University:
Santa Fe College
Course:
PHY 2004 | Applied Physics 1
Academic year:

2024
Views:

72

Pages:

5
Author:

Jabin F.

Friction Force: It acts between two solid surfaces in contact. It is of two types: (1) Kinetic Friction, (2) Static Friction. (1) Kinetic Friction: It acts when an object slides on a surface. F_k = μ_kN It acts opposite to motion or velocity. # N may or may not be equal to mg. (2) Static Friction: It acts when an object is at rest (or about to move) on a surface under the action of some applied forces. (F_s) F_s is adjustable force. It can adjust its value (magnitude as well as direction). It is calculated using equation of motion or concept of equilibrium. Static friction has a max. value (F_s)_max = μ_sN where μ_s = coefficient of static friction μ_k = " kinetic → μ_s and μ_k may be different, μ_s is slightly greater than μ_k. → If μ_s and μ_k are not separately given, then we assume the two to be equal μ_s = μ_k = μ conditionally. (F_s)_max = μ_sN is used only in those questions in which max. or min. value of something is to be calculated. → To slide an object on a surface, min. force required parallel to surface is (F_s)_max → If an object is at rest and max. or min. of something is to be calculated, then assume that (F_s)_max is acting on it and object is about to move (limiting equilibrium). → For equilibrium Σ Fx = 0 or Fleft = Fright Σ Fy = 0 or Fup = Fdown. Q-38 The equation of motion of a forced simple harmonic oscillator is x'' + ω²x = A cos(Ωt), where A is a constant. At resonance Ω = ω, the amplitude of oscillations at large times (a) saturates to a finite value (b) increases with time at t → ∞ (c) increases linearly with time (d) increases exponentially with time Solution: given eqⁿ: x'' + ω²x = A cos(ωt) solution of equation → x = CF + PI CF = C₁cos(ωt) + C₂sin(ωt) PI = 1/(D² + ω²) * A cos(ωt) but f(D²) = f(-ω²) = 0 ∴ PI = A/(D² + ω²) * cos(ωt) = A/(D² + ω²) * [cos(ωt) + (1/D)sin(ωt)] = (A/(-2ω)) * [cos(ωt)t + (1/ω)sin(ωt)] x = CF + PI = C₁cos(ωt) + C₂sin(ωt) - (A/2ω) * [cos(ωt)t + (1/ω)sin(ωt)] At larger time t → ∞, but sin(ωt), cos(ωt) ∈ [-1, 1] all terms are negligible except one term x ≈ -(A/2ω) * sin(ωt) → x ∝ t ∴ amplitude increases linearly with time at large time (C) Ans.