• Sep 1st 2006, 05:12 PM
1. A wheel has a 2cm diameter. the speed of a point on its rim is 11 m/s. What is its angular speed?
2. A horse on a merry-go-round is 7m from the center and travels at 10km/h. What is its angular speed?
3. A water wheel has a 10ft radius. The wheel revolves 16 times per minute. What is the speed of the river in mi/hr?
• Sep 1st 2006, 05:22 PM
Quick
Does angular speed mean how many degrees difference it turns in a second? :confused:

(for example, let's call the beginning point A the point after one second C and the center of the circle B... Are you trying to find angle ABC?)

This is my 600th post!!!!
• Sep 1st 2006, 05:26 PM
topsquark
Quote:

1. A wheel has a 2cm diameter. the speed of a point on its rim is 11 m/s. What is its angular speed?
2. A horse on a merry-go-round is 7m from the center and travels at 10km/h. What is its angular speed?
3. A water wheel has a 10ft radius. The wheel revolves 16 times per minute. What is the speed of the river in mi/hr?

All three of these involve the formula $v = r \omega$, where v is the tangential speed, r is the radius of the motion, and $\omega$ is the angular speed.

1. r = 1 cm = 0.01 m (NOT 2 cm!), v = 11 m/s (ALWAYS check your units!)
So $\omega = \frac{v}{r} = \frac{11}{0.01}$ rad/s = 1100 rad/s.

The other two are similar. Be careful of the last one...there are a number of unit changes you need to make. My suggestion is to get the speed of the river in ft/s, then convert to mi/hr. Also to find the angular speed in problem 3, note that 16 revolutions is equal to $16 \cdot 2 \pi$ rad.

-Dan
• Sep 1st 2006, 05:28 PM
topsquark
Quote:

Originally Posted by Quick
Does angular speed mean how many degrees difference it turns in a second? :confused:

(for example, let's call the beginning point A the point after one second C and the center of the circle B... Are you trying to find angle ABC?)

The angular speed is the time rate of change of the angle an object turns through. Similar to the definition of speed: $v = \frac{ \Delta x}{ \Delta t}$ angular speed is $\omega = \frac{ \Delta \theta}{ \Delta t}$ where $\theta$ represents the "angular position" of the rotating object.

Oh, and in case the units are obscure to you, 1 radian is defined as the angle marking out an arc length on a circle equal to the radius. We may calculate from this definition that $\pi$ radians are equivalent to 180 degrees.

-Dan
• Sep 1st 2006, 05:38 PM
Quick
I would recomend trying to figure out a formula yourself (I do it all the time)

Here's my method:

A wheel has diameter $d$, therefore it's circumference is $d\pi$

Now an object moves at a speed of $v$ around the circumference...

therefore the angular speed, $\omega$ for the object (in degrees) is: $\omega=360\left(\frac{v}{d\pi}\right)$

which can then be converted to radians: $\omega=360\left(\frac{v}{d\pi}\right)\times\frac{
\pi
}{180}=\frac{2v}{d}=\boxed{\frac{v}{r}}$

can you figure out how I got that answer?
• Sep 1st 2006, 06:22 PM
Soroban

Here's #3 . . .

Quote:

3. A water wheel has a 10ft radius. The wheel revolves 16 times per minute.
What is the speed of the river in mi/hr?

The circumference of the wheel is: . $C \:=\:2\pi R = 20\pi$ feet.

At 16 rev/min, a point on the wheel moves: . $16 \times 20\pi = 320\pi$ feet per minute.

We have: . $\frac{320\pi\;\text{feet}}{1\;\text{ minute}} \cdot \frac{60\;\text{minutes}}{\text{1 hour}} \cdot \frac{\text{1 mile}}{5280\;\text{feet}} \:=\:\frac{40\pi}{11}\text{ mph}$

• Sep 1st 2006, 07:48 PM