Problem 31 - College Physics

Question

On a single diagram, carefully sketch each force vector to scale and identify its magnitude and direction on your drawing: (a) \(60 \mathrm{lb}\) at \(25^{\circ}\) east of north, (b) \(40 \mathrm{lb}\) at \(\pi / 3\) south of west, (c) \(100 \mathrm{lb}\) at \(40^{\circ}\) north of west, (d) \(50 \mathrm{lb}\) at \(\pi / 6\) east of south.

Step-by-Step Solution

Answer

Draw each vector from the origin as described and label with magnitude and direction.

Step 1: Understand the Coordinate System

To sketch vectors on a 2D plane, imagine a standard coordinate system where North is typically up (positive y-axis), East is to the right (positive x-axis), South is down (negative y-axis), and West is to the left (negative x-axis).

Step 2: Sketch Vector (a)

Vector (a) is 60 lb at 25° east of north. From the origin, draw a vector towards the north and then rotate it 25° towards the east. Mark the vector to show it is 60 lb. The direction will make an angle of 25° from the vertical y-axis.

Step 3: Sketch Vector (b)

Vector (b) is 40 lb at \( \pi/3 \) (or 60°) south of west. From the origin, initially point towards the west (left) and rotate the vector 60° downwards towards the south. Mark the vector to show it is 40 lb.

Step 4: Sketch Vector (c)

Vector (c) is 100 lb at 40° north of west. From the origin, point first towards the west and rotate the vector 40° upward towards the north. Mark the vector as 100 lb. This vector will be in the second quadrant.

Step 5: Sketch Vector (d)

Vector (d) is 50 lb at \( \pi/6 \) (or 30°) east of south. From the origin, point downward towards the south and rotate it 30° towards the east. Mark this vector as 50 lb. This vector should be in the fourth quadrant.

Step 6: Finalize the Diagram

Ensure all vectors are properly labeled with their magnitudes and directions. Check that the angles are aligned correctly according to the cardinal points and magnitude is proportional to the vector length as closely as possible in your sketch.

Key Concepts

Vector Magnitude

Understanding vector magnitude is crucial for solving physics problems involving forces. The magnitude of a vector refers to its length in the coordinate system. For example, if we have a force vector of 60 lb, its magnitude is 60.

This magnitude tells you the strength or size of the vector. In physical terms, for force vectors, it denotes how strong the force is. When working with vectors, always ensure to represent them accurately according to their magnitude.

When sketching:

• Ensure the length of your vector line corresponds to its magnitude.
• Use a consistent scale for all vectors (e.g., 1 cm = 10 lb).

Visual consistency helps to quickly compare forces and understand their relationships within a vector diagram.

Vector Direction

Vector direction indicates where a vector is pointing. This is typically described using angles from a reference axis in your coordinate system.

In our exercise, each vector has a specific direction given by an angle and some cardinal directions like "north of west" or "east of south." For example, a vector direction of 25° east of north means:

• Start from the north (positive y-axis).
• Rotate 25° towards the east (towards the positive x-axis).

Understanding vector direction is essential for accurate vector addition and analysis, such as finding resultant vectors or balancing forces.

Coordinate System

The coordinate system serves as the framework for positioning vectors. In most physics problems, we use a simple Cartesian coordinate system where:

• North corresponds to the positive y-axis.
• East corresponds to the positive x-axis.
• South corresponds to the negative y-axis.
• West corresponds to the negative x-axis.

This consistent setup allows for easy interpretation of vector directions as angles relative to these axes. When sketching vectors on this plane, it's crucial to maintain accuracy in terms of directions and angles, as specified in the problem.

Force Diagram

A force diagram visually represents the different forces acting on an object, using arrows or vectors to show magnitude and direction. Creating a detailed force diagram is a key skill in physics as it helps to visualize the situation and allows for better problem-solving insights.
When making a force diagram:

• Start at a common point (usually the origin) for all vectors.
• Draw vectors to scale based on their magnitudes.
• Illustrate each vector's direction clearly, using angles provided.
• Label each vector with its magnitude for easy reference.

This diagram serves as the basis for applying physics laws, such as Newton's laws, to analyze forces and predict motion or balance.