# PracticeBig 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.

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

#### 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]
All that is needed is to plug this into the above formula.

• puremath

#### 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]