(6/11/26) Friction and Contact Forces

Introduction and Definitions of Friction

• Conceptual Definition: Friction is a force that resists motion or any force trying to be applied to an object. It essentially occurs when trying to rub two substances together.

• Mechanical Presence: Frictional force exists when two substances are in contact with each other. A common example provided is the interaction between a tire and the road.

• Relationship to Motion:

  • Friction always opposes the direction of motion, meaning it acts in the direction opposite to the velocity vector.

  • According to Newton's law of motion, the molecules of each surface interact to create this resistance.

• Static Scenario: If there is no motion, friction still exists. In this state, it opposes the sum of all other forces that are acting parallel to the surfaces in contact.

  • Stapler Example: If a stapler is resting in a hand and an attempt is made to move it, there is a frictional force between the stapler and the palm. An applied force must overcome the threshold of static friction before the object can begin to move.

  • Continuation of Friction: Even once an object is in motion, friction continues to act; the object must still overcome kinetic friction to maintain movement over a surface.

Classifications and Types of Friction

• Broad Classification: Friction can be categorized into two primary forms based on the medium of contact:

  • Fluid Friction: This occurs in fluids or between lubricated surfaces.

  • Dry Friction: This occurs between non-lubricated surfaces of solid objects. Examples include a pen writing on a board or a locker being moved across a floor.

• Dry Friction Sub-categories:

  1. Static Friction: The frictional resistance that must be overcome to move an object from a stationary position. It occurs when two surfaces are not moving relative to each other but a force is applied to move them.

  2. Dynamic (Kinetic) Friction: The frictional force acting between surfaces that are in motion relative to each other. Dynamic friction is generally lower in magnitude compared to the maximum static friction threshold.

Contact Forces and Vector Resolution

• Contact Force Definition: Force occurring specifically between objects that are in physical contact.

• Component Resolution: Contact forces can be resolved into two main perpendicular components:

  • Normal Force ($F_{n}$): This is the perpendicular component of the contact force acting relative to the point of contact. This is a crucial reference for calculating friction.

  • Parallel Component (Friction): The component of the contact force acting parallel to the surface.

• Example: Running Dynamics:

  • When a runner pushes off the ground, they push backward to move forward.

  • The Normal Force acts upwards on the runner from the ground. This force is responsible for accelerating the runner vertically (in the $y$ direction).

  • The Friction Force acts forward on the runner. This is the only force capable of moving the runner horizontally across the track (in the $x$ direction).

The Relationship Between Normal Force and Friction

• Proportionality: Maximum frictional force is directly proportional to the normal contact force.

• Mathematical Expression: An increase in the normal force ($F_{n}$) results in a proportional increase in the maximum possible friction ($F_{f}$).

• Molecular Theory: High normal force pushes molecules of the two surfaces closer together. This increase in physical compression enhances molecular interactions and "locking" between the surfaces.

• Weight Correlation: Higher weight results in a higher normal force, which consequently increases the maximum friction. A $10\,kg$ box will experience more friction than a $5\,kg$ box because its surfaces are more compressed together on a microscopic level.

• Surface Area and Friction:

  • Friction is not affected by the size of the surface area in contact.

  • If the normal force remains constant but the surface area increases, the normal force is simply spread out over a wider area.

  • The individual force on each molecule is reduced, but the total number of interacting molecules increases, resulting in the same total frictional force.

  • Note: Race car tires (slicks) are designed for maximum contact for performance reasons (like heat dissipation and grip consistency), but the basic law of friction states Area does not change the coefficient of friction itself.

Mathematical Modeling of Friction

• The Friction Formula: The maximum force of friction is calculated as:

  • $F_{f} = \mu \times F_{n}$

  • Where $F_{f}$ is the maximum force of friction, $\mu$ is the coefficient of friction, and $F_{n}$ is the normal force.

• The Coefficient ($\mu$):

  • This is a unitless constant that depends on the types of materials interacting (e.g., rubber on ice has a low $\mu$, while rubber on asphalt has a high $\mu$).

  • It also varies depending on whether the surfaces are static or moving relative to each other.

• Force Thresholds: Friction varies from negative maximum to positive maximum based on the direction of the applied force.

• Applied Force Graphing: As external applied force increases, the friction force increases linearly at a $1:1$ ratio until it reaches the static friction threshold. Once motion begins, the friction drops to a slightly lower, constant value known as kinetic/dynamic friction.

• Example Calculation (Block on Counter):

  • A block of wood with a mass $m = 5\,kg$ rests on a ceramic counter.

  • Coefficient of static friction $\mu_{s} = 0.4$.

  • Acceleration due to gravity $g = 9.8\,m/s^2$.

  • First, calculate Normal Force: $F_{n} = m \times g = 5\,kg \times 9.8\,m/s^2 = 49\,N$.

  • Next, calculate threshold Force: $F_{f} = \mu \times F_{n} = 0.4 \times 49\,N = 19.6\,N$.

  • Horizontal force necessary to move the block must be $≥ 19.6\,N$.

Dynamics and Case Studies

• The Horse and Cart Paradox: Newton's third law states action and reaction forces are equal and opposite (the horse pulls the cart, the cart pulls the horse).

  • The system moves because friction acts differently on the components.

  • The horse's hooves experience high friction against the ground due to smaller surface area and specific force application (high normal force/pressure), while the cart's wheels are designed to minimize friction.

  • The resultant force allowing forward motion is the difference in shared frictional forces within the system.

• Tug of War Case Study: Kevin vs. Phil:

  • Kevin: Mass ($m$) = $210\,kg$ ($\sim 460\,lbs$). Pull force = $3000\,N$.

  • Phil: Mass ($m$) = $120\,kg$ ($\sim 260\,lbs$). Pull force = $3000\,N$.

  • Common Factors: Both are $5'11"$, wearing footwear with $\mu = 1.5$, competing on a rubber floor.

  • Winner: Kevin wins because of his significantly higher mass. Higher mass results in a higher normal force ($F_{n} = m \times g$). Since $F_{f} = \mu \times F_{n}$, Kevin has a much higher maximum static friction ($F_{f} = 1.5 \times (210 \times 9.8) = 3087\,N$) than Phil ($F_{f} = 1.5 \times (120 \times 9.8) = 1764\,N$). Phil's friction cannot resist the $3000\,N$ force, causing him to slide.

• The Height Advantage in Pulling:

  • If Competitor A is taller than Competitor B, Competitor A pulls at an upward angle while B pulls at a downward angle.

  • Pulling upward on the rope reduces the Normal Force ($F_{n}$) of the opposing person because the upward component of the pull counteracts a portion of their weight. Since $F_{n}$ decreases, their frictional resistance also decreases, making it easier to pull them.

Questions & Discussion

• Question (Gravity Constants): Does it matter if we use $g = 10\,m/s^2$ or $g = 9.8\,m/s^2$ on exams?

  • Response: Both are acceptable. Unless specified that a certain value must be used, the instructor will not deduct marks for using $10\,m/s^2$ for calculations.