Weight and Normal Force - Physics

1. Fundamental Concepts

Definition: Weight is the force exerted on an object due to gravity, typically denoted as $$W = mg$$ where $$m$$ is the mass of the object and $$g$$ is the acceleration due to gravity.
Normal Force: The normal force ($$N$$) is the force that counteracts the weight of an object when it is resting on a surface. It acts perpendicular to the surface.
Equilibrium: An object is in equilibrium if the net force acting on it is zero, i.e., $$\sum F = 0$$.

2. Key Concepts

Basic Rule: $$W = mg$$

Normal Force: When an object is at rest on a flat surface, $$N = W$$ or $$N = mg$$

Application: Used in analyzing forces in static and dynamic systems

3. Examples

Example 1 (Basic)

Problem: A book with a mass of 2 kg is placed on a table. Calculate the normal force exerted by the table on the book.

Step-by-Step Solution:

Calculate the weight of the book using $$W = mg$$: $$W = 2 \cdot 9.8 = 19.6 \text{ N}$$
Since the book is at rest, the normal force equals the weight: $$N = W = 19.6 \text{ N}$$

Validation: The normal force must equal the weight for the book to be in equilibrium. Thus, $$N = 19.6 \text{ N}$$ ✓

Example 2 (Intermediate)

Problem: A 5 kg block is placed on an inclined plane at an angle of 30 degrees. Determine the normal force exerted by the plane on the block.

Step-by-Step Solution:

Calculate the weight of the block: $$W = 5 \cdot 9.8 = 49 \text{ N}$$
The normal force is the component of the weight perpendicular to the incline: $$N = W \cos(30^\circ) = 49 \cdot \cos(30^\circ) = 49 \cdot \frac{\sqrt{3}}{2} \approx 42.4 \text{ N}$$

Validation: Using trigonometry, the normal force should be approximately 42.4 N for the block to remain stationary on the incline. Thus, $$N \approx 42.4 \text{ N}$$ ✓

4. Problem-Solving Techniques

Free Body Diagrams: Draw a free body diagram to visualize all forces acting on the object.
Component Analysis: Break down forces into their components along and perpendicular to the direction of interest.
Equilibrium Check: Ensure that the sum of forces in any direction equals zero for objects in equilibrium.