open 30

#$&*

course phy 201

this was not difficult concept to understand

If your solution to stated problem does not match the given solution, you should self-critique per instructions at

http://vhcc2.vhcc.edu/dsmith/geninfo/labrynth_created_fall_05/levl1_22/levl2_81/file3_259.htm.

Your solution, attempt at solution.

If you are unable to attempt a solution, give a phrase-by-phrase interpretation of the problem along with a statement of what you do or do not understand about it. This response should be given, based on the work you did in completing the assignment, before you look at the given solution.

030. Rotational Motion

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Question: `q001. Note that this assignment contains 4 questions.

If an object rotates through an angle of 20 degrees in five seconds, then at what rate is angle changing?

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Your solution:

20 degree 5 seconds

find radians and divide by time

20/180 x pi = 1/9 pi / 5 = .022 m/s

confidence rating #$&*:

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Given Solution:

The change of 20 degrees in 5 seconds implies a rate of change of 20 degrees / (5 seconds) = 4 deg / sec. We call this the angular velocity of the object, and we designate angular velocity by the symbol `omega.

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Self-critique (if necessary):

why don’t we need to find radians?

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Self-critique rating: 2

@& 4 degrees / second is a perfectly good angular velocity. That solution was all that was required here. Be sure you understand this first.

We can go further with the units here, and it turns out that your instinct is a good one. The radian / sec is usually the better unit of angular velocity:

Since radians convert very easily to arc distance we're usually better off working in radians / second.

To find the number of radians per second we first note that 20 degrees is 20 / 180 * pi radians. Your calculation was correct; this could be expressed as 20 / 180 * pi since the radian is the default measure of angle.

However by not writing down the unit 'radian' your next step came out with incorrect units.

The object does rotate through 1/9 pi radians. This is to be divided by 5 seconds, not just by 5.

The correct calculation is ( 1/9 pi radians ) / 5 seconds = pi / 45 radians / second, which does approximate to about .022 radians / sec.

The unit m / s is not correct. The rotation is measured in radians and the angular velocity (i.e., rate of rotation) in radians / second.

We could also have converted 4 deg / sec to radians. 4 deg is 4 / 180 * pi radians, so

4 deg / sec = (4 / 180 * pi) radians / sec.

This reduces to pi/45 rad / sec, as before.*@

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Question: `q002. What is the average angular velocity of an object which rotates through an angle of 10 `pi radians in 2 seconds?

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Your solution:

10 pi radians / 2 seconds = 5 pi radians/ s

confidence rating #$&*:

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Given Solution:

The average angular velocity is equal to the angular displacement divided by the time required for that displacement, in this case giving us

`omega = `d`theta / `dt = 10 `pi radians / 2 seconds = 5 `pi rad/s.

STUDENT QUESTION

I write 5’pi radians as 15.7 radians. I know they equal each other, but would you rather see me write it as 5’pi??

INSTRUCTOR RESPONSE

5 pi is exact and 15.7 is not. The rounding error in the approximation 15.7 might or might not be significant in a given situation.

Also it's easy to see how 5 pi is related to the conditions of the problem; 15.7 is not as obviously related.

So in this case the multiple-of-pi notation is preferable, though either would be acceptable.

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Self-critique (if necessary):

when do you change to radians?

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Self-critique rating: 3

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Question: `q003. If an object begins with an angular velocity of 3 radians / sec and ends up 10 seconds later within angular velocity of 8 radians / sec, and if the angular velocity changes at a constant rate, then what is the average angular velocity of the object? In this case through how many radians this the object rotate and at what average rate does the angular velocity change?

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Your solution:

3 radians/ sec- 8 radians/ sec = 5 radians /sec

5 radians/sec / 10 secon = .5 radians /s ^2

confidence rating #$&*:

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Given Solution:

Starting at 3 rad/s and ending up at 8 rad/s, the average angular velocity would be expected to be greater than the minimum 3 rad/s and less than the maximum 8 rad/s. If the angular velocity changes at a constant rate, we would in fact expect the average angular velocity to lie halfway between 3 rad/s and 8 rad/s, at the average value (8 rad/s + 3 rad/s) / 2 = 5.5 rad/s.

Moving at this average angular velocity for 10 sec the object would rotate through 5.5 rad/s * 10 s = 55 rad in 10 sec.

The change in the angular velocity during this 10 seconds is (8 rad/s - 3 rad/s) = 5 rad/s; this change takes place in 10 seconds so that the average rate at which the angular velocity changes must be ( 5 rad / sec ) / (10 sec) = .5 rad/s^2. This is called the average angular acceleration.

Angular acceleration is designated by the symbol lpha. Since the angular velocity in this example changes at a constant rate, the angular acceleration is constant and we therefore say that

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Your solution:

only found the average angular acceleration. but not the average

confidence rating #$&*:

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Given Solution:

`alpha = `d `omega / `dt.

Again in this case `d`omega is the 5 rad/sec change in the angular velocity.

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Self-critique (if necessary):

I think this is missing the question

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Self-critique rating:

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Question: `q004. If an object starts out with angular velocity 14 rad/s and accelerates at a rate of 4 rad/s^2 for 5 seconds, then at what rate is the object rotating after the 5 seconds? Through how many radians will the object rotate during this time?

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Your solution:

vf = v0 + ads

vf = 14 rad/s + 4 rad/s^2 (5s)

vf= 34 rad/s

confidence rating #$&*:

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Given Solution:

Changing angular velocity at the rate of 4 rad/s^2 for 5 sec the angular velocity will change by (4 rad/s^2) (5s) = 20 rad/s.

Since the angular velocity was already 14 rad/s at the beginning of this time period, it will be 14 rad/s + 20 rad/s = 34 rad/s at the end of the time period.

The uniform rate of change of angular velocity implies that the average angular velocity is (14 rad/s + 34 rad/s) / 2 = 24 rad/s.

An average angular velocity of 24 radians/second, in 5 seconds the object will rotate through an angle `d`theta = (24 rad/s) ( 5 sec) = 120 rad.

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Self-critique (if necessary):

what’s wrong with using this formula vf = v0 + ads ?

@& Your idea is correct, and those formulas do pretty much apply. However we're talking about rotation here, so instead of v we use omega, and instead of a we use alpha. Instead of `ds we use `dTheta.

The equation vf = v0 + a `dt (note that your formula vf = v0 + a `ds is not correct; in your solution you actually did use `dt instead of `ds so your solution is fine) is therefore translated from vf = v0 + a `dt to

omega_f = omega_0 + alpha `dt

All the other equations of motion translate similarly. For example the equation vf^2 = v0^2 + 2 a `ds becomes omega_f^2 = omega_0^2 + 2 alpha `dTheta.*@

The uniform rate of change of angular velocity implies that the average angular velocity is (14 rad/s + 34 rad/s) / 2 = 24 rad/s.

didn’t include this part.

An average angular velocity of 24 radians/second, in 5 seconds the object will rotate through an angle `d`theta = (24 rad/s) ( 5 sec) = 120 rad.

so this is the same as finding the distance, right?

@& This is completely analogous to finding displacement. However we're now talking about angular displacement, so to remind us of the fact we put the relationships into angular notation.

omega_Ave = `dTheta / `dt, by the definition of aveage angular velocity as average rate of change of angular position with respect to clock time.

So by a simple rearrangement

`dTheta = omega_Ave * `dt.

In this date

`dTheta = 24 rad / s * 5 sec = 120 rad.*@

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Self-critique rating: 3

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#*&!

&#Good responses. See my notes and let me know if you have questions. &#