When solving systems of equations for first-year physics, I am told that to be able to find the solution for a system of $n$ unknowns (if one exists), I need at least $n$ equations.
In this system, I am looking for $a$. The known quantities are $\theta_0$ and $\theta_1$.
$$ T_0 \cos \theta_0 = mg $$
$$ T_0 \sin \theta_0 = \frac{m v ^{2}}{\ell \sin \theta_0} $$
$$ T_1 \cos \theta_1 – mg = ma $$
$$ T_1 \sin \theta_1 = \frac{m v ^{2}}{\ell \sin \theta_1} $$
In this system of equations, I had 5 unknown quantities: $T_0$, $T_1$, $m$, $\ell$, $a$. But I was able to successfully solve for $a$ with four equations. Where is the logical error in my understanding?
Best Answer
You can solve for $\ell$ from the start by plugging in the first equation to the second equation, so I don't think you made a mistake. Ie. from the first equation get an expression for m, plug into second equation, eliminate T0 and then just compute $\ell$.