2 more questions related to Circular Motion and Speed

[Kikki:)]

Homework Statement

The first question reads:

In riding a bicycle, it is noted that the 26 inch diameter wheel makes 15.0 revolutions in a time of 8.50sec. What is the angular speed of the wheel? What distance does the bicycle travel during this time? (in feet) (in rad/s)

Homework Equations

Circumference = 2 pi r

360 degrees = 1 revolution

Possibly radian = arc length/radius

The Attempt at a Solution

I do know that the radius in feet is 1.1ft and the diameter is 2.2ft. I also figured out that the amount of rads is 6.9rad by: 1.1rev. x 2pi rad/rev = 6.9 rad (the revs cancel out). But would you use one of the kinematic equations to find for the distance and speed? I get that you would probably use the fomula for average velocity = change in distance over change in time.

The second question:

Homework Statement

An electrical wire with a diameter of 0.75cm is wound on a spool with a radius of 30cm and a length of 24cm. Through how many radians must the spool be turned to wrap one even layer of wire? What is the length of this wound wire?

Homework Equations

C = 2 pi r

The Attempt at a Solution

I first drew it all out. then took 24cm divided by .75cm to equal 32cm. Is that how you find the length of the wire? I'm not exactly getting how to do this problem at all. Would finding the circumference of both be a good thing?

 

[ideasrule]

I do know that the radius in feet is 1.1ft and the diameter is 2.2ft. I also figured out that the amount of rads is 6.9rad by: 1.1rev. x 2pi rad/rev = 6.9 rad (the revs cancel out).

Where did you get 1.1rev from? The question says that the wheel turned through 15 revolutions.

But would you use one of the kinematic equations to find for the distance and speed? I get that you would probably use the fomula for average velocity = change in distance over change in time.

Angular speed is just the angle divided by time, exactly analogous to how linear speed is linear distance divided by time. So however many radians 15 revolutions is, that value divided by 8.5s gives you the angular speed.

As for distance traveled, how far does the bike travel for each revolution of its wheels?

Is that how you find the length of the wire? I'm not exactly getting how to do this problem at all. Would finding the circumference of both be a good thing?

You've already found that the wire must be wrapped around the spool 32 times. How many radians does this correspond to? How much wire is needed to wrap around the spool once?

 

[Kikki:)]

Where did you get 1.1rev from? The question says that the wheel turned through 15 revolutions.



Angular speed is just the angle divided by time, exactly analogous to how linear speed is linear distance divided by time. So however many radians 15 revolutions is, that value divided by 8.5s gives you the angular speed.

As for distance traveled, how far does the bike travel for each revolution of its wheels?



You've already found that the wire must be wrapped around the spool 32 times. How many radians does this correspond to? How much wire is needed to wrap around the spool once?

So take 15 rev x 2 x pi rad/rev = 94.2 rads. Then take 94.2rad divided by 8.5s = 11.1 rad/s.

Then for the distance you would take 11.1rads/s divided by 1.1ft.$$\alpha$$ = a/r formula. Or would you use $$\omega$$ = v/r ?

Since it was 32times to wrap it around to convert it to rads you take 32 x 2pi/rads = 200.96rads. But its asking to find it just for once around, so you take 200.96 divided by 60 to get 3.35rads? But for to finds the length of this wire you take the 3.35rads x 30cm = 100.5cm

 

[Kikki:)]

Much thanks for helping me figure it out! :]

 

[rwisz]

For the first question, the angular speed can be calculated by dividing the number of revolutions by the time taken (in seconds). In this case, it would be 15.0 rev/8.50 sec = 1.76 rev/s. To convert this to rad/s, we can multiply by 2π (since 1 revolution = 2π radians), giving us an angular speed of 11.06 rad/s.

To find the distance traveled, we can use the formula d = rθ, where d is the distance, r is the radius, and θ is the angle (in radians). In this case, the angle would be 15 revolutions (or 30π radians), since the wheel makes 15 revolutions. Therefore, the distance traveled would be 2.2ft x 30π = 207.35 feet.

For the second question, we can use the same formula d = rθ to find the number of radians the spool must be turned to wrap one even layer of wire. The radius of the spool is 30cm, so the circumference would be 60π cm. The length of the wire is 24cm, so the number of layers would be 24cm/60π cm = 0.4 layers. Since one layer is equivalent to 2π radians, the spool would need to be turned 0.4 x 2π = 0.8 radians to wrap one even layer of wire.

To find the length of the wire, we can use the formula for circumference (C = 2πr) to find the length of one layer of wire, and then multiply by the number of layers. In this case, the length of one layer would be 60π cm, and since there are 0.4 layers, the total length of the wire would be 60π cm x 0.4 = 24π cm.