Equilibrium and Friction: Forces in Physics

Equilibrium and Friction

Introduction

This section covers the concepts of equilibrium, net force, and friction, which are fundamental in understanding the motion of objects and their interactions with surfaces. The notes include definitions, equations, and practical examples relevant to college-level physics.

Net Force and Equilibrium

Net Force

The net force on an object is the vector sum of all forces acting on it. According to Newton's Second Law, the net force determines the object's acceleration.

• Equation: $\sum \vec{F}_x = m a_{\text{net},x}$

• Equation: $\sum \vec{F}_y = m a_{\text{net},y}$

• Forces must be resolved into components along the same axis before summing.

Example: A skydiver of mass 80.0 kg experiences a 200.0 N upward drag force while falling. The net acceleration is not $9.8\ \mathrm{m/s^2}$ because the drag force opposes gravity.

Equilibrium

An object is in equilibrium when the net force acting on it is zero, resulting in zero acceleration. This can occur when the object is at rest or moving with constant velocity.

• Condition for Equilibrium: $\vec{F}_{\text{net},x} = 0$ if $a_{\text{net},x} = 0$

• Equilibrium does not require the object to be at rest; it can also be in uniform motion.

Example: A submersible is lowered at a steady rate by a cable. The tension in the cable balances the weight and any drag force, resulting in equilibrium.

Concept Check: Equilibrium

• An object in equilibrium can be at rest or in uniform motion (constant velocity).

• If there is no net force, the object cannot accelerate, but it may still be moving.

Friction

Types of Friction

Friction is a force that opposes the relative motion or attempted motion between two surfaces in contact. There are two main types:

• Static Friction ($f_s$): The force that resists the initiation of sliding motion between two surfaces. It acts when the object is at rest relative to the surface.

• Kinetic Friction ($f_k$): The force that opposes the motion of two surfaces sliding past each other. It acts when the object is already moving.

Equations for Friction

• Static Friction: $f_s \leq \mu_s n$

• Kinetic Friction: $f_k = \mu_k n$

• Where $\mu_s$ and $\mu_k$ are the coefficients of static and kinetic friction, respectively, and $n$ is the normal force.

Properties of Friction

• Static friction adjusts up to its maximum value to prevent motion.

• Kinetic friction is usually less than the maximum static friction and remains constant for a given pair of surfaces.

• The direction of friction is always opposite to the direction of relative motion or attempted motion.

Table: Coefficients of Static and Kinetic Friction

The following table compares typical coefficients of friction for various material pairs:

Materials	Static $\mu_s$	Kinetic $\mu_k$
Rubber on concrete	1.00	0.80
Steel on steel (dry)	0.80	0.60
Steel on steel (lubricated)	0.10	0.05
Wood on wood	0.50	0.20
Wood on snow	0.12	0.06
Ice on ice	0.10	0.03

Friction and Wheels

• When a wheel rolls without slipping, the point of contact with the ground is momentarily at rest relative to the surface. Thus, static friction acts at the contact point.

• If the wheels lock and slide (e.g., during hard braking), kinetic friction acts instead.

• Static friction is responsible for the acceleration and deceleration of rolling vehicles.

Forces on an Inclined Plane (Ramp)

Analyzing Forces

When an object rests on an inclined plane, the forces acting on it include gravity, the normal force, and friction. The component of gravity parallel to the incline tends to slide the object down, while friction opposes this motion.

• The normal force $n$ is perpendicular to the surface.

• The frictional force can be static or kinetic, depending on whether the object is at rest or moving.

Critical Angle and Coefficient of Friction

• The critical angle $\theta_c$ is the angle at which the object just begins to slide.

• Relationship: $\mu_s = \tan(\theta_c)$

• For angles less than $\theta_c$, static friction prevents motion; for greater angles, the object slides and kinetic friction applies.

Example: A 2.0 kg box begins to slide at $30^\circ$ on a ramp, so $\mu_s = \tan(30^\circ) \approx 0.577$.

Practice Problems

• Calculate the normal and frictional forces for a box on a ramp at various angles.

• Determine the net acceleration when the box slides down the ramp.

Additional info: The notes infer standard values and relationships for friction and equilibrium, and expand on the context of practice problems for clarity.