Understanding Differential Angular Speeds in Wheel and Axle Systems

[kellyneedshelp]

I'm trying to figure out the following problem but I don't know where to begin:

Two wheels of diameter 0.78m are attached to opposite ends of an axle of length 1.6m. The wheels roll around a circular track of inside radius 9m.
a) Through what angle around the circular track must the axle assembly move so that the outer wheel makes one revolution more than the inner wheel?
b) What is the ratio of the angular speeds of spin of the two wheels? The differing angular speeds of the wheels is the reason for the differential in the drivetrain of a car or a truck.


I need help visualizing the wheels and axle, and the track, because right now the way I am thinking of the problem, the inside and outside wheel would always move at the same angle as one another, so clearly I must not be thinking of the problem correctly.

If anyone could help me understand the set-up then maybe I could solve the actual problem. Any help would be greatly appreciated.

Thanks!

 

[vaishakh]

The path made by the outer wheel is a bit more than the inner wheel in the circular path becoz the radius made by it is more. I mean it is 1.6m longer than the radius of the inner circle due to the length of the axle. So the distance covered by the outer wheel is greater than the inner wheel for the same angular displacement. For what angular displacement does the diiference become equal to the circumference of the wheel - this is the first question

 

[Doc Al]

Just to add to vaishakh's comments...

I need help visualizing the wheels and axle, and the track, because right now the way I am thinking of the problem, the inside and outside wheel would always move at the same angle as one another, so clearly I must not be thinking of the problem correctly.

You are thinking of the angle the wheels make with respect to the circular track, which is of course always the same for each wheel. But what the problem is asking about is the angle that each wheel turns as it rotates about its axle, not the center of the track; as vaishakh points out, since the wheels are moving at different speeds, they turn through different angles.

 

[kellyneedshelp]

ok i think i am understanding the question much better now.
thanks so much!

 

[kellyneedshelp]

i am still having some difficulties with this problem after all.

for part a) i got 1.39 radians as the answer but this is not correct.
this is how i got my answer:

Circumference inner circle = 2*pi*9m = 55.5m
Circumference outer circle = 2*pi*10.6m = 66.57m
Circumference of wheels = 2*pi*(.78/2) = 2.45m

(55.5m)/(2.45m) = 22.65306122 rotations of inside wheel per orbit
(66.57m)/(2.45m) = 27.17142857 rotations of outer wheel per orbit

then i noticed that the outer wheel rotates about 4.5 times more per orbit than the inner wheel does, so i figured at (1/4.5)*(2*pi), the outer wheel would have rotated one extra time. that is how i got 1.39 radians but this is not correct.

could anyone tell me what i am doing wrong? am i on the right track or not at all?thanks!

 

[Doc Al]

Your thinking is exactly right. I'll bet it's something silly. For example, check your arithmetic here:

Circumference inner circle = 2*pi*9m = 55.5m

 

[kellyneedshelp]

you're right, i was using the wrong number for that circumference and rounding too much. the correct answer was 1.53 radians.

thanks for the help!