Graphical Analysis Question - Torque, Angular Displacement

[Tempest Desh]

Homework Statement

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?

Homework Equations

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

Torque = (Moment of Inertia)(Angular Acceleration)

tangential speed = (radius)(angular speed)

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

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

 

[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?

 

[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 10^0.1 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, θ_f = 0.04 rad
t = 1 sec, θ_f = 1 rad
t = 2 sec, θ_f = 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 θ_f to give you three simultaneous equations. You also have three unknowns: (θ_i, ω_i 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.

 

[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 θ_i as 0.01 rad, with the t_i as 0.1 seconds. What am I doing wrong (i.e. how do I go about finding _i and ω_i)? I have two other equations: ω_f = ω_i + αt and ω_f²= ω_i² + 2α(θ_f - θ_i). Thanks again.

 

[collinsmark]

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 θ_i as 0.01 rad,

What leads you to think that θ_i = 0.01 rad?

with the t_i as 0.1 seconds.

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.]

What am I doing wrong (i.e. how do I go about finding _i and ω_i)? I have two other equations: ω_f = ω_i + αt and ω_f²= ω_i² + 2α(θ_f - θ_i). Thanks again.

There is no ω_f that is of concern. Don't bother trying to find ω_f for this particular problem.

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

*(But part of the problem is determining the θ_i and ω_i, 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.

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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: θ_i, ω_i, 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 θ_i and ω_i are both zero. Generally, you might need to solve for θ_i and ω_i. 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.)]