On a farm, you are pushing on a stubborn pig with a constant...

On a farm, you are pushing on a stubborn pig with a constant horizontal force with magnitude 30.0 N and direction 37.0$^\circ$ counterclockwise from the +$x$-axis. How much work does this force do during a displacement of the pig that is (a) $\overrightarrow{s} = (5.00 \, \mathrm{m})\hat{\imath}$; (b) $\overrightarrow{s} = - (6.00 \, \mathrm{m})\hat{\jmath}$; (c) $\overrightarrow{s} = - (2.00 \, \mathrm{m})\hat{\jmath} + (4.00 \, \mathrm{m}) \hat{\jmath}$?

Key Concepts

Force as a Vector

Force is a vector quantity that has both magnitude and direction. In problems involving work, it is crucial to resolve the force into its components, which allows for accurate determination of how much of the force is acting in the direction of the displacement. Understanding the vector nature of force assists in applying trigonometry to find the necessary components.

Displacement Vector

The displacement vector represents a change in position and, like force, has both magnitude and direction. In work calculations, expressing displacement in vector form enables the use of the dot product to combine it with the force vector. Analyzing problems in terms of vector components is essential for determining the work done when the force and displacement are not aligned.

Work

Work in physics is defined as the scalar product (or dot product) of the force and displacement vectors. It quantifies the amount of energy transferred by a force acting through a distance. The work done depends not only on the magnitudes of the force and displacement, but also on the cosine of the angle between them, indicating how much of the force contributes to moving an object in the direction of the displacement.

Dot Product of Vectors

The dot product is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors) and returns a single number. In the context of work, the dot product multiplies the corresponding components of the force and displacement vectors and sums the products; it inherently includes the cosine of the angle between the vectors, which determines the effective component of the force in the direction of the displacement.

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