The motion of a mass on a spring, with spring constant K is as shown in figure. The equation of motion is given by x ( t ) = A sin ω t + B cos ω t with ω = m K Suppose that at time t = 0 , the pos

Question

The motion of a mass on a spring, with spring constant K is as shown in figure.

The equation of motion is given by $x (t) = A sin ω t +$ B $cos ω t$ with $ω = \frac{K}{m}$

Suppose that at time $t = 0$ , the position of mass is $x (0)$ and velocity $v (0)$ , then its displacement can also be represented as $x (t) = Cos (ω t - ϕ)$ , where C and $ϕ$ are:

TestHub · Accepted Answer

x = A sin ω t + B cos ω t v = dt dx ​ = A ω cos ω t − B ω sin ω t At t = 0 , x ( 0 ) = B v ( 0 ) = A ω x = A sin ω t + B sin ( ω t + 9 0 ∘ ) A n e t ​ = A 2 + B 2 ​ tan α = A B ​ ⇒ cot α = B A ​ ⇒ x = A 2 + B 2 ​ sin ( ω t + α ) ⇒ x = A 2 + B 2 ​ cos ( ω t − ( 90 − α )) x = C cos ( ω t − ϕ ) ⇒ C = A 2 + B 2 ​ C = ω 2 [ v ( 0 ) ] 2 ​ + [ x ( 0 ) ] 2 ​ ϕ = 90 − α tan α = cos α = B A ​ ⇒ tan ϕ = x ( 0 ) ω v ( 0 ) ​ ϕ = tan − 1 ( x ( 0 ) ω v ( 0 ) ​ )

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