91Ó°ÊÓ

In physics, a force vector represents both the magnitude and direction of a force acting on an object. Vectors are drawn as arrows, where the length represents the magnitude and the direction of the arrow shows the direction of the force.

In our exercise, the force vector \(\vec{F}=(4.0 \mathrm{~N}) \hat{\mathrm{i}}+c \hat{\mathrm{j}}\) indicates that the force has two components:

• \(F_x = 4.0 \mathrm{~N}\): The force acting in the horizontal direction (i).
• \(F_y = c \, \mathrm{~N}\): The force acting in the vertical direction (j).

The value of \(c\) changes the vertical component, which is crucial for calculating work done, as force in different directions affects work differently. Understanding these vectors helps determine how forces interact with displacement to do work.

The dot product is a way to multiply two vectors, which results in a scalar (a single number). It is essential in physics to calculate work done by a force. The dot product of two vectors \(\vec{A}\) and \(\vec{B}\) is calculated as:
\[\vec{A} \cdot \vec{B} = A_x \cdot B_x + A_y \cdot B_y\]

This formula multiplies each component of the vectors in the same direction and sums them up.

For the given problem, \(\vec{F} \cdot \vec{d}= F_x \times d_x + F_y \times d_y\), the dot product allows us to find the work done by multiplying the components: \(F_x = 4.0 \mathrm{~N}\) with \(d_x = 3.0 \mathrm{~m}\), and \(F_y = c\) with \(d_y = -2.0 \mathrm{~m}\). Understanding the dot product helps when calculating interactions involving directions, as only the components aligned with each other contribute to the work.

Displacement is a vector that represents how far an object has moved and in what direction. Unlike distance, which is scalar, displacement takes into account the direction as well.

In our scenario, the displacement vector \(\vec{d}=(3.0 \mathrm{~m}) \hat{\mathrm{i}}-(2.0 \mathrm{~m}) \hat{\mathrm{j}}\) shows that the object moves:

• \(3.0 \mathrm{~m}\) in the positive x-direction (right).
• \(-2.0 \mathrm{~m}\) in the negative y-direction (down).

This movement is crucial for work calculation, as work depends on both how much and in which direction an object moves under the influence of force. By understanding displacement, you know what part of the force contributes to doing the work.

Work done by a force on an object involves both the force acting on the object and its movement, or displacement. It's calculated using the formula for the dot product as:
\[W = F_x \times d_x + F_y \times d_y\]

In the example, with the displacement \(\vec{d}=(3.0 \mathrm{~m}) \hat{\mathrm{i}}-(2.0 \mathrm{~m}) \hat{\mathrm{j}}\) and the force \(\vec{F} = (4.0 \mathrm{~N}) \hat{\mathrm{i}} + c \hat{\mathrm{j}}\), the calculation becomes:
\[W = (4.0 \times 3.0) + (c \times -2.0)\]

This results in an expression for work: \(12.0 - 2.0c\). Work done can be positive, negative, or zero and is defined by how force is applied relative to movement. When conditions such as zero or specific work values (like 17 J or -18 J) are given, we adjust the force component \(c\) to meet the condition. These calculations show how forces modify energy and movement in practical terms.