2nd derivative of angular displacement wrt time

  • #1
Prez Cannady
21
2
If ##\theta## is angular displacement, does ##\frac{d^2\theta}{dt^2} = (\frac{d\theta}{dt})^2##? Proof?
 

Answers and Replies

  • #2
Prez Cannady
21
2
Parameterized, I think not. A contradiction:

$$ \theta = sin(t) $$
$$ \frac{d\theta}{dt} = cos(t) $$
$$ \frac{d^2\theta}{dt^2} = -sin(t) $$
$$ \left(\frac{d\theta}{dt}\right)^2 = (cos(t))^2 $$
 
  • #3
olivermsun
Science Advisor
1,268
135
How about this:
$$
\begin{align}
\theta(t) &= t \\
\frac{d\theta}{dt} &= 1 \\
\frac{d^2\theta}{dt^2} &= 0 \neq \left( \frac{d\theta}{dt} \right)^2
\end{align}
$$
 
  • #4
Prez Cannady
21
2
Yeah. Dimensionally they agree because ##\theta## is dimensionless, but they're not equivalent. Thanks.
 
  • #5
FactChecker
Science Advisor
Gold Member
7,302
3,138
Try a couple of sanity checks of the proposed equation:
1) If the rotation rate is not changing, the second derivative is zero. Does that mean that it is not rotating at all?
2) Since the right hand side is always positive, does that mean that the rotation rate can only get more positive?
 
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