Equilibrium

Static and Dynamic Equilibrium

In physics, equilibrium refers to the state of an object when the net force acting on it is zero. There are two main types:

• Static Equilibrium: The object is at rest and remains at rest.

• Dynamic Equilibrium: The object moves in a straight line at a constant speed.

In both cases, there is no net force acting on the object. This can be mathematically expressed as:

Thus, the sums of the x- and y-components of the forces are zero in equilibrium.

Finding the Tension in a Rope While Towing

Example Problem: Car Towed at an Angle

Consider a car with a mass of 1500 kg being towed at a steady speed by a rope held at a 20° angle from the horizontal. A friction force of 320 N opposes the car’s motion. The goal is to find the tension in the rope.

• The car is in dynamic equilibrium (), so .

• Forces acting on the car: tension (), friction (), normal force (), and gravity ().

The free-body diagram shows these forces, with the tension at an angle .

Component Analysis

Break the tension into components:

• (horizontal component)

• (vertical component)

Write Newton’s second law in component form:

To solve for the tension:

Plug in the values:

Key Point: When the rope is at an angle, the required tension is greater than the friction force because only the horizontal component of the tension balances friction.

Table: Force Components

Summary

• Equilibrium problems require the net force in each direction to be zero.

• For objects pulled at an angle, resolve forces into components and apply Newton’s second law in each direction.

• Only the component of tension in the direction of motion balances the opposing force (e.g., friction).

Section 5.2: Dynamics and Newton’s Second Law

Newton’s Second Law

Newton’s Second Law relates the net force acting on an object to its acceleration:

In component form:

This law is the foundation for analyzing dynamics problems, connecting forces to the resulting motion.

Problem-Solving Approach for Dynamics

1. Identify knowns and unknowns: List all given quantities and what you are asked to find.

2. Draw diagrams: Include a force identification diagram, a free-body diagram, and a motion diagram if needed.

3. Establish a coordinate system: Define axes and directions for forces and motion.

4. Apply Newton’s second law: Write equations for each component and solve for the unknowns.

5. Check your answer: Ensure units are correct and the result is physically reasonable.

Example: If a force is known, use Newton’s second law to find acceleration, then use kinematics to find position or velocity. If acceleration is known from kinematics, use it to find the net force.

Additional info: The notes continue with further examples and applications of Newton's laws, including friction, interacting objects, and ropes and pulleys, as indicated by the topic wheel in the second image.