Applying Newton’s Laws: Forces, Friction, and Circular Motion

Applying Newton’s Laws

Learning Goals

This chapter focuses on the application of Newton’s laws of motion to solve a variety of problems in classical mechanics. The main objectives are:

• To use Newton’s first law to solve problems involving forces acting on a body in equilibrium.

• To use Newton’s second law to solve problems involving forces acting on an accelerating body.

• To understand the nature of different types of friction forces and how to solve problems involving these forces.

• To solve problems involving forces acting on a body moving along a circular path.

Introduction to Newton’s Laws of Motion

Newton’s three laws of motion provide the foundation for classical mechanics. While the laws themselves are simple, applying them to real-life situations requires careful analysis and systematic problem-solving techniques.

• Equilibrium problems involve analyzing forces on a body at rest or moving with constant velocity.

• Non-equilibrium problems require understanding the relationship between force and motion for accelerating bodies.

Newton’s First Law and Equilibrium

Definition and Application

Newton’s First Law (Law of Inertia) states that a body remains at rest or moves with constant velocity unless acted upon by a net external force. A body is in equilibrium when:

• It is at rest, or

• It moves with constant velocity in an inertial frame of reference.

The essential physical principle is:

• The sum of all forces acting on the body is zero: $\sum \vec{F} = 0$

• In component form: $\sum F_x = 0$ $\sum F_y = 0$

Problem-Solving Strategy for Equilibrium

1. Identify the relevant concept: Use Newton’s first law.

2. Draw a sketch of the physical situation.

3. Draw a free-body diagram for each body in equilibrium. Include only forces acting on the body (e.g., weight $w = mg$).

4. Choose coordinate axes and include them in your diagram.

5. Find components of each force along the axes.

6. Set up equations: $\sum F_x = 0$ $\sum F_y = 0$

7. Solve for unknowns and check your answer for consistency.

Newton’s Second Law: Dynamics of Particles

Definition and Application

Newton’s Second Law relates the net force on a body to its acceleration:

• If the net force is not zero, the body accelerates: $\sum \vec{F} = m\vec{a}$

• In component form: $\sum F_x = m a_x$ $\sum F_y = m a_y$

Problem-Solving Strategy for Dynamics

1. Identify the relevant concept: Use Newton’s second law.

2. Draw a sketch showing each moving body.

3. Draw a free-body diagram for each body, showing all forces (label each force, e.g., weight $w = mg$).

4. Choose coordinate axes for each body.

5. Identify additional equations if bodies are connected (e.g., by a rope).

6. Find force components along each axis.

7. List knowns and unknowns; identify target variables.

8. Write equations for each component: $\sum F_x = m a_x$ $\sum F_y = m a_y$

9. Solve the equations for the target variables.

10. Evaluate your answer for physical reasonableness.

Frictional Forces

Nature of Friction

Friction is a force that opposes the relative motion between two surfaces in contact. It acts parallel to the surface and arises from molecular interactions at the contact points.

• The normal force acts perpendicular to the surface.

• The friction force acts parallel to the surface.

• Both are components of the overall contact force between surfaces.

Types of Friction

• Static friction ($f_s$): Acts when there is no relative motion. It can vary up to a maximum value: $f_s \leq \mu_s n$

• Kinetic friction ($f_k$): Acts when a body slides over a surface. It has a constant value: $f_k = \mu_k n$

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

Example: Box on a Surface

• No applied force: No friction ($f = 0$).

• Weak applied force: Static friction balances applied force ($f_s < \mu_s n$).

• Stronger applied force: Static friction reaches maximum ($f_s = \mu_s n$), box is about to move.

• Box slides: Kinetic friction acts ($f_k = \mu_k n$).

Coefficients of Friction: Typical Values

The coefficients of friction depend on the materials in contact. The following table summarizes typical values:

Materials	Coefficient of Static Friction ($\mu_s$)	Coefficient of Kinetic Friction ($\mu_k$)
Steel on steel	0.74	0.57
Aluminum on steel	0.61	0.47
Copper on steel	0.53	0.36
Brass on steel	0.51	0.44
Zinc on cast iron	0.85	0.21
Copper on cast iron	1.05	0.29
Glass on glass	0.94	0.40
Copper on glass	0.68	0.53
Teflon on Teflon	0.04	0.04
Teflon on steel	0.05	0.04
Rubber on concrete (dry)	1.0	0.8
Rubber on concrete (wet)	0.3	0.25

Example: Windshield Wipers

• On dry glass, static friction can cause the wiper to stick and squeak.

• On wet glass, friction is reduced, allowing smooth motion.

Fluid Resistance and Terminal Speed

Definition and Application

When a body moves through a fluid (such as air or water), it experiences a drag force that opposes its motion. The magnitude of this force depends on the speed of the body.

• At low speeds, drag force is often proportional to velocity ($F_{drag} \propto v$).

• At higher speeds, drag force is often proportional to the square of velocity ($F_{drag} \propto v^2$).

• A falling body reaches terminal speed when the drag force equals its weight: $F_{drag} = mg$

Dynamics of Circular Motion

Uniform Circular Motion

For a particle moving in a circle at constant speed, both the acceleration and the net force are directed toward the center of the circle (centripetal direction).

• The net force required for uniform circular motion is: $F_{net} = m \frac{v^2}{R}$

• Velocity is always tangent to the circle; acceleration and net force point toward the center.

Example: Ball on a String

• If the string breaks, no net force acts on the ball, so it moves in a straight line at constant velocity (Newton’s first law).

Misconceptions: Centrifugal Force

• There is no real force called centrifugal force in an inertial frame of reference.

• The quantity $m v^2 / R$ is not a force to be included in a free-body diagram; it is the required net force for circular motion.

Fundamental Forces of Nature

All known forces in nature are manifestations of four fundamental interactions:

• Gravitational interaction

• Electromagnetic interaction

• Strong interaction

• Weak interaction

Physicists aim to unify these interactions into a single comprehensive theory, often referred to as a "theory of everything."