Applying Newton's Laws: Equilibrium, Dynamics, Apparent Weight, and Friction

Applying Newton's Laws

Equilibrium Problems

Equilibrium in physics refers to the state where the net force acting on an object is zero. This can result in either no motion (static equilibrium) or constant velocity motion (dynamic equilibrium). Both types are physically equivalent in terms of net force.

• Definition of Equilibrium: An object is in equilibrium if the vector sum of all forces acting on it is zero: $\vec{F}_{\text{net}} = 0$

• Static Equilibrium: The object is at rest; all forces cancel each other.

• Dynamic Equilibrium: The object moves with constant velocity; net force is still zero.

• Example (Static): An 8.0 kg block hangs from the ceiling, supported by two ropes at angles. The forces from the ropes and gravity sum to zero.

• Example (Dynamic): A sled of mass m is pulled at an angle θ above the horizontal, opposed by friction fk. If the sled moves at constant velocity:

  • a) Find the tension in the rope.

  • b) Find the normal force on the sled.

• Incline Problem: A box of mass m rests on a smooth incline at angle θ, held by a rope. Determine the magnitude of each force acting on the box.

Simple Dynamics Problems

When the net force is not zero, the object accelerates according to Newton's Second Law. In two dimensions, this leads to two equations for the components of force and acceleration.

• Newton's First Law: $\vec{F}_{\text{net}} = 0$ (equilibrium)

• Newton's Second Law: $\vec{F}_{\text{net}} = m\vec{a}$

• Component Form: $F_{\text{net},x} = ma_x$ $F_{\text{net},y} = ma_y$

• Procedure: Find acceleration from net force, then use kinematic equations to find velocity and displacement.

• Example: A hockey puck (mass 50.0 g) moves east at 2.00 m/s. It receives a 3.00 N northward kick for 30.0 ms. Find its velocity just after the kick. Solution outline:

  • Calculate impulse: $J = F \Delta t$

  • Find change in velocity: $\Delta v = J/m$

  • Combine initial and final velocity vectors.

Apparent Weight and Weightlessness

Apparent weight is the normal force or tension a scale reads, which may differ from the true gravitational weight. In free fall or gravity-free environments, apparent weight can be zero, leading to the sensation of weightlessness.

• True Weight: $F_g = mg$

• Apparent Weight: The normal force or tension exerted on a scale. Key Point: In free fall, normal force is zero, so you feel weightless.

• Example: Standing on a scale in an elevator (mass = 60 kg):

  • (a) At rest: Apparent weight = $mg$

  • (b) Moving upward at constant velocity: Apparent weight = $mg$

  • (c) Accelerating upward at $2.0\ \text{m/s}^2$: Apparent weight = $m(g + a)$

  • (d) Accelerating downward at $2.0\ \text{m/s}^2$: Apparent weight = $m(g - a)$

  • (e) In free fall: Apparent weight = $0$

• What does a scale measure? The normal force or tension you exert on the scale, not necessarily your true weight.

Dynamics Problems with Two Objects

When two objects are connected by a string over a pulley, their motions are coupled. The tension in the string is the same in magnitude for both objects, and they share the same acceleration.

• System Setup: Two masses, $m_1$ and $m_2$, connected by a massless string over a massless pulley.

• Key Points:

  • Tension is equal in magnitude for both masses.

  • Both masses accelerate together.

  • Free-body diagrams (FBDs) are essential for analysis.

• Special Cases:

  • Atwood Machine: Pulley with two masses hanging vertically ($\alpha = 90^\circ$).

  • Half-Atwood Machine: One mass on a horizontal surface, one hanging ($\alpha = 0^\circ$).

Frictional Forces

Friction is a resistive force that opposes motion between surfaces. It arises from microscopic electrostatic interactions and can be reduced by lubricants. There are several types of friction, each with distinct properties.

• Types of Friction:

  • Static Friction ($f_s$): Opposes the initiation of motion. Maximum value: $f_{s,\text{max}} = \mu_s F_N$

  • Kinetic Friction ($f_k$): Opposes motion once it has started. $f_k = \mu_k F_N$

  • Rolling Friction: Resists motion of rolling objects (e.g., wheels).

• Coefficients:

  • $\mu_s$ = coefficient of static friction (no units)

  • $\mu_k$ = coefficient of kinetic friction (no units)

• Key Properties:

  • Static friction is generally greater than kinetic friction.

  • Both are proportional to the normal force.

  • Values depend on the materials in contact.

• Example: A wooden box on a ramp: At what angle does it start to slide? Solution outline:

  • Set $f_{s,\text{max}} = mg \sin \theta$

  • Find $\theta$ where $mg \sin \theta = \mu_s mg \cos \theta$

  • So, $\tan \theta = \mu_s$

• Example: Box slides down a ramp with kinetic friction. Find speed at bottom using energy or kinematics.

Fluid Friction (Drag)

Fluid friction, or drag, is the resistive force experienced by objects moving through fluids (liquids or gases). Unlike solid friction, fluid friction depends on velocity and object shape.

• General Form: $F_{\text{drag}} = kv^n$ where $n$ is between 1 and 2, $k$ depends on fluid properties and object shape.

• Low Speed: $F_{\text{drag}} \approx kv$ (linear)

• High Speed: $F_{\text{drag}} \approx kv^2$ (quadratic)

• Air Resistance: $F_{\text{drag}} = \frac{1}{2} C_d \rho A v^2$ where $C_d$ is the drag coefficient, $\rho$ is air density, $A$ is cross-sectional area.

• Terminal Velocity: When drag force equals gravitational force: $mg = kv$ Solve for $v$ to find terminal velocity.

• Example: Find the terminal velocity of a 10.0 kg ball of lead falling through air.

Table: Comparison of Friction Types

Type	Formula	Depends on	Typical Value
Static Friction	$f_{s,\text{max}} = \mu_s F_N$	Normal force, surface type	$\mu_s$ between 0 and 1
Kinetic Friction	$f_k = \mu_k F_N$	Normal force, surface type	$\mu_k$ between 0 and 1
Fluid Friction (Drag)	$F_{\text{drag}} = kv^n$	Velocity, fluid properties, shape	$k$ varies

Additional info: The notes infer standard equations and procedures for solving equilibrium and dynamics problems, as well as the physical meaning of apparent weight and friction types, to provide a self-contained study guide.