Animated Solution for Physics - Work, Energy, and Power: A force $\mathbf{F}=(5\hat{\mathbf{i}}+3\hat{\mathbf{j}}+2\hat{\mathbf{k}})\text{ N}$ is applied over a particle which displaces it from its origin to the point $\mathbf{r}=(2\hat{\mathbf{i}}-\hat{\mathbf{j}})\text{ m}$. The work done on the particle in joule is

Calculating Work Done

Work is a fundamental concept in physics, but it is often misunderstood. It is not just about exerting effort; it is about the successful transfer of energy. When a force pushes an object, only the component of the force that perfectly aligns with the object's displacement actually contributes to the work done. Any force pushing perpendicular to the motion does absolutely zero work!

Imagine a particle resting at the origin of a 3D coordinate system. Suddenly, a constant force $\mathbf{F}=5\hat{\mathbf{i}}+3\hat{\mathbf{j}}+2\hat{\mathbf{k}}$ Newtons acts upon it. This force pushes the particle, displacing it to a new position vector $\mathbf{r}=2\hat{\mathbf{i}}-\hat{\mathbf{j}}$ meters.

Notice that the displacement vector has no $\hat{\mathbf{k}}$ component. This means the particle did not move at all in the z-direction, even though the force had a z-component pushing it that way.

To calculate the work done by a constant force, we use the dot product (or scalar product) of the force vector and the displacement vector. The dot product is the perfect mathematical tool for this scenario because it automatically multiplies the parallel components and completely ignores the perpendicular ones.

The formula is elegantly simple:

Let's substitute our given vectors into the equation. To avoid any silly mistakes, we can explicitly write out the zero component for the displacement:

Now, we execute the atomic computations by multiplying the corresponding components: - X-components: $5\times 2=10$ - Y-components: $3\times (-1)=-3$ - Z-components: $2\times 0=0$

Summing these up gives us the total work done:

The final answer is $7\text{ Joules}$. This positive value indicates that, overall, the force transferred energy into the particle, aiding its motion.