Animated Solution for Physics - Work, Energy, and Power: A force of $\mathbf{F}=(5y+20)\hat{\mathbf{j}}\text{ N}$ acts on a particle. The work done by this force when the particle is moved from $y=0\text{ m}$ to $y=10\text{ m}$ is ...... J.

Imagine a particle constrained to move along a straight line, specifically the y-axis. It starts its journey at the origin, where $y=0$, and travels upwards until it reaches $y=10$.

During this journey, it experiences a force given by $\mathbf{F}=(5y+20)\hat{\mathbf{j}}$.

Notice how the force is not a simple constant number. Because it contains the variable $y$, the magnitude of the force changes as the particle changes its position. This is the defining characteristic of a variable force.

When dealing with a constant force, we can simply multiply force by displacement. However, for a variable force, we must sum up the infinitesimally small amounts of work done over tiny displacements.

This requires the power of integration. The fundamental formula for work done by a variable force is:

Since our particle is moving exclusively along the y-axis, our infinitesimal displacement vector is simply $d\mathbf{r}=dy\hat{\mathbf{j}}$.

Substituting our given force and displacement into the integral, and setting our limits from the starting position to the final position, we get:

Because the dot product of $\hat{\mathbf{j}}\cdot \hat{\mathbf{j}}$ is exactly $1$, the vector equation beautifully simplifies into a standard scalar integral:

Now, we perform the mathematical heavy lifting. We integrate the polynomial term by term.

The integral of $5y$ with respect to $y$ is $\frac{5y^2}{2}$. The integral of the constant $20$ is simply $20y$.

Applying these rules, our expression becomes:

The final step is to evaluate this expression at our upper and lower limits. We substitute $y=10$ into the expression first, and then subtract the expression evaluated at $y=0$.

The lower limit evaluates entirely to zero, making our lives easier. For the upper limit, $10^2$ is $100$.

Multiplying $100$ by $\frac{5}{2}$ gives $250$. Adding this to $20\times 10$, which is $200$, we arrive at our final destination.