Question #336995

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(C1C12+C22cosw˙t+C2C12+C22sinw˙t)y(t)=\sqrt{C_1^2+C_2^2}(\frac{C_1}{\sqrt{C_1^2+C_2^2}}\cos\.wt+\frac{C_2}{\sqrt{C_1^2+C_2^2}}\sin\.wt). Point out that 1C1C12+C221-1\leq\frac{C_1}{\sqrt{C_1^2+C_2^2}}\leq1 and (C1C12+C22)2+(C2C12+C22)2=1\left(\frac{C_1}{\sqrt{C_1^2+C_2^2}}\right)^2+\left(\frac{C_2}{\sqrt{C_1^2+C_2^2}}\right)^2=1 Therefore, there is such ϕ0\phi_0 that C1C12+C22=cosϕ0\frac{C_1}{\sqrt{C_1^2+C_2^2}}=\cos\phi_0 and C2C12+C22=sinϕ0\frac{C_2}{\sqrt{C_1^2+C_2^2}}=\sin\phi_0. It can be obtained from the properties of functions arccos\arccos, arcsin\arcsin. We receive: y(t)=(cosϕ0cosw˙t+sinϕ0sinw˙t)=cos(wtϕ0)y(t)=(\cos\phi_0\,\cos\.wt+\sin\phi_0\sin\.wt)=\cos(wt-\phi_0). Thus, we get: y(t)=Acos(˙wtϕ0)y(t)=A\,\cos\.(wt-\phi_0), where A=C12+C22A=\sqrt{C_1^2+C_2^2} . After the change: ϕ0ϕ1\phi_0\rightarrow-\phi_1 we receive formula b). After the change www\rightarrow-w, ϕ0ϕ0π2\phi_0\rightarrow\phi_0-\frac{\pi}2 in formula y(t)=Acos(˙wtϕ0)y(t)=A\,\cos\.(wt-\phi_0) we receive: y(t)=Acos(˙π2wtϕ0)=Asin(wt+ϕ0)y(t)=A\,\cos\.(\frac{\pi}2-wt-\phi_0)=A\sin(wt+\phi_0). It is formula a). .


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