# Lecture 6: Friction and Circular Motion ## Physics I — Mechanics & Thermodynamics --- ## Learning Objectives By the end of this lecture, students will be able to: 1. Distinguish between static and kinetic friction and apply the friction force equations 2. Solve problems involving friction on flat and inclined surfaces 3. Describe uniform circular motion and calculate centripetal acceleration 4. Apply Newton's second law to circular motion problems 5. Analyze banked curves, conical pendulums, and vertical loops --- ## Lecture Content ### I. Friction — Overview Friction is a **contact force** that opposes the relative motion or tendency of motion between two surfaces. At the microscopic level, friction arises because surface irregularities interlock and molecules at contact points adhere to one another. Friction always acts **parallel** to the contact surface and comes in two varieties: **static friction** (when there is no sliding) and **kinetic friction** (when sliding occurs). The magnitude of friction depends on two factors: the nature of the surfaces in contact, captured by the coefficient of friction, and the normal force pressing the surfaces together. Counterintuitively but as verified by experiment, friction does **not** depend on the apparent area of contact. ### II. Static Friction **Static friction** (f_s) prevents an object from starting to slide. It is a **variable** force that adjusts to match the applied force, up to a maximum value: 0 <= f_s <= mu_s N, where mu_s is the **coefficient of static friction** (dimensionless) and N is the normal force. As long as the applied force does not exceed f_s,max = mu_s N, the object remains stationary. Once the applied force exceeds this maximum, the object begins to slide and kinetic friction takes over. Static friction is responsible for everyday phenomena such as walking, driving (where tires grip the road), and holding objects in place. ### III. Kinetic Friction **Kinetic friction** (f_k) acts on an object that is already sliding, with magnitude f_k = mu_k N, where mu_k is the **coefficient of kinetic friction**. Its direction is always opposite to the direction of sliding. Typically mu_k < mu_s, which means it takes more force to start an object sliding than to keep it sliding. At moderate speeds, kinetic friction is approximately constant for a given pair of surfaces. An important consequence of kinetic friction is energy dissipation: it converts kinetic energy into thermal energy. A graph of friction force vs. applied force for a block on a surface. The x-axis is the applied horizontal force, the y-axis is the friction force. The line rises at 45 degrees (f_s = F_applied) in the static region up to f_s,max = mu_s N (marked with a peak). Then the line drops to a constant value f_k = mu_k N in the kinetic region, illustrating that kinetic friction is less than the maximum static friction. Annotations label the static and kinetic regions. ### IV. Friction on Inclined Planes For a block on an incline at angle theta, the weight component along the incline is mg sin(theta) and the normal force is N = mg cos(theta). The maximum static friction force is therefore f_s,max = mu_s mg cos(theta). The block remains stationary as long as mg sin(theta) <= mu_s mg cos(theta), which defines a critical angle for impending slip: tan(theta_c) = mu_s, or theta_c = arctan(mu_s). If the block slides, the acceleration down the incline is a = g(sin theta - mu_k cos theta). If sin theta < mu_k cos theta, friction exceeds the gravitational pull along the slope, and the block decelerates and eventually stops. ### V. Uniform Circular Motion An object moving at **constant speed** along a circular path of radius r undergoes **uniform circular motion**. Although the speed is constant, the **velocity** is continuously changing direction, which means the object is accelerating. This acceleration is directed toward the center of the circle and is called **centripetal acceleration**: a_c = v^2 / r = omega^2 r. The **period** T is the time for one complete revolution, given by T = 2 pi r / v. The **frequency** f is the number of revolutions per second, with f = 1/T and units of Hz (s^-1). The angular velocity is omega = 2 pi f = 2 pi / T = v / r, measured in rad/s. ### VI. Centripetal Force By Newton's second law, the centripetal acceleration requires a **net inward force**: F_c = m a_c = m v^2 / r. It is essential to understand that **centripetal force is not a new type of force**. It is simply the name given to whatever real force or combination of forces provides the inward acceleration. This could be a string's tension (for a ball on a string), gravity (for a satellite orbiting Earth), a normal force (in a loop-the-loop), static friction (for a car turning on a road), or any combination of forces with a net radial component. There is **no outward "centrifugal force"** in an inertial frame. The perceived outward push is a fictitious force that appears only in the rotating (non-inertial) reference frame. Panel A: Top-down view of a car making a circular turn on a flat road. The velocity vector is tangent to the circle, and static friction points radially inward toward the center providing centripetal force. Panel B: A ball on a string being swung in a horizontal circle (conical pendulum). The tension is resolved into a vertical component (balancing weight) and a horizontal component (providing centripetal force). The geometry of the cone angle is labeled. ### VII. Applications of Circular Motion On a **flat curve (unbanked road)**, static friction provides the centripetal force: f_s = m v^2 / r. The maximum safe speed is v_max = sqrt(mu_s g r), and if the car exceeds this speed, it skids outward. For a **banked curve** designed so that no friction is needed, the road is tilted at angle theta so the normal force has a horizontal component. The conditions N sin(theta) = m v^2 / r and N cos(theta) = mg combine to give the ideal banking angle: tan(theta) = v^2 / (rg). In **vertical circular motion**, such as a roller coaster loop, the forces vary around the circle. At the top, both gravity and the normal force point toward the center, giving mg + N = m v^2 / r. At the bottom, the normal force points inward while gravity points outward, giving N - mg = m v^2 / r. The minimum speed at the top to maintain contact with the track is v_min = sqrt(rg), which occurs when N = 0. For a **conical pendulum**, a mass on a string traces a horizontal circle with the string making angle theta to the vertical. The horizontal component of tension provides the centripetal force (T sin(theta) = m v^2 / r) while the vertical component supports the weight (T cos(theta) = mg). The period works out to T_period = 2 pi sqrt(L cos(theta) / g), where L is the string length. A roller coaster car at three positions on a vertical circular loop: bottom, side, and top. At each position, a free-body diagram shows the weight (always downward) and the normal force (always pointing toward the center of the loop). At the top, both forces point downward. At the bottom, weight points down and normal points up. Newton's second law equation for the radial direction is written at each position. The minimum speed condition at the top (N = 0) is highlighted.