Another angular velocity problem

[Imperil]

A Ferris wheel makes two rotations per minute. The radius of the Ferris wheel is 25 m.

a) Determine the angular velocity in rad/s
b) A ride lasts 4 min. How far will a rider travel in 4 min?

My Work:

a) C = 2Pi(25 m) = 50Pi m
50Pi m / 25 m = 2Pi rad

2 rotations/min x 2Pi = 4Pi rad/min

Therefore the angular velocity is Pi / 15 rad/s

b) 4 x 4Pi = 16Pi
16Pi = a / 25 m
a = 25(16Pi)
a = 400Pi
a = 1256.6 m

I believe I am correct but I am hoping someone could check over my approach to the problem as my text was not very clear at all and I have tried to figure this out mainly on my own.

 

[HallsofIvy]

A Ferris wheel makes two rotations per minute. The radius of the Ferris wheel is 25 m.

a) Determine the angular velocity in rad/s
b) A ride lasts 4 min. How far will a rider travel in 4 min?

My Work:

a) C = 2Pi(25 m) = 50Pi m
50Pi m / 25 m = 2Pi rad

Why did you calculate this? Surely you knew from the start that there are $2\pi$ radians in any circle!

2 rotations/min x 2Pi = 4Pi rad/min

Therefore the angular velocity is Pi / 15 rad/s

Yes, although I think I would have said "one rotation in 30 seconds is $2\pi/30= \pi/15$ radians per second", this is fine.

b) 4 x 4Pi = 16Pi
16Pi = a / 25 m
a = 25(16Pi)
a = 400Pi
a = 1256.6 m

You had already calcuated in (a) (where it wasn't necessary!) that the circumference of the wheel is $50\pi$ meters. Now you calculate it again! In 4 minutes you will have made 8 rotations: $8(50\pi)= 400\pi$ m just as you say. I think I would leave it as $400\pi m$ and not change to an approximate answer.

I believe I am correct but I am hoping someone could check over my approach to the problem as my text was not very clear at all and I have tried to figure this out mainly on my own.

You've done some unnecessary work but you have the correct answers.

 

[Architeuthis Dux]

Your approach to the problem is correct. Here are some explanations for your calculations:

a) To find the angular velocity, we use the formula w = 2πf, where w is the angular velocity in radians per second, and f is the frequency in rotations per second. In this case, the frequency is given as 2 rotations per minute, which can be converted to rotations per second by dividing by 60 (since there are 60 seconds in a minute). So, the frequency is 2/60 = 1/30 rotations per second. Plugging this into the formula, we get w = 2π(1/30) = π/15 rad/s.

b) To find the distance traveled by a rider in 4 minutes, we use the formula d = rθ, where d is the distance traveled, r is the radius of the Ferris wheel, and θ is the angular displacement. In this case, the rider will complete 2 rotations per minute, so in 4 minutes, they will complete 8 rotations. This corresponds to an angular displacement of 8(2π) = 16π radians. Plugging this into the formula, we get d = (25)(16π) = 400π m ≈ 1256.6 m. So, your answer is correct.

Overall, your approach to the problem is correct and your calculations are accurate. Good job!