Math Puzzle from February 15, 2019

Can you prove the Law of Sines using physics?

Written on February 15, 2019 by Noble Mushtak

The following was a puzzle presented to Marshwood GT students on February 15, 2019. Have fun doing math!

In AP Physics, you've probably learned about conservation of angular momentum, which says that angular momentum within a system of objects does not change if no external force acts upon the system. Today, we will use conservation of angular momentum to prove the Law of Sines.

Part 1:

$Diagram of forces acting on two point-masses$

In the above diagram, there is a system of two point-masses of mass $m_1$ and $m_2$. Furthermore, angular momentum and torque are being measured relative to the origin $O$, and masses $m_1, m_2$ are $r_1, r_2$ away from the origin. There is a force $\vec{F}$ acting on $m_1$ and a force $-\vec{F}$ acting on $m_2$. In other words, the force acting on $m_2$ is equal and opposite to that acting on $m_1$.

Given that there are no external forces acting on the system, explain why the force acting on $m_2$ must be equal and opposite to that acting on $m_1$. Moreover, explain why both force vectors must be parallel to the line between $m_1$ and $m_2$.

Part 2:

$Diagram of perpendicular components of forces acting on two-point masses$

In the new diagram, $F_{1,\perp}$ represents the component of $\vec{F}$ perpendicular to $r_1$ and $F_{2,\perp}$ represents the component of $-\vec{F}$ perpendicular to $r_2$. Using your knowledge of torque and conservation of angular momentum, show that:

$$r_1F_{1,\perp}-r_2F_{2,\perp}=0\implies r_1F_{1,\perp}=r_2F_{2,\perp}$$

Part 3:

$Diagram of angles between force vectors acting on two point-masses$

Finally, we've labeled four of the angles in the diagram: $A,B,C,D$. Using your knowledge of geometry and trigonometry, show that:

$$F_{1,\perp}=\lvert \vec{F}\rvert\sin A$$ $$F_{2,\perp}=\lvert \vec{F}\rvert\sin B$$

You do not need Law of Sines to prove either of these equations. Note that $\lvert \vec{F}\rvert$ represents the length of the force vector $\vec{F}$.

Part 4: Combining Parts 2 and 3, we now have:

$$r_1\lvert \vec{F}\rvert\sin A=r_2\lvert \vec{F}\rvert\sin B$$

Use this equation to prove the Law of Sines:

$$\frac{\sin A}{r_2}=\frac{\sin B}{r_1}$$