91Ó°ÊÓ

When a coin moves from one point to another on a plane, it undergoes a displacement. The displacement vector helps us describe this movement in terms of both direction and magnitude. In this example, the coin moves from the origin \(0, 0\) to \(3.0 \, m, 4.0 \, m\). So, the displacement vector \( \vec{d} \) is \( 3.0 \, \hat{i} \, + \, 4.0 \, \hat{j} \, m \). This means the coin moves 3 meters in the x-direction and 4 meters in the y-direction.
By finding the magnitude of this vector using the Pythagorean theorem, we can comprehend the total distance traveled: \[ |\vec{d}| = \sqrt{(3.0 \, m)^2 + (4.0 \, m)^2} \approx 5.0 \, m \].
This information is valuable as it gives us a clear idea of how far the coin has moved from its starting point.

Forces that act at an angle need to be broken down into their respective components to better understand their effects. The force \( \vec{F} \) acting on our coin has a magnitude of 2.0 N and is directed at 100° counterclockwise from the positive x-axis.
Using trigonometry, we resolve this force into its x and y components:
\[ F_x = \vec{F} \, \cos(100°) \approx -0.342 \, N \] \[ F_y = \vec{F} \, \sin(100°) \approx 1.880 \, N \]
The negative sign in \( F_x \) indicates that the force acts in the negative x-direction, while the positive value in \( F_y \) shows it acts in the positive y-direction. Understanding these components is key to analyzing how the force affects the motion of the coin.

Trigonometry plays a crucial role in physics, especially when breaking down vectors into their components. In this scenario, we used trigonometric functions like cosine and sine to resolve the force acting on the coin into its x and y components. Here’s how:
\[ F_x = 2.0 \, N \cos(100°) \approx -0.342 \, N \] \[ F_y = 2.0 \, N \sin(100°) \approx 1.880 \, N \]
The angle given (100°) is with respect to the positive x-axis, so cosine is used to determine the adjacent (x direction) component and sine for the opposite (y direction) component.
Using these functions helps us accurately break forces and displacements into manageable parts, simplifying complex problems in physics.

The dot product is a mathematical operation that tells us how much one vector goes in the direction of another. When computing work done by a force, we often use the dot product between the force vector \( \vec{F} \) and the displacement vector \( \vec{d} \).
In our problem, work done \( W \) is calculated as:
\[ W = \vec{F} \cdot \, \, \, \vec{d} \]
This can be expressed in terms of components:
\[ W = F_x \cdot d_x + F_y \cdot d_y \]
By substituting in our values:
\[ W = (-0.342 \, N \cdot 3.0 \, m) + (1.880 \, N \cdot 4.0 \, m) \approx -1.026 \, J + 7.520 \, J = 6.494 \, J \]
The result tells us the amount of energy transferred to the coin through the force during its displacement.

A coordinate system is like a map that helps us understand the position and movement of objects. In this example, we use an x-y coordinate system to trace the coin's movement from the origin (0,0) to (3.0 m, 4.0 m). Understanding how to navigate this system is essential.
Here’s a quick refresher:

• The x-component represents horizontal movement.
• The y-component represents vertical movement.

By plotting these points, the displacement vector \( \vec{d} \) appears, showing us the direct path the coin travels.
In physics, coordinate systems help us break down problems into smaller, more manageable pieces. This way, we can solve them step-by-step, just as we did with the coin problem.