Animated Solution for Physics - Laws of Motion: Three forces start acting simultaneously on a particle moving with velocity $\mathbf{v}$. These forces are represented in magnitude and direction by the three sides of a $\Delta ABC$ (as shown). The particle will now move with velocity

A Particle Under Multiple Forces

Imagine a particle cruising through space with a constant velocity $\mathbf{v}$. Suddenly, three distinct forces start acting on it simultaneously. At first glance, this might seem like a recipe for chaos. However, the problem gives us a crucial piece of geometric information: these three forces can be represented in magnitude and direction by the three sides of a triangle $ABC$, taken in the same order.

What does "taken in the same order" mean? It means if you place the vectors tip-to-tail, they form a continuous loop. For our triangle $ABC$, the forces are $\overrightarrow{AB}$, $\overrightarrow{BC}$, and $\overrightarrow{CA}$.

To understand the combined effect of these forces, we need to find their vector sum, which is the net force ${\mathbf{F}}_{net}$ acting on the particle.

According to the polygon law (or triangle law) of vector addition, if a set of vectors can be arranged to form a closed polygon with their tips and tails connecting in a continuous sequence, their resultant vector is exactly zero.

Mathematically, we can write this as:

Since the head of the last vector ($\overrightarrow{CA}$) meets the tail of the first vector ($\overrightarrow{AB}$), the total displacement in vector space is zero. Therefore:

Now that we know the net force acting on the particle is zero, we can determine its subsequent motion using Newton's Second Law of Motion. The law states that the net force on an object is equal to its mass times its acceleration:

Since ${\mathbf{F}}_{net}=0$, and the mass $m$ is non-zero, the acceleration $\mathbf{a}$ must be zero:

Acceleration is defined as the rate of change of velocity ($\frac{d\mathbf{v}}{dt}$). If the acceleration is zero, it means the velocity is not changing with respect to time.

Because the velocity is not changing, the particle will continue to move with its original velocity $\mathbf{v}$. The three forces perfectly balance each other out, maintaining the particle in a state of dynamic equilibrium.