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 = K m . Suppose that at time t = 0 , the posi

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 $ω = \sqrt{\frac{K}{m}}$ .
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 $x (t) = C \cos (ω t - ϕ),$ where $C$ and $ϕ$ are

TestHub · Accepted Answer

x = A s i n ω t + B c o s ω t v = d x d t = A ω cos ω t - B ω sin ω t At t = 0 , x ( 0 ) = B v ( 0 ) = A ω x = A sin ω + B sin ω t + 90 ° A net = A 2 + B 2 tan α = B A ⇒ cot α = A B ⇒ 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 = [ v ( 0 ) ] 2 ω 2 + [ x ( 0 ) ] 2 ϕ = 90 - α tan α = cos α = A B ⇒ tan ϕ = v ( 0 ) x ( 0 ) · ω ϕ = tan - 1 v ( 0 ) x ( 0 ) ω

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