A loaded grocery cart is rolling across a parking lot in a...

A loaded grocery cart is rolling across a parking lot in a strong wind. You apply a constant force $\vec{F}=(30 \mathrm{N}) \hat{\imath}-(40 \mathrm{N}) \hat{\mathrm{J}}$ to the cart as it undergoes a displacement $\vec{s}=(-9.0 \mathrm{m}) \hat{\boldsymbol{\imath}}-(3.0 \mathrm{m}) \hat{\boldsymbol{J}}$ . How much work does the force you apply do on the grocery cart?

Key Concepts

Vector Components

Breaking vectors into components involves expressing them in terms of unit vectors along standard axes. This process simplifies calculations by allowing operations on each dimension independently. It is crucial for analyzing forces and displacements in different directions and for applying operations like the dot product effectively.

Constant Force

A constant force is one whose magnitude and direction do not change over time. When such a force acts over a displacement, the work done can be computed straightforwardly using the dot product, as the force remains uniform throughout the motion. This concept is fundamental in resolving many problems in mechanics.

Dot Product

The dot product is an algebraic operation that takes two equal-length vectors and returns a single number. In physics, it is used to calculate work by multiplying the magnitudes of the force and displacement vectors and the cosine of the angle between them. This operation helps isolate the component of the force that is effective in doing work.

Work

Work represents the transfer of energy resulting from a force acting over a displacement. Defined as the dot product of the force and displacement vectors, it quantitatively describes how much of the force contributes to movement along its direction. It is a scalar quantity and can be positive, negative, or zero depending on the alignment of the force and motion.

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