What is the formula for finding angular velocity from a graph?

[goonking]

Homework Statement

Homework Equations

The Attempt at a Solution

the displacement should be the area under the line , it was traveling backwards for 3 seconds, then it would take an additional 3 seconds to get back to the starting position. From there, the wheel rotates forward for 4 more seconds for a total of 10 seconds.

So do I need to find the area of the entire triangle (from seconds 0 to 10) then subtract the smaller triangle (from seconds 0 to 6) to be left with only the area of (seconds 6 to 10)?

 

[goonking]

I tried starting from second 6 (because it would be at position = 0)

then the wheel rotates for 4 seconds , and that should be the displacement, is that correct?

 

[Simon Bridge]

was traveling backwards for 3 seconds

You mean it was rotating in the negative direction ... maybe the negative direction was "clockwise", we are not told.

So do I need to find the area of the entire triangle (from seconds 0 to 10) then subtract the smaller triangle (from seconds 0 to 6) to be left with only the area of (seconds 6 to 10)?

You can do it either way - if you do it the second way, notice that the angular velocity at t=6s will not be zero.
The area inside 6s<t<10s will give you the total displacement.

You can also do it by figuring the equation of the line and/or using suvat equations.
Basically pick one ... if you are uncertain, try two methods and see if they agree.

 

[goonking]

You mean it was rotating in the negative direction ... maybe the negative direction was "clockwise", we are not told.
You can do it either way - if you do it the second way, notice that the angular velocity at t=6s will not be zero.
The area inside 6s<t<10s will give you the total displacement.

You can also do it by figuring the equation of the line and/or using suvat equations.
Basically pick one ... if you are uncertain, try two methods and see if they agree.

I did the first way:

area of whole triangle = 10 x 30 x .5 = 150

area of small triangle = 6 x 18 x .5 = 54

whole triangle minus small triangle = 150 - 54 = 96
is this what I'm suppose to get?
is that 96 already in radians?

 

[NascentOxygen]

area of whole triangle = 10 x 30 x .5 = 150

area of small triangle = 6 x 18 x .5 = 54

You won't get the right answer.

I think it is better to go:
triangle of positive area - triangle of negative area
7 x 21 x 0.5 - 3 x 9 x 0.5

 

[goonking]

You won't get the right answer.

I think it is better to go:
triangle of positive area - triangle of negative area
7 x 21 x 0.5 - 3 x 9 x 0.5

yes, your way is correct but i can't figure why the way I calculated mine came out to be wrong.

 

[NascentOxygen]

yes, your way is correct but i can't figure why the way I calculated mine came out to be wrong.

Sketch the graph and attach it here. Geometry will show where you are wrong.

 

[goonking]

tried my best to draw out the graphs with only ms paint.

so this is what I was suppose to do, the green triangle should be above the x axis. then subtract the red triangle from the green.

here is what I did, which was wrong, I took the area of the whole triangle, and subtracted the red triangle. As you can see, the area of green is much bigger.

 

[jbriggs444]

tried my best to draw out the graphs with only ms paint.

Note that the red triangle that you evaluated the area of is the wrong one. You want the area above the slanted line and below the x axis.

That is a cosmetic error, however, because the area of both those triangles will be equal.

 

[NascentOxygen]

Very good.

So, starting with the area of that monster triangle, subtract the red triangle, then subtract it again, then deduct the rectangular area of green that you don't want, should, in a roundabout way, give the answer, too!

Much better to sketch the graph right from the start so that you can see what you are doing.

 

[Simon Bridge]

Just to be clear...
The area in "the displacement is the area under the graph" is between the graph and the time axis.
Thus the triangle inside 0<t<3s has height -9rad/s and base 3s, so it's area is -9(rad/s)x3(s)/2 = -13.9 rad.
Notice how the units multiply just the same as the numbers do, so (rad/s) times (s) = (rad)?

Note: since you were not given "the whole triangle" as part of the problem statement, you had to do some working to derive/deduce it.
You should show that part of your working in your written answer too.

 

[BvU]

Dear king of goons,

I think it's good to remind you of Simon's post #3: once all the artwork is done and you have an answer you feel confident about, you should take the extra step and find the same answer "figuring the equation of the line and/or using suvat equations". It'll give you a good experience to feel the power of abstraction.