An umbrella with a diameter of 1 m turns 22 rev every 44 seconds. If it's 1.5 m from the ground, calculate the time it takes for a drop to reach the ground and the distance away from the umbrella a person should stand so as to not get wet.
For this problem I made a = (0, -g) and used the equation Vyo= 1.5 - g(t^2)/2. And made Vyo=0 so, solving for t, I got sqrt(3/g).
Then v=2pir/T,so (22x2pi)/44s = pi rev/s
so w= pi rev/s
Then, using the equation v=Rw , v = (.5m)(pi rev/s) = pi/2 m/s
After that I didn't know what to do, so if I made any errors above, please let me know. I think for finding the distance I would use the equation D = sqrt(R^2 = Xmax^2) but I don't know how I could incorporate that into the problem
Thank you
The drops begin to move with horizontal speed
v(x)=2•π•n•R=2•π•N•R/t=
=2•π•22•0.5/44=1.57 m/s
The time for falling down is
t=sqrt(2h/g) = sqrt(2•1.5/9.8)=0.55 s.
L=v(x) •t =1.57•0.55 =0.87 m
To calculate the time it takes for a raindrop to reach the ground, and the distance away from the umbrella a person should stand to not get wet, let's go step by step:
Step 1: Calculate the time it takes for a raindrop to reach the ground
You mentioned that you used the vertical motion equation Vy0 = 1.5 - (g * t^2) / 2. This equation is correct for finding the vertical position of the raindrop at any given time.
To find the time it takes for the raindrop to reach the ground, you need to set the vertical position (Vy0) equal to zero and solve for time (t):
0 = 1.5 - (g * t^2) / 2
Rearranging the equation, you get:
(g * t^2) / 2 = 1.5
Multiplying both sides by 2 / g:
t^2 = 3 / g
Taking the square root on both sides:
t = sqrt(3 / g)
So, the time it takes for a raindrop to reach the ground is sqrt(3 / g) seconds.
Step 2: Calculate the distance away from the umbrella a person should stand to not get wet
Next, let's calculate the distance away from the umbrella a person should stand to avoid getting wet. We'll use the relationship between linear velocity (v), radius (R), and angular velocity (w): v = R * w.
You correctly found that the angular velocity (w) is pi rev/s.
Now, let's calculate the linear velocity (v):
v = (R * w)
v = (0.5 m) * (pi rev/s)
v = (0.5 m) * (pi / 2) m/s
v = pi / 4 m/s
So, the linear velocity of a point on the edge of the umbrella is pi / 4 m/s.
To find the distance away from the umbrella, we can use the equation D = v * t, where D is the distance, v is the linear velocity, and t is the time it takes for a raindrop to reach the ground.
D = (pi / 4 m/s) * (sqrt(3 / g) seconds)
This gives you the distance away from the umbrella a person should stand to avoid getting wet.
Remember to plug in the appropriate value for the acceleration due to gravity (g) in m/s^2 to get the final answer.
I hope this helps! Let me know if you have any more questions.