linear and angular velocity

[sayyadina]

A wheel has a diameter of 25 inches. Find the radian angle the wheel rolls over 16 ft.

Arc = angle/rod or angle = arc/rod

arc= 16ft

rod= 25 inches/2 =12.5 inches (but it needs to be in feet too?) so 12.5/12 ft

16 / (12.5/12) = 15.36 radians


I'm not sure if I should be making sure that I'm working with all the same unit (feet versus feet and inches), or just work with inches and feet.


Object A moves around object B in a 32ft radius circle. Find the linear velocity of object A if it completes a full revolution every 3.5 seconds.

So it goes 2pi radians in 3.5 seconds.

32(2pi/3.5) = 57.44626567 ft/sec

Is this right, or am I missing something?

 

[Romsek]

A wheel has a diameter of 25 inches. Find the radian angle the wheel rolls over 16 ft.

Arc = angle/rod or angle = arc/rod

arc= 16ft

rod= 25 inches/2 =12.5 inches (but it needs to be in feet too?) so 12.5/12 ft

16 / (12.5/12) = 15.36 radians


I'm not sure if I should be making sure that I'm working with all the same unit (feet versus feet and inches), or just work with inches and feet.


Object A moves around object B in a 32ft radius circle. Find the linear velocity of object A if it completes a full revolution every 3.5 seconds.

So it goes 2pi radians in 3.5 seconds.

32(2pi/3.5) = 57.44626567 ft/sec

Is this right, or am I missing something?

what is rod? Is that the distance of 1 radian? New term for me then. (* updated: I guess rod = radius seeing how that's the length of 1 radian *)

The key here is that when you roll 2pi radians you roll 1 circumference.

So all you do is divide the distance rolled by the circumference of the wheel (in the same units) and multiply by 2pi.

For the 2nd problem you are correct that A goes 2pi radians every 3.5 seconds. So compute what that translates to in feet/s given you are moving on a 32 ft radius circle. That gives you some length/3.5 seconds. Do the division to come up with the linear speed. (it's not the velocity, velocity is a vector, poorly worded problem)

 

[stapel]

I think you've posted two different exercises, so I'll edit below to clarify my understanding of your post.

Question 1) A wheel has a diameter of 25 inches. Find the radian angle the wheel rolls over 16 ft.

My answer) Arc = angle/rod or angle = arc/rod

How is the old-timey linear measure of "rods" coming into play?

arc= 16ft

rod= 25 inches/2 =12.5 inches (but it needs to be in feet too?) so 12.5/12 ft

Are you perhaps using "rod" to stand for "radius"? If so, then I think you mean:

. . . . .total arc length: 16 feet = (16 feet)*(12 inches / foot) = 192 inches
. . . . .diameter: 25 inches
. . . . .radius: (1/2)(25 inches) = 12.5 inches
. . . . .arc-length formula: s = (angle measure, in radians)*(radius)
. . . . .192 = (angle measure)*(12.5)

. . . . .To find the number of radians, solve
. . . . ."192 = (angle measure)*(12.5)" for the
. . . . .angle measure:

. . . . .(192/12.5) = (angle measure) = 15.36

Question 2) Object A moves around object B in a 32ft radius circle. Find the linear velocity of object A if it completes a full revolution every 3.5 seconds.

My answer) So it goes 2pi radians in 3.5 seconds.

32(2pi/3.5) = 57.44626567 ft/sec

Is this right, or am I missing something?

You appear to be attempting to do many computations all in one step. It might help if you wrote things out clearly, so you're sure of your steps:

. . . . .angular velocity: (1 revolution)/(3.5 seconds) = (2/7) rev/sec
. . . . .arc length of one revolution: (2pi)*(32 ft) = 64pi feet
. . . . .linear velocity: (64 pi feet / rev)*(2/7 rev / sec) = (128pi/7) rev/sec

The decimal form is the same as what you got, but the clarity of the steps makes plain that the computations and units are valid. As a result, you can be confident of the solution.