2D cirlces

Question

A two wheeled carriage, whose axle tree is 1.22m long is driven around a circle, the outer wheel makes 3/2 revolutions for every 1 revolution of inner wheel. The wheels are each 0.9 m high. What is circumference of circle described by outer wheel?

Answer 1

 

The wheels of the carriage make two concentric circles which are 1.22 m apart.

The wheels are 0.9 m high, or have a diameter of 0.9 m.

The diameter of the inner circle is unknown, call it x.

The diameter of the outer circle is twice the axle tree plus the diameter of the inner circle.

The diameter of the outer circle is 1.22+1.22+x=(2.44+x).

The circumference of the outer circle is C=(pi)(2.44+x)

The wheels each have a circumference of C=(pi)(0.9).

The circumference of the inner circle is C=(pi)(x).

When the carriage makes one complete revolution, you have the two concentric circles.

During that one complete revolution, the inner circle goes around completely n number of times.

During that one complete revolution, the outer wheel goes around 3/2 n number of times.

We can write the circumference of the concentric circles in different ways.

The wheels each have a circumference of C=(pi)(0.9)

The circumference of the outer circle is the circumference of the outer wheel times n times 3/2.

C=(pi)(0.9)(n)(3/2)

The circumference of the inner circle is the circumference of the inner wheel times n times 1.

C=(pi)(0.9)(n)

Set the circumferences of the outer circles equal to each other and set the circumferences of the inner circles equal to each other.

(pi)(0.9)(n)(3/2)=(pi)(2.44+x)

(pi)(0.9)(n)(1)=(pi)(x)

multiply the first equation by (2/3)

(pi)(0.9)(n)=(2/3)(pi)(2.44+x)

(pi)(0.9)(n)=(pi)(x)

both right-hand sides equal (pi)(0.9)(n), so set them equal to each other

(pi)(x)=(2/3)(pi)(2.44+x)

divide both sides by (pi)

x=(2/3)(2.44+x)

x=(2/3)(2.44)+(2/3)(x)

x-(2/3)(x)=(2/3)(2.44)

(1/3)(x)=(2/3)(2.44)

multiply both sides by (3)

(3)(1/3)(x)=(3)(2/3)(2.44)

x=(2)(2.44)

x=4.88 m for the diameter of the inner concentric circle

the outer concentric circle has a diameter of 4.88+1.22+1.22=7.32 m

C=(pi)(d)

C=3.14159*7.32

C=22.99644 m for the circumference of the outer circle

To check the solution, find the circumference of each circle two different ways.

n=5.422, the number of times the inner wheel rotates (I found this by dividing my answer by measurements given)

C=(pi)(4.88) and C=(5.422)(2.82743) (# of revolutions*circumference of the wheel)

C=15.331 m for the inner circle

C=(5.422)(2.82743)=15.33 m again for the inner circle

for the outer circle...

C=# of revolutions*circumference of the wheel*(3/2)

C=(5.422)(2.82743)(1.5)

C=22.99548 m (the other way gave us 22.99644, approximately equal !!)

 

 

 

Answer 2

Hi Shivam,

 

I understand (from Wikipedia) that axle tree to mean the difference in distance between the two concentric circles the wheels make. With that I proceeded to look for the speed of the two wheels

 

v₁ = distance/time = 2∏r/t

 

v₂ = 2∏(r+1.22)/t

 

We know also that v₁:v₂ is 2:3

 

=> 3[2∏r] = 2[2∏(r+1.22)]

 

=> r = 2.44 this is the radius of the circle made by the inner wheel. We add the axle tree's length to get to the radius of the outer circle

 

r = 2.44 + 1.22 = 3.66

 

The circumference of the outer circle, C=2∏*3.66

I hope this helps

  

Answer 3

Wow. this is a great problem. It looks a lot harder than it really is. 

 

Let's draw a figure to help us. Draw two concentric circles. The inside one has a radius = x, the outside one has a radius = x + 1.22

 

Do you see why? The inside circle is traced by the inside wheel and the outside (bigger) circle is traced by the outside wheel. The 1.22 is the axle distance between the two wheels.

 

Now, the problem tells us that the outside wheel turns 3/2 turn for every 1 turn of the inside wheel. Think about if we could put paint on the wheels and they painted their path as they turned. The length traced out on the ground by the outside wheel would be 3/2 as long as the inside track. Now extend this to the whole circumference. Doesn't this mean that the circumference of the outside circle is 3/2 of the inside circle? Does the information about the diameter of the wheel (the 0.9m value) even matter? Not to me, it doesn't. 

 

What we know: 

C₂ = C₁*3/2

So:

2πr₂ = (3/2)2πr_{1 (there's a factor of 2pi on both sides, let's cancel)}

r₂ = (3/2)r₁

Substitute values for the radii:

(x + 1.22) = 3/2 x

multiply through by 2:

2x + 2.44 = 3x

subtract 2x from both sides:

2.44 = x

and

2.44 +1.22 = 3.66 

 

I say the outside radius is 3.66 Meters. 

so the outside circumference is 2*pi*(3.66m) = 7.32*pi meters