Chapter 5: Applying Newton's Laws – Equilibrium and Force Analysis

Chapter 5: Applying Newton's Laws

5.1 Equilibrium

Equilibrium in physics refers to the state in which the net force acting on an object is zero. This can occur when an object is at rest (static equilibrium) or when it moves in a straight line at constant speed (dynamic equilibrium).

• Static Equilibrium: The object is at rest and all forces balance.

• Dynamic Equilibrium: The object moves at constant velocity; forces still balance.

• In both cases, there is no net force acting on the object.

• For equilibrium in two dimensions (x and y): $\sum F_x = m a_x = 0$ $\sum F_y = m a_y = 0$ This means the sums of the x- and y-components of the force are zero.

Problem-Solving Approach: 5.1 Equilibrium Problems

To solve equilibrium problems, follow these steps:

• SOLVE: Apply Newton's second law for each component: $\sum F_x = m a_x = 0$ $\sum F_y = m a_y = 0$

• Identify force components from a free-body diagram.

• Solve the equations for unknown forces.

• ASSESS: Check units, reasonableness, and completeness of your answer.

Example: Finding the Forces – Orangutan Hanging from a Rope

Consider an orangutan weighing 500 N hanging from a vertical rope. What is the tension in the rope?

• The orangutan is in static equilibrium, so the net force is zero.

• Forces acting: upward tension ($\vec{T}$) and downward weight ($\vec{w}$).

• Free-body diagram shows: $\vec{T}$ (up), $\vec{w}$ (down), $\vec{F}_{net} = 0$

• Only y-components matter: $T_y + w_y = m a_y = 0$ Since $T_y = T$ (up) and $w_y = -w$ (down): $T - w = 0$ $T = w = 500\,\text{N}$

• Conclusion: The tension in the rope equals the weight of the orangutan.

Example: Forces in Static Equilibrium – Rod on Frictionless Ice

A rod slides on a frictionless sheet of ice, held by a string. If the rod is at rest, which string angle keeps it in equilibrium?

• Forces acting: weight (down), tension (along string), normal force (up from ice).

• Free-body diagrams for different string angles show that only the configuration where the net force is zero (no horizontal component) is correct.

• Key Point: Tension acts along the string, weight acts down, normal force acts perpendicular to the surface.

• If friction were present, other configurations could be possible, but on frictionless ice, only the correct angle keeps the rod at rest.

Definitions and Concepts

• Equilibrium: State where the sum of all forces (and torques, if considering rotation) is zero.

• Free-body diagram: A graphical representation showing all forces acting on an object.

• Tension: Force transmitted through a string, rope, cable, or similar object.

• Weight: The gravitational force acting on an object, $w = mg$.

Additional info:

• In equilibrium problems, always resolve forces into components and use Newton's laws in each direction.

• Free-body diagrams are essential for visualizing and solving force problems.