Practice Big Wheel, Small Wheel

puremath

New member
Suppose that a belt drives two wheels of radii r and R. (The picture cannot be uploaded. It is too large to be uploaded). The radius r of the small wheel is 6 cm and the radius R of the big wheel is 10 cm. The angular speed of the larger wheel is 100 rpm.

Determine the following:

A. The angular speed of the larger wheel in radians per minute.

I got 200(pi) rad/min

B. The linear speed of a point on the circumference of the larger wheel.

I got 2000(pi) cm/sec

C. The angular speed of the smaller wheel in radians per minute.

Hint Given:

Because of the belt, the linear speed of a point on the circumference of the larger wheel is equal to the linear speed of a point on the circumference of the smaller wheel.

D. The angular speed of the smaller wheel in rpm.

I got a and b right. I cannot figure out c and d. Can you set c and d up for me?
 

MarkFL

La Villa Strangiato
Math Helper
Let's let \(v_R\) be the linear speed of the larger wheel and \(v_r\) be the linear speed of the smaller wheel. As suggested by the hint, if there is no slipping of the belt these two linear speeds must be the same:

[MATH]v_R=v_r[/MATH]
Now, using the fact that:

[MATH]v=r\omega[/MATH]
We may then write:

[MATH]R\omega_R=r\omega_r\implies \omega_r=\frac{R}{r}\omega_R[/MATH]
This should makes sense, as this implies the smaller wheel will have a larger angular speed than the larger wheel.
 

puremath

New member
Let's let \(v_R\) be the linear speed of the larger wheel and \(v_r\) be the linear speed of the smaller wheel. As suggested by the hint, if there is no slipping of the belt these two linear speeds must be the same:

[MATH]v_R=v_r[/MATH]
Now, using the fact that:

[MATH]v=r\omega[/MATH]
We may then write:

[MATH]R\omega_R=r\omega_r\implies \omega_r=\frac{R}{r}\omega_R[/MATH]
This should makes sense, as this implies the smaller wheel will have a larger angular speed than the larger wheel.

I am confused with all these subscript variables. Can you use single letters in place of v_R and v_r?
 

MarkFL

La Villa Strangiato
Math Helper
I find it less confusing to use subscripts for variables that represent similar things, like radii and angular velocities. You need to get comfortable fully with the language of mathematics. :)
 

puremath

New member
I find it less confusing to use subscripts for variables that represent similar things, like radii and angular velocities. You need to get comfortable fully with the language of mathematics. :)

Please answer C and D for my notes. I have been away from Cohen for more than a week.
 

MarkFL

La Villa Strangiato
Math Helper
Above I posted:

[MATH]\omega_r=\frac{R}{r}\omega_R[/MATH]
We are given:

[MATH]R=10\text{ cm}[/MATH]
[MATH]r=6\text{ cm}[/MATH]
[MATH]\omega_R=100\text{ RPM}=200\pi\frac{\text{rad}}{\text{min}}[/MATH]
All that is needed is to plug this into the above formula.
 

puremath

New member
Above I posted:

[MATH]\omega_r=\frac{R}{r}\omega_R[/MATH]
We are given:

[MATH]R=10\text{ cm}[/MATH]
[MATH]r=6\text{ cm}[/MATH]
[MATH]\omega_R=100\text{ RPM}=200\pi\frac{\text{rad}}{\text{min}}[/MATH]
All that is needed is to plug this into the above formula.

Very good. When I return to math from the new place, I will read each lesson in parts and answer questions in parts. Right now, no math. I just finished packing most of the few items I own, including all math books. I have an appointment with a room agency tomorrow. I hope to be situated at a new room soon.
 
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