Show that simple harmonic motion y(t) = C1 cos ωt + C2 sin ωt can be written as:
a. y(t) = A sin(ωt + ϕ0)
b. y(t) = A cos(ωt + ϕ1)
1
Expert's answer
2022-05-04T15:06:38-0400
At first, rewrite the function in the following way: y(t)=C12+C22(C12+C22C1cosw˙t+C12+C22C2sinw˙t). Point out that −1≤C12+C22C1≤1 and (C12+C22C1)2+(C12+C22C2)2=1 Therefore, there is such ϕ0 that C12+C22C1=cosϕ0 and C12+C22C2=sinϕ0. It can be obtained from the properties of functions arccos, arcsin. We receive: y(t)=(cosϕ0cosw˙t+sinϕ0sinw˙t)=cos(wt−ϕ0). Thus, we get: y(t)=Acos(˙wt−ϕ0), where A=C12+C22 . After the change: ϕ0→−ϕ1 we receive formula b). After the change w→−w, ϕ0→ϕ0−2π in formula y(t)=Acos(˙wt−ϕ0) we receive: y(t)=Acos(˙2π−wt−ϕ0)=Asin(wt+ϕ0). It is formula a). .
Comments