I 2nd derivative of angular displacement wrt time

If ##\theta## is angular displacement, does ##\frac{d^2\theta}{dt^2} = (\frac{d\theta}{dt})^2##? Proof?
 
Parameterized, I think not. A contradiction:

$$ \theta = sin(t) $$
$$ \frac{d\theta}{dt} = cos(t) $$
$$ \frac{d^2\theta}{dt^2} = -sin(t) $$
$$ \left(\frac{d\theta}{dt}\right)^2 = (cos(t))^2 $$
 

olivermsun

Science Advisor
1,244
117
How about this:
$$
\begin{align}
\theta(t) &= t \\
\frac{d\theta}{dt} &= 1 \\
\frac{d^2\theta}{dt^2} &= 0 \neq \left( \frac{d\theta}{dt} \right)^2
\end{align}
$$
 
Yeah. Dimensionally they agree because ##\theta## is dimensionless, but they're not equivalent. Thanks.
 

FactChecker

Science Advisor
Gold Member
2018 Award
4,827
1,649
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?
 

Want to reply to this thread?

"2nd derivative of angular displacement wrt time" You must log in or register to reply here.

Physics Forums Values

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving

Hot Threads

Top