Graphical Analysis Question - Torque, Angular Displacement

1. Apr 25, 2012

Tempest Desh

1. The problem statement, all variables and given/known data

The graph shows the behavior of an object with the moment of inertia of 18 kgm^2
a) Calculate the magnitude of the constant torque that acted on the system.
b) Through what angle did the object rotate between t=2 and t=20 seconds?

2. Relevant equations

http://homework.sdmesa.edu/cmona/195/PS/195THQ08S12.pdf [Broken]

Torque = (Moment of Inertia)(Angular Acceleration)

theta(final) = theta(initial) + (initial angular speed)(time) + (.5)(angular acceleration)(time^2)

3. The attempt at a solution

Since I'm given the moment of inertia, all I need to do is determine the angular acceleration of the object and substitute it into the equation for torque, along with the value for moment of inertia. I'm given a graph of the log of angular displacement vs. log time (the link I've provided...question 2). I'm not sure how to use this to determine the value(s) I'm looking for. Thanks in advance

Last edited by a moderator: May 5, 2017
2. Apr 25, 2012

Tempest Desh

Okay, so I've found the equation of the graph to be f(x) = x^2, as the y-values are squares of the x-values. I'm not sure where to go from here to find the angular speed and thence the angular acceleration. Any ideas?

3. Apr 25, 2012

collinsmark

I appreciate your confusion with this problem. The graph you were given is poorly labeled and is misleading.

The plot, as it was given to you, implies that the values given are logt and logθ. But this is not true. The values given are obviously t and θ directly. The tick marks, and format are displayed logarithmically, but given numbers on the axis are not logarithms!

For example, on the x-axis, the numbers given are 0.1, 1, 10, 100. Since the label on the x-axis is "log of time", the implication that the numbers given are 0.1 = log(t), 1 = log(t), 10 = log(t), 100 = log(t). Or another way of looking at it, is the numbers given are 100.1 sec, 101 sec, 1010 sec, 10100 sec. But this is not right!

The numbers given on the x-axis (i.e. 0.1, 1, 10, 100) are in units of seconds. The numbers given on the y-axis (0.01, 0.1, 1, 10, 100, 1000) are in units of radians.

There are two ways this graph can be fixed (one of the two but not both):
1. The title and labels should be reworded, getting rid of the "log of" in their description. The x-axis should be labeled "time (seconds)" and the y-axis should be labeled "angular displacement (radians)". The logarithmic nature of the plot is obvious by the tick-marks.
2. Or the labels should stay as they are already, but the number that are displayed should be changed to the log of the current numbers. In other words, the x-axis should have the numbers -1, 0, 1, 2, and the y-axis numbers should be replaced with -2, -1, 0, 1, 2, 3.

I prefer the former. But anyway, the confusion isn't your fault. It's the fault of whomever mislabeled the plot. You might want to print out this thread and bring it to your instructor to keep such mislabeling from happening in the future (it confuses students.)

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Okay, now let's get to the problem. The numbers on the x-axis are time in seconds, and the numbers on the y-axis are angular displacement in radians. Notice that there are a few places where the line crosses the tick marks:

t = 0.2 sec, θf = 0.04 rad
t = 1 sec, θf = 1 rad
t = 2 sec, θf = 4 rad

You also know (from your original post) that

$$\theta_f = \theta_i + \omega_i t + \frac{1}{2} \alpha t^2$$

So plug in each pair of t and θf to give you three simultaneous equations. You also have three unknowns: (θi, ωi and α). Three equations, three unkowns. You should be able to solve for the three unknowns. (Don't worry, some of the unknowns turn out to be trivial in this case making it easier.) That will give you the angular acceleration, α, that you are trying to find.

Last edited: Apr 25, 2012
4. Apr 26, 2012

Tempest Desh

Thanks for helping clear up that confusion. The problem states that the torque is constant, which implies that α should be constant, but I'm getting different accelerations at different times. I'm taking θi as 0.01 rad, with the ti as 0.1 seconds. What am I doing wrong (i.e. how do I go about finding i and ωi)? I have two other equations: ωf = ωi + αt and ωf2= ωi2 + 2α(θf - θi). Thanks again.

5. Apr 26, 2012

collinsmark

No, t is the independent variable. t0 = 0. There isn't even a t0 in the $\theta_f = \theta_i + \omega_i t + \frac{1}{2} \alpha t^2$ formula. Assume that at time 0, t = 0.

[Edit: okay, the truth is you could modify everything such that you use the equation $\theta_f = \theta_i + \omega_i (t-t_0) + \frac{1}{2} \alpha (t - t_0)^2$. But that is just making this problem way too difficult. I've worked out the problem, and I'm convinced it was meant to be a fairly simple problem. The bad part is that the given plot was labeled horribly making everything seem harder than it was meant to be. I believe that this problem was originally a very simple problem, and it technically still has rather simple solution. Unfortunately, it is needlessly complicated by a poorly made [and poorly labeled] plot showing the data. But I suggest trying to avoid the bad plot problems from making this problem more difficult than it needs to be. Think of it as a rather simple problem with a poor plot.]
There is no ωf that is of concern. Don't bother trying to find ωf for this particular problem.

The variable you're really concerned about is α. Concentrate on that.*

*(But part of the problem is determining the θi and ωi, so you can't really ignore those. The problem statement wording didn't give you the assurance that they are 0, even if they turn out to be 0. So you'll have to at least assure yourself what they are [even if they are 0], one way or another.

----------------

Okay, I'll help get you started. Recall

$$\theta_f = \theta_i + \omega_i t + \frac{1}{2} \alpha t^2$$

I'll plug in the first point:

$$0.04 \ \mathrm{[rad]} = \theta_i + \omega_i (\mathrm{0.2 \ [sec]}) + \frac{1}{2} \alpha (\mathrm{0.2 \ [sec]})^2$$

You plug in the other two.

You have three unknowns: θi, ωi, and α. Three unknowns. How many simultaneous equations do you need to solve for three unknowns? I just gave you the first one.

[Edit: there is a shortcut you might take here. It doesn't always work except for special cases. If you can find an α such that it satisfies $\theta_f = \frac{1}{2} \alpha t^2$ for all t, it essentially shows that θi and ωi are both zero. Generally, you might need to solve for θi and ωi. But if you assume that they they're zero and it fits with the given data, then that shows they're zero. (Reminder: You've already found that the plot shows y = x2, as you mentioned in your previous post, so keep going with that.)]

Last edited: Apr 26, 2012