For some reason, it seems fairly common for textbooks to turn to issues of angular velocity, linear velocity, and revolutions per minute (rpm) shortly after explaining circle sectors, their areas, and their arc lengths.
An arc's length is the distance partway around a circle; and the linear distance covered by, say, a bicycle is related to the radius of the bike's tires. If you mark one point on the bike's front tire (say, the spot opposite the tire valve) and count the number of times the wheel revolves, you can find the number of circle-circumferences that the marked point moved.
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If you "unwind" these circumferences to get a straight line, then you'll have found the distance that the bike traveled. This sort of relationship between the different measures is, I think, why this topic often arises at this point in one's studies.
First, we need some technical terminology and definitions.
"Angular velocity" is a measure of turning per time unit. It tells you the size of the angle through which something revolves in a given timespan. For instance, if a wheel rotates sixty times in one minute, then it has an angular velocity of 120π radians per minute. Then the angular velocity is measured in terms of radians per second, the Greek lowercase omega (ω) is often used as its name.
"Linear velocity" is a measure of distance per time unit. For instance, if the wheel in the previous example has a radius of 47 centimeters, then each pass of the circumference is 94π cm, or about 295 cm. Since the wheel does sixty of these revolutions in one minute, then the total length covered is 60 × 94&pi = 5,640π cm, or about 177 meters, in one minute. (That's about 10.6 kph, or about 6.7 mph.)
"Revolutions per minute", usually abbreviated as "rpm", is a measure of turning per time unit, but the time unit is always one minute. And rather than give the angle measure of the turning, it just gives the number of turnings. When you're looking at the tachometer on a vehicle's dashboard, you are looking at the current rpm of the vehicle's engine. In the example above, the rpm would be simply "60".
"Frequency" f is a measure of turning (or vibrations) per time unit, but the time unit is always one second. The unit for frequencies is the "hertz", which is denoted as Hz.
The relationship between frequency f (in Hz), rpm, and angular velocity ω (in radians) is demonstrated below (all of the elements in any one row are equivalent):
ω (in rad/sec) |
f (in Hz) |
rpm |
However, you may find that "angular velocity" being used interchangeably (but only informally; not by scientists) with rpm or frequency. Also, some (such as physicists) would hold that "angular velocity" is a vector quantity and ω is a scalar quantity called "angular frequency".
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Please don't bother memorizing these potential conflations or worrying about what "vectors" or "scalars" might be. I'm telling you about this in order to warn you that you should pay very close attention to how your particular textbook and your particular instructor define the various terms for that particular class. And know that, in your next class, the terms and definitions may very well be different.
The "rpm" is the number of times the wheel revolves per minute. To figure out how many times this wheel spins in one minute, I'll need to find the (linear, or straight-line) distance covered (per minute) when moving at 45 kph. Then I'll need to find the circumference of the wheel, and divide the total per-minute (linear) distance by this "once around" distance. The number of circumferences which fit inside the total distance is the number of times the wheel revolves in that time period.
First, I'll convert the (linear) velocity of the cart from kph to "centimeters per minute", using what I've learned about converting units. (Why "centimeters per minute"? Because I'm looking for "revolutions per minute", so minutes are a better time unit than hours. Also, the diameter is given in terms of centimeters, so that's a better length unit than kilometers.)
So the distance covered in one minute is 75,000 centimeters. The diameter of the wheel is 100 cm, so the radius is 50 cm, and the circumference is 100π cm. How many of these circumferences (or wheel revolutions) fit inside the 75,000 cm? In other words, if I were to peel this wheel's tread from the cart and lay it out flat, it would measure a distance of 100π cm. How many of these lengths fit into the entire distance covered in one minute? To find out how many of (this) fit into so many of (that), I must divide (that) by (this), so:
Then, rounding to the nearest whole revolution (that is, rounding the answer to a whole number), my answer is:
239 rpm
Note: This speed isn't as fast as it might appear: it's just under four revolutions per second. You can do that on your bike without breaking a sweat. Here's another note: The source from which I'd gotten my framework for the above exercise used "angular velocity" and "ω" for "the number of revolutions per minute". Yes, an algebra textbook used the wrong units.
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The previous exercise gave the speed of a vehicle and information about the wheel. From this, we found the revolutions per minute. We can go the other way, too; we can start with the revolutions per minutes (plus information about a wheel), and find the speed of the vehicle.
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The linear velocity will be the straight-line distance that the bike moves during a defined period of time. They've given me the number of times the wheel revolves each minute. A fixed point on the tire (say, a pebble in the tire's tread) moves the length of the circumference for each revolution. Unrolling this distance onto the ground, the bike will move along the ground the same distance, one circumference at a time, for each revolution. So this question is asking me to find the circumference length, and then use this to find the total distance covered per minute.
Since the diameter is 78 cm, then the circumference is C = 78π cm. Unwinding the tire's path into a straight line on the ground, this means that the bike moves 78π cm forward for each revolution of the tire. There are 120 such revolutions per minute, so:
(78π cm/rev)×(120 rev/min) = 9,360π cm/min
Now I need to convert this from centimeters-per-minute to kilometers-per-hour:
The bike is moving at about 17.6 kph.
...or about eleven miles an hour.
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The velocity will be the (linear, or equivalent straight-line) distance traveled in one second, divided by the one second. They gave me information for one year, so I'll start there. The circumference of the circle with r = 93,000,000 miles will be the linear distance that the Earth covers in one year.
This is the number of miles covered in one year, but I need the number of miles covered in only one second. There are twenty-four hours in a day, sixty minutes in an hour, and sixty seconds in a minute, so the total number of seconds for that year is:
Then the linear velocity, being the total linear distance divided by the total time and expressed as a unit rate, is:
Then, rounded to one decimal place, the linear velocity of the Earth is:
18.5 miles per second
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"Hey!" I hear you cry. "When are we gonna use angle measures for anything?" While many ("most"?) of the exercises in your book will probably be similar to the above, you may on occasion find yourself dealing with actual radians and degrees.
"A curve of radius 3000 feet" means that, if I'd tried to fit a circle snugly inside the curve, the best fit would have been a circle with a radius of r = 3000 feet. In other words, I can use circle facts to answer this question.
Since the radius of the curve is in feet and since I need to find the angle traversed in one minute, I'll start by converting the miles-per-hour speed to feet-per-second:
The amount of the curved track that the train covers is also a portion of the circumference of the circle. So this 880 feet is the arc length, and now I need to find the subtended angle of the (implied) circle sector:
But this value is in radians (because that's what the arc-length formula uses), and I need my answer to be in degrees, so I need to convert:
The train turns through an angle of about:
17°
Imagine that you were to stand at the center of that imaginary circle (that is, three thousand feet away from the curve, more than half a mile away) and watched the train moving along the curve. If you held your hand out at arm's length, made a tight fist, and, while firmly holding the middle fingers down with your thumb, raised your pinkie and index fingers, the distance between them would be about fifteen degrees. The train would move hardly more than that. Were you to hold your fist at arm's length and extend your pinkie and thumb, the distance would be about twenty-five degrees. The train would not exit your fingers in the time allotted.
(I sometimes learn the coolest stuff when I'm researching word problems. Then again, my definition of "cool" may be a bit sad....)
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