Extreme Position(Amplitude) of a particle performing SHM
Example : A particle moves along y-axis according to the equation y (in cm)=3sin100πt+8sin250πt−6.Find whether the motion is simple harmonic or not.Also, calculate amplitude of particle and its mean position. Solution: The given equation can be written as y=3sin100πt+(4−4cos100πt)−6 =3sin100πt+4sin(100πt+π/2)−2 or y=5sin(100π−53∘)−2 ymax=5−2=3 ymin=−5−2=−7 Mean position=2ymax+ymin=−2 cm
Use phase to understand relative position between two SHMs
Example: What is the minimum phase difference between two SHMs y1=sin(π/6)sin(ωt)+sin(π/3)cos(ωt) ; y2=cos(π/6)sin(ωt)+cos(π/3)cos(ωt) ?
Solution: y1=sin(ωt)sin(6π)+cosωtsin3π =sinωtcos(3π)+cosωtsin3π =sin(ωt+3π) y2=sin(ωt+6π) Thus phase difference is
Find time taken to travel between two given positions in SHM
Example: A particle executes SHM along a straight line with mean position at x=0 and with a period of 20 sec and amplitude of 5 cm. Find the shortest time taken by it to go from x=4cm to x=−3cm ?
Solution: x=Asin( ωt+ϕ) ω=202π=202π=10πsecrad let at t=0,x=4, thus 4=5 sin ϕ sin−154=ϕ Now for at t=t1, let x=−3, thus we have 5−3=sin(ωt1+ϕ) or sin−1(5−3)−sin−1(54)=ωt1 10π(−0.643−0.927)=t1 Solving we get t1=5sec
Displacement as a function of time is a simple harmonic motion
Standard equation of simple harmonic motion is: a=−w2x Any general equation satisfying the above criterion represents a simple harmonic motion. i.e. x=Asinwt
Angular displacement as a function of time
In angular SHM equation of motion is given by:
τ=−kθ General equation for angular displacement: θ=θosin(wt+ϕ)
Shift of displacement-time plot with change in phase
In the given plot, phase difference is π/4 x1(t)=Asin(ωt) x2(t)=Asin(ωt+π/4)