Newton's Laws and Mechanics

Newton’s 2nd Law

• Concept: F = ma where F is the force, m is mass, and a is acceleration.

• Kinetic Energy:

  • Formula: KE = 1/2 mv²

• Work-Energy Theorem:

  • Relation: KE_f + PE_f = KE_i + PE_i + external work

Key Physics Concepts to Remember

• Mechanical Work:

  • Formula: W = F • x (where F is force and x is displacement)

  • Description: Defined as the scalar product of force and displacement.

• Power:

  • Formula: P = work/time

  • Description: Measures the rate at which work is done.

• Circular Motion:

  • Formula: θ = 2π (unit of angle in radians)

Simple Harmonic Motion

Spring Oscillations

• Periodic Motion:

  • Definition: Motion that oscillates back and forth over the same path with the same cycle time.

  • Mass-Spring System: A fundamental model for studying periodic systems.

Spring Dynamics

• Frictionless Surface:

  • Equilibrium Position:

    • Definition: The point where the spring length is neither stretched nor compressed (x = 0).

• Spring Force:

  • Relationship: F = -kx (where k is the spring constant and x is displacement).

  • Differential Equation:

    • F = m * a given by the equation m(d²x/dt²) + kx = 0, which describes harmonic motion.

• Solution:

  • Displacement as a function of time: x(t) = A cos(ωt)

  • Where ω = √(k/m).

Motion Characteristics

Oscillation Parameters

• Velocity and Acceleration:

  • Can be expressed as functions of time from the harmonic motion equations.

• Period (T):

  • Definition: Time taken for one full cycle of oscillation.

  • Amplitude (A): Maximum displacement from the equilibrium position.

Time Relationships

• Circular motion and oscillations interconnected through the angular frequency. Here, ω² relates to frequency and amplitude as ω = √(k/m) correlates with T and A.

Period and Frequency

Definitions and Units

• Period (T): Time required to make one complete cycle (measured in seconds).

• Frequency (f): Number of cycles per second (measured in Hertz, Hz).

• Relationships:

  • Units: T = seconds, f = 1/T = 1/s = 1 Hz.

Modifications in Vertical Oscillation

Vertical Spring Dynamics

• Restoring Force Balance:

  • New equilibrium position exists where spring force equals gravitational force when spring hung vertically.

Energy in Simple Harmonic Motion

Energy Calculations

• Potential Energy (PE):

  • Formula: PE = ½ kx²

• Total Mechanical Energy:

  • The energy remains conserved in a frictionless system: Total Energy = PE + KE.

• Energy Distribution:

  • At limits of motion: All energy is potential.

  • At the equilibrium: All energy is kinetic.

Restoring Forces and Pendulum Motion

Conditions for Simple Harmonic Motion (SHM)

• Force Proportionality:

  • For SHM, restoring force must be proportional to negative displacement. Formula: F = mg sin θ.

  • For small angles, sin θ ≈ θ simplifies the relationship.

Example of the Simple Pendulum

• Differential Equation:

  • m(d²θ/dt²) + (mg/l) sin θ = 0

Energy in Motion

Period and Frequency Relations

• SHM Characteristics:

  • Period formula: T = 2π√(l/g) (for pendulum)

  • Frequency can be calculated as f = 1/T

  • SHM characteristics are sinusoidal in nature.

Detailed Example of SHM Calculation

Problem Setup

• Given Data: 885-g mass (0.885 kg), k = 184 N/m, initial speed = 2.26 m/s.

• Analyze parameters:

  1. Period: T = 0.436 s; f = 1/T = 2.29 Hz

  2. Amplitude (A):

     • Compute via A = v_initial/ω = 0.157 m

  3. Maximum Acceleration (a_max):

     • Formula: a_max = ω²A = 32.6 m/s²

  4. Total Energy: E = ½ kA² = 2.26 J

  5. Kinetic Energy at x = 0.4 A:

     • KE = Total Energy - PE = 2.26 J - (½ k(0.4A)²) = 1.90 J.