Chapter 5: Applying Newton’s Laws – Physics with Calculus Study Guide

Applying Newton’s Laws

Overview

This chapter explores the application of Newton’s Laws of Motion to various physical situations, focusing on equilibrium, friction, forces in contact, ropes, pulleys, and drag. The concepts are foundational for understanding dynamics and statics in Physics with Calculus.

Equilibrium

Static and Dynamic Equilibrium

Equilibrium occurs when the net force on an object is zero. Static equilibrium refers to objects at rest, while dynamic equilibrium refers to objects moving at constant velocity.

• Static Equilibrium: Object is at rest; $\ ext{acceleration} = 0$.

• Dynamic Equilibrium: Object moves at constant speed in a straight line; $\ ext{acceleration} = 0$.

• In 1D: $\sum F_x = m a_x = 0$

• In 2D: $\sum F_{xy} = 0$

Equilibrium Problem-Solving Approach

• Draw a diagram.

• Identify all forces acting on the object.

• Draw a free-body diagram.

• Determine known and unknown quantities.

• Write equations from Newton’s second law.

• Solve for the unknowns.

Mass and Weight

Definitions

Mass (m): The amount of matter in an object, measured in kilograms (kg). Mass is invariant regardless of location.

Weight (W): The gravitational force acting on an object, measured in Newtons (N). Weight depends on the local gravitational field.

• On the Moon: $W_{\text{Moon}} = \frac{1}{6} W_{\text{Earth}}$

• Formula: $W = m g$

Apparent Weight

The apparent weight is the normal force exerted by a supporting surface, which can differ from true weight when accelerating.

• Accelerating upward: $w_{\text{app}} = n = w + m a_y$ (heavier)

• Accelerating downward: $w_{\text{app}} = n = w - m a_y$ (lighter)

• Constant speed: $w_{\text{app}} = n = w$

Weightlessness

Weightlessness occurs when there is no supporting force, such as in free fall. The apparent weight is zero.

• Free fall: $w_{\text{app}} = n = 0$

Normal Force

Normal Force on Flat and Inclined Surfaces

The normal force is the perpendicular supporting force from a surface. On an incline, it is less than the object's weight.

• Flat surface: $n = mg$

• Inclined surface: $n = mg \cos \theta$

Friction

Static Friction

Static friction prevents motion between surfaces. It acts opposite to the direction of attempted motion and has a maximum value:

• Maximum static friction: $f_s \leq \mu_s n$

• $\mu_s$ is the coefficient of static friction.

Kinetic Friction

Kinetic friction opposes motion once objects are sliding. It is given by:

• $f_k = \mu_k n$

• $\mu_k$ is the coefficient of kinetic friction.

Rolling Friction

Rolling friction occurs when a wheel rolls on a surface, typically much less than kinetic friction.

• $f_r = \mu_r n$

• $\mu_r$ is the coefficient of rolling friction.

Coefficients of Friction Table

The table below compares static, kinetic, and rolling friction coefficients for various materials.

Material	Static ($\mu_s$)	Kinetic ($\mu_k$)	Rolling ($\mu_r$)
Rubber on concrete	1.00	0.80	0.02
Steel on steel (dry)	0.80	0.60	0.002
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	—

Objects in Contact

Solving Contact Force Problems

When objects are in contact, analyze each separately using free-body diagrams and Newton’s laws. Action-reaction pairs have equal magnitude and opposite direction.

• Draw separate free-body diagrams.

• Write Newton’s second law for each object.

• Equate action/reaction force magnitudes.

• Objects in contact share the same acceleration.

Ropes and Tension

Tension in Ropes

When a rope pulls an object, the tension is the force transmitted through the rope. For massless ropes, tension is the same throughout.

• Tension $T$ is equal to the force the rope exerts on the object.

• Action/reaction pairs: $T = F_{\text{box on rope}}$

Comparing Tension Forces

When multiple objects are pulled by a rope, the tension can differ depending on mass and acceleration.

Pulleys

Massless, Frictionless Pulley

For a massless, frictionless pulley, the tension in the string is the same on both sides.

Drag Forces

Drag Force and Reynolds Number

Drag force opposes motion through a fluid and increases with speed. The Reynolds number determines the nature of drag (laminar or turbulent).

• Reynolds number: $\text{Re} = \frac{\rho v L}{\eta}$

• High Re (>1000): Turbulent drag (cars, planes)

• Low Re (<1): Laminar drag (small particles)

Drag Coefficients and Cross-Sectional Area

Drag depends on the object's shape and area. Typical drag coefficients are listed below.

Object	Drag Coefficient ($C_D$)
Commercial airliner	0.024
Toyota Prius	0.24
Pitched baseball	0.35
Racing cyclist	0.88
Running person	1.2

Terminal Speed

Definition and Conditions

Terminal speed is reached when the drag force equals the weight force, resulting in zero acceleration.

• At terminal speed: $D = W$

• Acceleration: $a = 0$

Sample Problems and Applications

Curling Stone Example

In curling, friction and initial speed determine the stone's final position. Sweeping the ice can adjust the coefficient of friction, affecting speed and trajectory.

• Force to accelerate stone: $F = m a$

• Friction force: $f = \mu_k n$

• Increasing mass does not change distance if friction is proportional to weight.

• Sweeping decreases the coefficient of friction, lengthening travel.

Summary Table: Problem-Solving Approaches

Type	Steps
Equilibrium	Draw diagram, identify forces, free-body diagram, write equations, solve.
Dynamics	Draw diagram, identify forces, free-body diagram, write equations, use kinematics, solve.
Contact Forces	Separate diagrams, Newton’s law for each, equate action/reaction, same acceleration.

Additional info: Academic context was added to clarify definitions, formulas, and problem-solving steps, and to ensure completeness for exam preparation.