Linear and Angular Velocities

In the study of circular motion, we encounter two essential measures: linear velocity and angular velocity.

Linear Velocity is the measure of how fast an object travels along the circle's edge. It's the distance a point covers, divided by the time it takes, algebraically represented as:

\( v = \cfrac{s}{t} \)

• \(s\) represents distance
• \(t\) represents time

It tells us the speed of an object on the path it follows, giving us insight into the journey along the circumference.

Conversely, Angular Velocity focuses on rotation, measuring how quickly an object spins around the circle’s center. This rate of rotation is measured by the angle swept out over time, formulated as:

\( \omega = \cfrac{\theta}{t} \)

• \( \theta \) represents the angle in radians
• \(t\) represents time

Angular velocity provides a perspective on the object's rotational movement, describing the rapidity of its circular dance around a fixed point.

The table below summarizes all the major variables and formulas we will focus on this lesson:

What are some of the possible units used to measure each of the following? Click to reveal the answers.

Time:   sec, min, hr

Arc Length:   mm, cm, m, km

Angle:   degrees, revolutions, radians

Linear Velocity:   cm/sec, km/hr

Angular Velocity:   ° /sec, rev/sec, rad/sec, rpm

Converting Between Radians and Revolutions

Radians can be converted into Revolutions and vice versa. The conversion factor used to convert between radians and revolutions is:

\(2\pi\;[\text{rads}] = 1\;[\text{revs}]\)

We can use this conversion factor to convert radians into revolutions and vice versa:

\(\theta_{\text{revs}} = x [\text{rads}] \times \cfrac{1\;[\text{revs}]}{2\pi\;[\text{rads}]}\)

\(\theta_{\text{rads}} = x [\text{revs}] \times \cfrac{2\pi\;[\text{rads}]}{1\;[\text{revs}]}\)
Example

Convert \(2.4 \; [\text{revolutions}]\) to radians.

In order to convert revolutions to radians, we can use the appropriate formula and substitute the pertinent value(s):

\(\theta = x [\text{revs}] \times \cfrac{2\pi\;[\text{rads}]}{1\;[\text{revs}]}\)

\(\theta = 2.4 [\text{revs}] \times \cfrac{2\pi\;[\text{rads}]}{1\;[\text{revs}]}\)

\(\theta = 4.8\pi \; [\text{rads}]\)

We can now simplify the expression:

\(\theta \approx 15.1 \; [\text{rads}]\)

Therefore, we can determine that \(2.4 \; [\text{revs}]\) converted to radians is \(\boldsymbol{\approx 15.1 \; [\textbf{rads}]}\).

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Solving Word Problems

There are a few strategies we can consider when solving word Problems involving angular displacement, velocity, and acceleration:

• Identify Known and Unknown Variables. This will help determine what formula you need to use to determine the value of the unknown variable
• Make Proper Unit Conversions. Identify the unit of the current variable and multiply it by the appropriate ratio to properly convert it into a different unit. As we stated above, if you want to convert radians to degrees, multiply your current value by \(180^{\circ}\pi\). Likewise, if you want to convert degrees to radians, multiply the current value by \(\pi/180\) radians.
• Include Units in Final Answer. It’s also helpful to keep track of what units your variables use while performing calculations.

Example

Suppose a \(150\;[\text{cm}]\) diameter wheel is rotating at \(25 \;[\text{rpm}]\). At what rate is the wheel moving along the road in \(\cfrac{\text{m}}{\text{hr}}\)?

First, we can define the given variables. In this instance, we can identify that \(d = 150\;[\text{cm}]\); therefore, the radius, \(r\), is half that, \(r = 75\;[\text{cm}]\). We can also identify that the angular velocity, \(ω = 25\;[\text{rpm}]\).

Next, we can calculate the linear velocity:

\(v = \omega r \)

\(v = 25\cfrac{[\text{rev}]}{[\text{min}]} \times 75\;[\text{cm}])\)

\(v = 1875\cfrac{[\text{rev} \cdot \text{cm}]}{[\text{min}]}\)

Then, we can convert \([\text{revs}]\) to \([\text{rads}]\), \([\text{cm}]\) to \([\text{m}]\) and \([\text{min}]\) to \([\text{hr}]\) to measure the velocity in \([\text{m}]/[\text{hr}]\):

\(v = 1875 \cfrac{[\text{rev} \cdot \text{cm}]}{[\text{min}]} \times \cfrac{2\pi\;[\text{rads}]}{1\;[\text{rev}]} \times \cfrac{1\;[\text{m}]}{100\;[\text{cm}]} \times \cfrac{60\;[\text{min}]}{1\;[\text{hr}]}\)

\(v = 2250\pi \cfrac{[\text{m}]}{[\text{hr}]}\)

Finally, we can simplify the final value to get the final velocity:

\(v = 2250\pi \left[\cfrac{\text{m}}{\text{hr}}\right]\)

\(v \approx 7068 \left[\cfrac{\text{m}}{\text{hr}}\right]\)

Therefore, we can determine that the wheel is moving at a speed of \(\boldsymbol{\approx 7068 \left[\cfrac{\textbf{m}}{\textbf{hr}}\right]}\) along the road.

Example

A bicycle has a \(70\;[cm]\) wheel diameter; how many rotations per second does the cyclist have to achieve to push the bicycle along a flat surface at \(25[\text{km}/\text{hr}]\)?

First, we can define the given variables. In this instance, we can identify that \(d = 70[cm]\), so the radius, \(r\), is half that, \(r = 35\;[\text{cm}]\). We can also identify that the linear velocity, \(v = 25\;[\text{km}/\text{hr}]\).

Next, we can convert \(25\;[\text{km}]/[\text{hr}]\) to \([\text{cm}]/[\text{s}]\):

\(v = 25\; \cfrac{[\text{km}]}{[\text{hr}]} \times \cfrac{100000\; [\text{cm}]}{[\text{km}]} \times \cfrac{1\;[\text{hr}]}{3600\; [\text{sec}]}\)

\(v = \cfrac{1250000\;[\text{cm}]}{1800\;[\text{s}]}\)

Then, we can calculate the angular velocity:

\(\omega = \cfrac{v}{r}\)

\(\omega = \cfrac{\cfrac{1250000\;[\text{cm}]}{1800\;[\text{s}]}}{35\;[\text{cm}]}\)

\(\omega = \cfrac{1250000\;\cancel{[\text{cm}]}}{1800\;[\text{s}]} \times \cfrac{1}{35\;\cancel{[\text{cm}]}}\)

\(\omega = \cfrac{19.84 \; [\text{rads}]}{[\text{sec}]}\)

After, we can convert the velocity from radians to revolutions:

\(\omega = \cfrac{19.84 \; [\text{rads}]}{[\text{sec}]} \times \cfrac{1 \; [\text{rev}]}{2\pi \;[\text{rads}]}\)

\(\omega \approx 3.16\cfrac{[\text{revs}]}{[\text{sec}]}\)

Therefore, we can determine that the cyclist has to achieve \(\boldsymbol{\approx 3.16 \; \left[\cfrac{\textbf{revs}}{\textbf{sec}}\right]}\) to push his biycle at the intended speed.

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