(CH5) Applying Newton's Laws: Equilibrium, Multiple Objects, and Friction

Chapter 5: Applying Newton's Laws

Equilibrium of Particles

When analyzing systems in which objects are at rest or moving with constant velocity, the concept of equilibrium is essential. In equilibrium, the net force acting on a particle is zero, resulting in no acceleration.

• Equilibrium Condition: The sum of all forces acting on a particle is zero: $\sum \vec{F} = 0$

• Example: Traffic Light Supported by Cables

  • Visualize the traffic light as a particle at rest, suspended by three cables.

  • Assume cables do not break and nothing is moving.

  • Model the system as a particle in equilibrium.

  • Draw a free-body diagram showing the tensions ($T_1$, $T_2$, $T_3$) and the gravitational force ($F_g$).

  • Apply equilibrium equations in both x and y directions to solve for unknown tensions.

• Equilibrium Equations: $\sum F_x = 0$ $\sum F_y = 0$

• Application: Used to analyze static structures such as bridges, signs, and suspended objects.

Inclined Planes

Objects on inclined planes experience forces that can be resolved into components parallel and perpendicular to the surface. This analysis is crucial for understanding motion and equilibrium on slopes.

• Forces Acting on the Object:

  • Normal Force ($n$): Acts perpendicular to the plane.

  • Gravitational Force ($mg$): Acts vertically downward.

• Coordinate System:

  • Choose x-axis along the incline and y-axis perpendicular to the incline.

  • Resolve gravity into components: $mg \sin \theta$ (parallel to incline) $mg \cos \theta$ (perpendicular to incline)

• Modeling:

  • Use Newton's Second Law for the x-direction (motion along the incline).

  • Apply equilibrium in the y-direction (no motion perpendicular to the incline).

• Example: Car parked on a hill, block sliding down a ramp.

Multiple Objects and Newton's Laws

When two or more objects are connected or in contact, Newton's laws can be applied to each object individually or to the system as a whole. This approach is essential for analyzing systems such as pulleys and connected masses.

• Key Points:

  • Each object may experience different forces, but if connected, their accelerations are related.

  • Free-body diagrams are crucial for visualizing forces on each object.

  • Newton's laws can be applied to solve for unknowns such as tension and acceleration.

• Example: Atwood's Machine

  • Consists of two masses ($m_1$, $m_2$) connected by a string over a pulley.

  • Forces: Tension (same for both objects if the string and pulley are ideal), gravitational force.

  • Both masses have the same magnitude of acceleration.

  • Equations: For $m_1$: $T - m_1g = m_1a$ For $m_2$: $m_2g - T = m_2a$

  • Solving these equations yields the acceleration and tension.

• Application: Used to study mechanical advantage and dynamics in pulley systems.

Forces of Friction

Friction is a resistive force that opposes the relative motion between two surfaces in contact. It arises from microscopic interactions and is crucial in everyday phenomena and engineering applications.

• Types of Friction:

  • Static Friction ($f_s$): Prevents motion when a force is applied but the object remains at rest.

  • Kinetic Friction ($f_k$): Acts when the object is in motion.

• Frictional Force Equations: $f_s \leq \mu_s n$ $f_k = \mu_k n$ Where $\mu$ is the coefficient of friction and $n$ is the normal force.

• Properties:

  • Static friction is generally greater than kinetic friction.

  • Direction of frictional force is opposite to the direction of motion or impending motion.

  • Coefficients of friction depend on the materials in contact and are nearly independent of contact area.

• Application: Essential in analyzing motion on surfaces, vehicle dynamics, and machinery.

Coefficients of Friction: Table

The coefficients of friction vary depending on the materials in contact. The following table summarizes typical values for static and kinetic friction coefficients:

Materials	Static Friction ($\mu_s$)	Kinetic Friction ($\mu_k$)
Rubber on concrete	1.0	0.8
Steel on steel	0.74	0.57
Aluminum on steel	0.61	0.47
Glass on glass	0.94	0.40
Copper on steel	0.53	0.36
Wood on wood	0.25–0.50	0.20
Waxed wood on wet snow	0.14	0.10
Waxed wood on dry snow	0.04	0.02
Metal on metal (lubricated)	0.15	0.06
Teflon on Teflon	0.04	0.04
Ice on ice	0.10	0.03
Synovial joints in humans	0.01	0.003

Note: All values are approximate and may vary depending on conditions.

Summary of Particle Models

Two primary models are used in Newton's Laws problems:

• Particle Under a Net Force:

  • Experiences a non-zero net force, resulting in acceleration.

  • Described by Newton's Second Law: $\sum \vec{F} = m\vec{a}$

  • Often used with constant acceleration kinematic equations.

• Particle in Equilibrium:

  • Maintains constant velocity (including zero).

  • All forces balance: $\sum \vec{F} = 0$

Additional info: These notes expand on the brief points in the slides, providing full academic context, definitions, and examples for each topic. The table of coefficients of friction is reconstructed and values are inferred from standard physics references.