Resultants of forces and velocities

Adding forces and velocities as vectors, the resultant of two forces using components or the cosine rule, forces in equilibrium, and boats and planes moving through currents and winds.

Resultant forces

Forces are vectors, so the resultant of several forces is their vector sum. A particle is in equilibrium when the resultant is $\underset{\sim}{0}$.

Resultant of two forces at angle θ

$$|\underset{\sim}{F}|^2 = a^2 + b^2 + 2ab\cos\theta$$

This comes from $|\underset{\sim}{a} + \underset{\sim}{b}|^2 = |\underset{\sim}{a}|^2 + 2\,\underset{\sim}{a} \cdot \underset{\sim}{b} + |\underset{\sim}{b}|^2$. Notice the plus sign: $\theta$ is the angle between the forces, not the angle in the triangle.

Example 1●●

Forces of $6$ N and $8$ N act on a particle at $60^\circ$ to each other. Find the magnitude of the resultant.

$|\underset{\sim}{F}|^2 = 36 + 64 + 2 \times 6 \times 8 \times \tfrac12 = 148$.

$|\underset{\sim}{F}| = 2\sqrt{37} \approx 12.17$ N.

Example 2●

Forces $\begin{pmatrix} 3 \\ 5 \end{pmatrix}$ N and $\begin{pmatrix} -1 \\ 2 \end{pmatrix}$ N act on a particle. Find the force that keeps it in equilibrium.

The sum is $\begin{pmatrix} 2 \\ 7 \end{pmatrix}$, so the third force must be $\begin{pmatrix} -2 \\ -7 \end{pmatrix}$ N.

Combining velocities

A boat moving through water that is itself moving has a resultant velocity equal to the boat's velocity relative to the water plus the velocity of the current. Planes in wind work the same way.

Example 3●●

A boat heads straight across a river at $4$ m/s. The current flows downstream at $3$ m/s. Find the boat's actual speed, and the angle its path makes with the bank.

Speed $= \sqrt{4^2 + 3^2} = 5$ m/s.

$\tan\alpha = \tfrac43$, so the path makes about $53^\circ$ with the bank.

Flashcards

Resultant

The vector sum of all the forces (or velocities).

Equilibrium

The resultant force is $\underset{\sim}{0}$.

Two forces $a$ and $b$ at angle $ heta$

$|\underset{\sim}{F}|^2 = a^2 + b^2 + 2ab\cos\theta$

Practice

Forces of $6$ N and $8$ N act at $60^\circ$ to each other. Find the magnitude of the resultant. Give an exact answer or a decimal.
Forces of $6$ N and $8$ N act at right angles. Find the magnitude of the resultant.
Forces of $6$ N and $8$ N act in opposite directions. Find the magnitude of the resultant.
Find the force that balances $\begin{pmatrix} 3 \\ 5 \end{pmatrix}$ N and $\begin{pmatrix} -1 \\ 2 \end{pmatrix}$ N. Enter it as $x, y$.
A boat heads across a river at $4$ m/s, and the current flows at $3$ m/s. Find the boat's actual speed.
Forces of $3$ N and $5$ N act at $60^\circ$ to each other. Find the magnitude of the resultant.
A plane flies at $300$ km/h due north into a wind of $50$ km/h blowing due south. Find its ground speed in km/h.
Three forces keep a particle in equilibrium. Their sum is:
The largest possible resultant of forces of $5$ N and $12$ N is: