Chapter 5: Using Newton's Laws – Friction, Uniform Circular Motion, and Drag Forces

5.1 Using Newton's Laws With Friction

Introduction to Friction

Friction is a fundamental force that arises whenever two solid surfaces slide against each other. It is a contact force caused by the microscopic roughness and interactions at the interface of materials. Although the detailed microscopic mechanisms are complex and not fully understood, friction plays a crucial role in everyday phenomena and engineering applications.

• Kinetic friction occurs when surfaces slide past each other.

• Static friction acts when surfaces are at rest relative to each other, preventing motion.

Kinetic Friction

Kinetic friction opposes the motion of a moving object and is characterized by the following equation:

• Formula: $F_k = \mu_k F_N$

• $F_N$ is the normal force (perpendicular to the surface).

• $\mu_k$ is the coefficient of kinetic friction, which depends on the materials in contact.

• This equation gives the magnitude of the frictional force, not its direction.

Static Friction

Static friction prevents the initiation of motion between surfaces at rest. It adjusts up to a maximum value to resist applied forces:

• Formula: $F_s \leq \mu_s F_N$

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

• It is generally harder to start moving an object than to keep it moving ($\mu_s > \mu_k$).

Properties of Friction Forces

• Friction is a contact force caused by material roughness.

• The friction force vector is always parallel to the surface.

• Kinetic friction opposes motion; static friction prevents motion.

• Friction opposes relative motion and acts opposite to the velocity of the sliding surface.

• Frictional forces are parallel to the surface and perpendicular to the normal force.

• Microscopic models of friction include roughness, sticking, scraping, and lubrication.

Table: Coefficients of Friction

Surfaces	Coefficient of Static Friction ($\mu_s$)	Coefficient of Kinetic Friction ($\mu_k$)
Wood on wood	0.4	0.2
Ice on ice	0.1	0.03
Metal on metal (lubricated)	0.15	0.07
Steel on steel (unlubricated)	0.7	0.6
Rubber on dry concrete	1.0	0.8
Rubber on wet concrete	0.7	0.5
Rubber on other solid surfaces	1–4	—
Teflon on Teflon in air	0.04	0.04
Lubricated ball bearings	<0.01	<0.01
Synovial joints (human limbs)	0.01	0.01

Friction Force vs. Applied Force Graph

• Static friction increases with applied force up to $F_s = \mu_s F_N$.

• Once motion starts, kinetic friction takes over: $F_k = \mu_k F_N$.

• Generally, $\mu_s \geq \mu_k$.

Example: Friction on a Box

A 10.0-kg box on a horizontal floor with $\mu_s = 0.40$ and $\mu_k = 0.30$. Find the friction force for applied forces of 0, 10, 20, 28, and 40 N.

• For $F_A < F_s^{max}$, friction matches applied force.

• For $F_A > F_s^{max}$, friction is kinetic: $F_k = \mu_k F_N$.

Example: Pulling Against Friction

A 10.0-kg box is pulled with a 40.0 N force at 30° above horizontal. $\mu_k = 0.30$. Calculate acceleration.

• Resolve forces into horizontal and vertical components.

• Find net force and use Newton's second law: $a = F_{net}/m$.

Example: Push or Pull a Sled?

Comparing the force required to push or pull a sled at the same angle $\theta$ on flat ground. The vertical component of the force affects the normal force and thus the friction.

• Pushing increases normal force, increasing friction.

• Pulling decreases normal force, reducing friction.

5.6 Velocity-Dependent Forces: Drag and Terminal Velocity

Introduction to Drag Forces

When objects move through fluids (liquids or gases), they experience a drag force that depends on their velocity. Drag is a resistive force that opposes motion and increases with speed.

• At low velocities: $F_D \propto v$

• At high velocities: $F_D \propto v^2$

Terminal Velocity

Terminal velocity is the constant speed reached by a falling object when the drag force equals the gravitational force.

• Formula: $v_T = \frac{mg}{b}$ (for $F_D = bv$)

• Before reaching $v_T$, acceleration decreases as drag increases.

• At $v_T$, net force is zero and the object falls at constant speed.

Cases for Falling Objects with Drag

• Case 1: Not at terminal velocity: $-mg + bv = ma_y$

• Case 2: At terminal velocity: $-mg + b v_T = 0$

Example: Ship Pulled by Tug Boats

A ship is pulled at constant velocity by two tug boats, each with tension 200,000 N at 28° to the direction of motion. Find the magnitude of the drag force.

