Chapter 4 newtons second law

Friction and Drag

Friction

Friction is a force that opposes the relative motion or tendency of such motion of two surfaces in contact. It plays a crucial role in everyday phenomena and engineering applications.

• Direction: Always acts opposite to the direction of motion or intended motion.

• Magnitude: Depends on the nature of the surfaces and the force pressing them together.

Types of Friction:

• Static Friction: The frictional force that prevents two surfaces from sliding past each other. It must be overcome to initiate motion.

• Kinetic Friction: The frictional force acting when two surfaces are sliding past each other.

It generally takes more force to start moving an object (overcome static friction) than to keep it moving (overcome kinetic friction).

Coefficients of Friction

The coefficient of static friction (\( \mu_s \)) and coefficient of kinetic friction (\( \mu_k \)) are dimensionless quantities that characterize the frictional properties of surfaces.

Materials	\( \mu_s \)	\( \mu_k \)
Steel on steel	0.74	0.57
Aluminum on steel	0.61	0.47
Copper on steel	0.53	0.36
Rubber on concrete (dry)	1.0	0.8
Rubber on concrete (wet)	0.3	0.25
Wood on wood	0.25–0.5	0.2
Glass on glass	0.94	0.4
Teflon on Teflon	0.04	0.04
Teflon on steel	0.04	0.04
Waxed wood on wet snow	0.14	0.1
Waxed wood on dry snow	0.10	0.04
Metal on metal (lubricated)	0.15	0.06
Ice on ice	0.1	0.03
Synovial joints in humans	0.01	0.003
Very rough surfaces		1.5

Key Equations:

• Maximum static friction: $f_{s,\text{max}} = \mu_s N$

• Kinetic friction: $f_k = \mu_k N$

Where N is the normal force.

Drag

Drag is a resistive force experienced by objects moving through fluids (liquids or gases). It acts opposite to the direction of motion and increases with speed.

• Drag Force Equation:

$D = \frac{1}{2} C_d \rho A v^2$

• \( C_d \): Drag coefficient (depends on shape)

• \( \rho \): Density of the fluid

• \( A \): Frontal area

• \( v \): Speed of the object

Mass and Weight

Definitions and Units

• Mass (m): A measure of the amount of matter in an object; also a measure of inertia. SI unit: kilogram (kg). Scalar quantity.

• Weight (W): The force exerted on an object due to gravity. SI unit: newton (N). Vector quantity.

Relationship:

$W = mg$

• Where g is the acceleration due to gravity (≈ 9.8 m/s2 on Earth).

The mass of an object is constant everywhere, but its weight depends on the local gravitational field.

Newton's Second Law of Motion

Statement and Equation

Newton's Second Law states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. The direction of acceleration is the same as the direction of the net force.

Mathematical Form:

$\vec{a} = \frac{\vec{F}_{\text{net}}}{m}$

or equivalently,

$\vec{F}_{\text{net}} = m \vec{a}$

• \( \vec{F}_{\text{net}} \): Net force (N)

• \( m \): Mass (kg)

• \( \vec{a} \): Acceleration (m/s2)

Key Points:

• Acceleration increases with greater net force (for constant mass).

• Acceleration decreases with greater mass (for constant force).

Examples and Applications

• If the net force on an object doubles and mass remains constant, acceleration doubles.

• If the mass of an object doubles and the net force remains constant, acceleration halves.

Free Fall and Nonfree Fall

Free Fall

When an object falls under the influence of gravity alone (no air resistance), it is in free fall. All objects in free fall near Earth's surface accelerate downward at the same rate, regardless of mass.

$a = g$

Nonfree Fall (with Air Resistance)

When air resistance is significant, the net force on a falling object is reduced, and acceleration is less than g. As speed increases, air resistance increases until it balances the weight of the object. At this point, the object falls at a constant speed called terminal velocity.

• Terminal Velocity: The constant speed reached when the force of air resistance equals the force of gravity.

$F_{\text{net}} = 0, \ a = 0, \ v = \text{constant}$

Free Fall vs. Nonfree Fall

• In a vacuum: No air resistance; all objects fall with the same acceleration (g).

• In air: Lighter objects reach terminal velocity quickly and fall slowly; heavier objects accelerate longer before reaching terminal velocity.

Example: A coin and a feather dropped in a vacuum fall together, but in air, the feather quickly reaches terminal velocity and falls slowly, while the coin continues to accelerate.

Additional info: The study of friction, drag, and Newton's Second Law is foundational for understanding dynamics in physics, including engineering, biomechanics, and atmospheric science.