A simple harmonic oscillation is represented by the equation y= 5cos(2pi×t + pi/4 ) in SI units. Compute the 1. Displacement 2. Speed 3. Acceleration due to gravity at t=1.5s.
Answers
Given,
A simple harmonic oscillation is represented by the equation, y = 5cos(2πt + π/4) in S.I units.
To find,
- displacement
- speed
- acceleration at t = 1.5 sec
standard equation of simple harmonic motion is given by, y = Acos(ωt + Φ)
on comparing we get,
Amplitude, A = 5 m
angular frequency, ω = 2π rad/s
phase constant, Φ = π/4 rad
1. displacement of simple harmonic oscillation, y = 5cos(2π × 1.5 + π/4)
= 5cos(3π + π/4) = 5cos(π + π/4)
= -5cos(π/4) = -5/√2 m
2. speed, v = dy/dt = -5 × 2π sin(2πt + π/4)
at t = 1.5 sec , v = -10π sin(3π + π/4) = 10/√2 = 5√2 m/s
3. acceleration, a = d²y/dt² = -5 × (2π)² cos(3π + π/4)
= -20π² cos(π + π/4) = 20π²/√2 = 10√2 π² m/s²
Explanation:
Maximum displacement is the amplitude A. The angular frequency ω , period T, and frequency f of a simple harmonic oscillator are given by ω=√km ω = k m , T=2π√mk,andf=12π√km T = 2 π m k , and f = 1 2 π k m , where m is the mass of the system and k is the force constant.