• Resolve the tension forces into the direction of motion.

• Sum the components to find total drag force.

5.2 Uniform Circular Motion – Kinematics

Definition and Properties

Uniform circular motion (UCM) refers to motion in a circle of constant radius at constant speed. The instantaneous velocity is always tangent to the circle.

• Speed is constant, but velocity direction changes continuously.

• Angular displacement is measured in radians.

Key Quantities in UCM

• Period (T): Time for one revolution.

• Frequency (f): Number of revolutions per unit time.

• Angular speed ($\omega$): $\omega = \frac{\Delta \theta}{\Delta t} = 2\pi f = \frac{2\pi}{T}$

• Speed: $v = \frac{2\pi R}{T} = \omega R$

Units

• Period: seconds (s)

• Frequency: hertz (Hz)

• Angular speed: radians per second (rad/s)

Speed, Velocity, and Acceleration in UCM

• Speed ($v$) is constant: $v = \omega R$

• Velocity changes direction, always tangent to the circle.

Acceleration in UCM

Although speed is constant, the direction of velocity changes, requiring an acceleration called centripetal acceleration:

• Magnitude: $a = \frac{v^2}{R} = \omega^2 R$

• Direction: Always toward the center of the circle.

Example: Acceleration of a Revolving Ball

A 150-g ball revolves in a horizontal circle of radius 0.600 m at 2.00 revolutions per second. Find its centripetal acceleration.

• Calculate $v$ using $v = 2\pi R f$.

• Find $a$ using $a = v^2 / R$.

5.3 Dynamics of Uniform Circular Motion

Net Force in UCM

For an object to maintain uniform circular motion, a net force must act toward the center of the circle (centripetal force).

• Formula: $\sum F_R = m a_R = m \frac{v^2}{r} = m r \omega^2$

• Direction: Always inward, toward the center.

Strategy for Circular Dynamics Problems

• Draw a free-body diagram (FBD) with axes oriented appropriately for horizontal or vertical circles.

• Identify forces acting in the plane and perpendicular to the circle.

• Apply Newton's second law: $F_{net} = m a_{centripetal}$

• Ensure net force points toward the center.

Equations Used in UCM

• $T = 1/f$

• $\omega = \Delta \theta / \Delta t = 2\pi f = 2\pi / T$

• $v = 2\pi f R = (2\pi R)/T = \omega R$

• $a = v^2 / R = \omega^2 R$

• $F_{net} = m v^2 / R = m \omega^2 R$

Centrifugal Force Misconception

• There is no real centrifugal force acting outward; the tendency to move straight is due to inertia.

• If centripetal force vanishes, the object moves tangentially to the circle.

Example: Revolving Ball in a Vertical Circle

A 0.150-kg ball on a 1.10-m cord is swung in a vertical circle. Find the minimum speed at the top to maintain circular motion, and the tension at the bottom if moving at twice that speed.

• At the top, tension plus gravity must provide centripetal force.

• At the bottom, tension must overcome gravity and provide centripetal force.

5.4 Highway Curves: Banked and Unbanked

Forces on a Car in a Curve

When a car rounds a curve, a net force toward the center is required for circular motion. On a flat road, this force is supplied by friction.

• If friction is insufficient, the car will skid and move in a straight line.

• Static friction keeps tires from slipping; kinetic friction takes over if slipping occurs, which is less effective.

Static vs. Kinetic Friction in Curves

• Kinetic friction is smaller than static friction.

• Static friction can point toward the center; kinetic friction opposes motion, making control difficult.

Example: Skidding on a Curve

A 1000-kg car rounds a flat curve of radius 50 m at 15 m/s. Will it skid if $\mu_s = 0.60$ (dry) or $\mu_s = 0.25$ (icy)?

• Calculate required centripetal force and compare to maximum static friction.

Example: Banking Angle

Find the angle at which a road should be banked for a car to round a curve of radius $r$ at speed $v$ without friction. Apply this to a 50 m radius curve at 50 km/h.

• Use force components and Newton's laws to derive the banking angle formula.

Summary of Chapter 5

• Kinetic friction: $F_k = \mu_k F_N$

• Static friction: $F_s \leq \mu_s F_N$

• An object moving in a circle at constant speed is in uniform circular motion.

• Centripetal acceleration: $a = v^2 / R = \omega^2 R$

• Centripetal force: $\sum F_R = m a_R = m \frac{v^2}{r} = m r \omega^2$