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Can someone help me please? I don't really understand Angular and Linear Velocity. Thank you!

A back wheel on a tricycle has a radius of 8cm and rotates at a rate of 200 times per minute. Approximately what are the angular velocity of the wheel in radians per second and the linear velocity of a point on the wheel in centimeters per second?

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    (Average) angular velocity will be the "total angle" angle your wheel revolves through, in your case $200(2\pi)$, divided by the time it takes to go through that angle, which is $60$ seconds. Now, as for linear velocity, when the wheel rotates through an angle of $1$ radian, then a point on the outside of the wheel will move a distance equal to the length of the radius (this is the definition of radian measure!) and so the linear velocity $v$ will be equal to $\omega \cdot r$, where $\omega$ is the angular velocity. – 2017-02-12

2 Answers 2

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Angular velocity $\omega$ is measured in radians per second, so in this case $$\omega=2\pi\times\frac {200}{60}$$

Linear velocity $v$ of a fixed point at distance $r$ from the centre of the circle is given by $$v=r\omega$$

I hope this helps

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The angular velocity corresponds to the radians that the radius sweeps in a second. Let's introduce an intermediate quantity: the period $T$. That is the time it takes for the wheel to complete one spin.

If we know that the wheel takes $T$ seconds to make a complete rotation, then calculating the angular velocity $\omega$ is easy: the angle swept in a whole rotation is $2\pi$ radians. Then we have $$\omega = \frac{2\pi}{T}\quad\text{[rad/s] or [$\mathrm s^{-1}$]}$$

In your problem there's a different quantity: the frequency. It is measured in $\mathrm s^{-1}$ and indicates how many whole rotations occur in one second. From the problem we know that $f = 200/60\ \mathrm s^{-1}$. You can verify that between frequency and period there exists the relation $T = f^{-1}$. Therefore we obtain another formula for $\omega$: $$\omega = 2\pi f\quad\text{[rad/s] or [$\mathrm s^{-1}$]}$$

Finally, tangential velocity is directly proportional to the angular velocity, and the constant of proportionality is the radius, i.e. $$v = \omega r\quad\text{[$\mathrm m\mathrm s^{-1}$]}$$