44 m:
0.44 = (0.40)*(ω^2) - (1/2)*(9.8)*(ω^2)
For the drop that reaches a height of 0.58 m:
0.58 = (0.40)*(ω^2) - (1/2)*(9.8)*(ω^2)
Now, let's solve these two equations simultaneously to find the value of ω.
From equation 1:
0.44 = 0.40ω^2 - 4.9ω^2
0.44 = -4.5ω^2
From equation 2:
0.58 = 0.40ω^2 - 4.9ω^2
0.58 = -4.5ω^2
Simplifying these equations, we get:
ω^2 = -0.44/4.5
ω^2 = -0.58/4.5
Taking the square root of both sides, we get:
ω = √(-0.44/4.5)
ω = √(-0.58/4.5)
Since we are dealing with angular acceleration, the values of ω will be positive. Therefore, we can discard the negative solutions and choose the positive values of ω.
Now that we have obtained the values of ω, we can calculate the angular acceleration using the formula:
α = ω / Δt
where α is the angular acceleration and Δt is the time interval between the two observations. Since it is not mentioned in the question, we cannot determine the exact value of Δt and therefore cannot calculate the angular acceleration.