Tension and Spring Force - Physics

1. Fundamental Concepts

Definition: Tension is the force exerted by a rope, string, or cable when it is pulled tight.
Spring Force: The elastic force exerted by a spring that returns it to its equilibrium position after being stretched or compressed.
Hooke's Law: States that the force exerted by a spring is directly proportional to the displacement from its equilibrium position, expressed as $$F = k \cdot x$$ where $$k$$ is the spring constant and $$x$$ is the displacement.

Tension in a Rope: $$T = m \cdot g$$

Where $$T$$ is the tension, $$m$$ is the mass of the object, and $$g$$ is the acceleration due to gravity.

Hooke's Law Application: $$F = -k \cdot x$$

The negative sign indicates that the force is always directed opposite to the displacement.

Equilibrium Condition: $$\sum F = 0$$

In equilibrium, the net force acting on an object is zero.

Problem: A 5 kg mass hangs from a rope. Calculate the tension in the rope.

Identify the forces: Gravity ($$F_g = m \cdot g$$) and tension ($$T$$).
Calculate the gravitational force: $$F_g = 5 \cdot 9.8 = 49 \text{ N}$$.
Since the mass is at rest, the tension must equal the gravitational force: $$T = 49 \text{ N}$$.

Validation: Substitute values → Original: $$F_g = 5 \cdot 9.8 = 49 \text{ N}$$; Simplified: $$T = 49 \text{ N}$$ ✓

Problem: A spring with a spring constant of 200 N/m is stretched by 0.5 m. Calculate the force exerted by the spring.

Validation: Substitute values → Original: $$F = 200 \cdot 0.5 = 100 \text{ N}$$; Simplified: $$F = 100 \text{ N}$$ ✓

Free Body Diagrams: Draw a free body diagram to visualize all forces acting on an object.
Isolate Variables: Isolate the variable you are solving for in the equation.
Check Units: Ensure that all units are consistent throughout the problem.