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help me with this trig question

T

TheSiege

Diamond Member
verbatim:

Angular Speed: A car with a wheel radius of 14 inches is moving with a speed of 55 mph
find the angular speed in radians per second...
 
this is what i think it should look like

14*55*2(Pi)/3600

14 = wheel base
55 = speed
2(Pi) = 1 revolution
3600 = seconds in an hour
 
The best method for unit conversions is the "multiply by one" technique... as long as the numerator equals the denominator in a fraction, the value of the fraction is one. Thus, multiplying by 3feet/1yard is multiplying by one, or multiplying by 1yard / 3feet is also multiplying by one. Make sure your units cancel out.

Since you're starting with 55miles/hour, you can multiply by 1 hour / 360 seconds, or you can start by multiplying by 5280 feet / 1 mile.
The order really doesn't matter... just so that in the end, all the units cancel except the units you want.


This is also helpful for beginning trig... to convert radians to degrees or degrees to radians, you'd multiply by Pi radians / 180 degrees or multiply by 180 degrees / Pi radians. In the final answer, when working in radians, "radians" is not expressed.
 
p.s. after you have an answer, post it and I or someone else will verify if it's correct or not.

*cue person who comes in here and just does all the work for the OP*
 
Originally posted by: TheSiege
verbatim:

Angular Speed: A car with a wheel radius of 14 inches is moving with a speed of 55 mph
find the angular speed in radians per second...
Click to expand...

linear_speed = 55 mph = b inch/sec (use online conversion website to find b)

linear_speed = angular_speed x radius

so, angular_speed = w = linear_speed / radius

w = (b inch/sec) / (14 inch) = ? radians/sec
 
55MPH = 55 / (2*pi*R) revolutions / hour = [55 / (2*pi*R)] * (2*pi) radians / hour = 55 / (3600*R) radians / second.

If you do the dimensional analysis, this works out as follows:

(55 miles / hour) * 1 / (2*pi*R miles / revolution) = (55 miles / hour) * ((1 / 2*pi*R) revolutions / mile) = 55 / (2*pi*R) revolutions / hour

We also have 1 revolution = 2*pi radians, so that's why we multiply by 2*pi (X revolutions / hour * (2*pi) radians / revolution = X radians / hour). Then divide by the number of seconds in an hour (3,600) and you're done. All you have to do is convert 14 inches into miles, and since 1 mile = 63,360 inches (if you recall, 1 mile = 5280 feet = 12*5280 inches), then 14 inches * 1 mile / 63,360 inches = 14 / 63,360 miles. So the answer is just:

55 / (3600 * 14 / 63360) radians per second, which Google says is 69.1428571 rad / s.
 
Just think units. For the distance, convert miles to revolutions, then revolutions to radians. For the time, convert hours to seconds. If you do both steps properly, you'll get it right.

this is what i think it should look like

14*55*2(Pi)/3600

14 = wheel base
55 = speed
2(Pi) = 1 revolution
3600 = seconds in an hour
Click to expand...

Consider your proposed solution. If you work out the units here, you have:

[inches] * [miles / hour] * [radians / revolution] / [seconds / hour]

= [inches] * [miles] * [radians / revolution] / [seconds]

Doesn't make any sense, does it?
 
Here's a trick you'll remember forever. It's called dimensional analysis. Don't let the big words fool you. Someone else mentioned it above. You can always multiply a number by 1, which means you can multiply by any fraction so long as the numerator equals the denominator. 1 yard = 3 feet, therefore 1 yard / 3 feet = 1 = 3 feet / 1 yard

Do algebra first using this basic formula

v = r*w (v is linear velocity, r is radius, w is angular velocity)
w = v/r

Convert mph into inches per second using dimensional analysis

55 (miles / hour) =
55 (miles / hour) * (1 hour / 3600 sec) =
55 (miles / hour) * (1 hour / 3600 sec) * (63360 in / 1 mile) =
968 in/sec

The beauty about dimensional analysis is that you can see your units as you're working. Each time you get the same unit in the numerator and denominator, you can cancel them out like in algebra. So 55 (miles / hour) * (1 hour / 3600 sec) = 55/3600 (miles/sec) because the "hour" cancels out on the top and bottom

w = v/r
v = 968 in/sec
r = 14 in

w = 69.1 rad/sec (radians are considered dimensionless)
 
here's another way to think about it, since you already have the 968 feet per second that the wheel travels:

How many times does the wheel have to roll to reach 968 feet?

That would be 968 divided by the circumference of the wheel = 968/(2*14*Pi)
So, it does that many rotations per second. How many radians is that? 1 rotation = 2Pi radians, so multiply it by 2Pi. Your final answer is 968/14.

Looking at it the way I had it above, if you had used S=r*theta, you'd have 968 feet/second = 14 feet * theta.
Dividing by 14 feet, the feet would cancel out and you'd be left with the 69.1 per second. As I said somewhere up there, radians are an implied unit, not always written.
 
I thought for a few minutes and realized that you may not know the v = r*w formula and that you may be expected to use an alternate approach.

55 mph = how many revolutions per second? You can find this from the circumference of the tire and by converting hours to seconds.

55 (miles / hour) = 55 (miles / hour) * (1 hour / 3600 sec) * (63360 in / 1 mile) * (1 revolution / 2*pi*14 in) * (2*pi rad / 1 revolution) = 69.1 rad/sec

Just remember that 1 revolution is equal to the circumference of the tire, and 1 revolution is also equal to 2pi radians. The rest should be obvious conversions (63360 inches in a mile, 3600 sec in an hour)

You'll notice that this is exactly the same solution as we found with the radial velocity equation (v = r*w) using nothing but dimensional analysis. If you do anything with math and science, you should always remember the power of units and what you can do with them.
 
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