Use Linear and Angular Speed Formulas

Angular speed concerns speed of rotation. For objects rotating quickly, such as the crankshaft in a car engine, the speed of rotation is given as a number of revolutions per minute. For slower rotations, such as a moon orbiting a planet, the angular speed might be measured in degrees per hour.

In both cases, angular speed is an idea that relates an angle with a unit of time.

In more advanced work, three things are involved: angle, time and direction, and the concept is called angular velocity, which is a treated as a vector quantity.

A point on a rotating object has a certain speed as it moves through space. Because its direction is always changing, we refer to its speed rather than its velocity. However, we can speak of its instantaneous velocity.

Problems involving angular speeds usually require the formula connecting the radius of a circle with its circumference: $C=2\pi r$C=2πr. It is usual to employ the Greek letter $\omega$ω to stand for an angular speed. The letter $s$s is frequently used for distance and $v$v for velocity.

An angular speed of $\omega$ω radians per second is the same as $\frac{\omega}{2\pi}$ω2π​ revolutions per second. Then, the magnitude of the instantaneous velocity of a point on the circumference must be $v=C\times\frac{\omega}{2\pi}=r\omega$v=C×ω2π​=rω.

Examples

Example 1

Consider the moon as it orbits the earth. The moon's orbit is really an ellipse but its eccentricity is low so that it can be approximated by a circular orbit. The radial distance from the earth is, on average, $384000$384000 km and each complete orbit takes $27.32$27.32 days. 

The length of one complete orbit must be $2\times\pi\times384000\approx2413000$2×π×384000≈2413000 km. This distance is covered in $27.32$27.32 days, which is $24\times27,32\approx656$24×27,32≈656 hours. Therefore, relative to the earth, the moon has an orbital speed of $2413000\div656\approx3680$2413000÷656≈3680 km/h.

This instantaneous velocity is about $1.02$1.02 kilometres per second in magnitude.

Example 2

A cyclist is riding at $14$14km/h on a bicycle with wheels whose radius is $33$33 cm. How many times does the wheel turn in one second? What is the angular speed of the wheel relative to the hub?

The speed, $14$14 km/h, is $14\times1000\times100=1400000$14×1000×100=1400000 cm/h or $388.9$388.9 cm/s. The circumference of the wheel is $33\times2\times\pi=207.3$33×2×π=207.3 cm. Therefore, in one second the wheel turns $388.9\div207.3\approx1.87$388.9÷207.3≈1.87 times.

The angular speed could be given as $1.87$1.87 revolutions per second or, in terms of an angle, $1.87\times360^\circ=675.3$1.87×360°=675.3 degrees per second. In radian measure it would be $1.87\times2\pi=11.75$1.87×2π=11.75 radians per second.

 

More Worked Examples

Question 1

Question 2

Question 3