1. I must have made a mistake, can you guys help me find it?

Imagine a turntable, of fixed radius, rotating with constant angular acceleration.

I was trying to plot the motion of an object stuck to the edge of the turntable. So let's consider the displacement-time graph of the object in the x-direction.

We know that the graph of radians-time is quadratic. The displacement-time graph is the "sine" of the radians-time graph. But in doing so, the gradient of the sine graph abruptly changes value.

This implies a sudden change in velocity, and thus infinite acceleration.

But of course, we know that we can angularly accelerate turntables without infinite acceleration.

I haven't been able to resolve this issue.  2.

3. let me see if I understand you:
if graph of radians-time is quadratic say w=t^2 (ignoring multiplying and adding constants), then the displacement-time graph is x=sin(t^2)?
And your problem is that the gradient of x=sin(t^2) abruptly changes values?

But the gradient of x=sin(t^2) doesn't abruptly change values...
dx/dt= 2t cos(t^2)

Not sure if that helps, but maybe if you give a little bit more information someone might be better able to help.  4. So angular acceleration is constant, angular velocity is increasing linearly and radians quadratically.

Now, I "chopped" the quadratic graph into segments. That way, when I "sine" each segment, I get a sine wave that has increasing frequency. Otherwise my GC gives me nonsense.

And where each segment of the sine wave meets the x-axis, the gradient changes abruptly.  5. Ok. I think the problem might be with the way you used your graphics calculator. I don't really understand what you mean by "chopped" into segments.
If you type "plot sin(x^2)" into Google you get a plot of what is a continuous and differentiable graph. The gradient of this graph does not change abruptly when meeting the x-axis?
(By change abruptly at a point, I mean in a way that the gradient is undefined at that point)  6. Is the graph supposed to have a constant amplitude as well?

Maybe my TI-84 fails. I can only plot what google gives me if I chop the graph into segments. When I do the math, the gradient changes abrubtly.  7. The graph is meant to have a constant amplitude. Because the y-axis is the displacement. So if you only consider a single dimension of the motion of an object on your turntable, the object oscillates. And the gradient of the motion can be found by differentiation.

As in my previous post, if x=sin(t^2) then dx/dt= 2t cos(t^2).
2t cos(t^2) is a continuous function. So doing the maths, the gradient does not change abruptly as far as I can tell.  8. cool. so i messed it up by "chopping up" my graph. too bad my ti-84 can't plot it like google does.  Bookmarks
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