A mass (M) is attached to the end of a spring of constant (k) and set into a mot

Question

A mass (M) is attached to the end of a spring of constant (k) and set into a motion by pulling it a distance (+A) horizontally and then releasing it. Let's agree to start our record of time the first time the mass (M) passes through the equilibrium position.
Given A = 0.1 meters, f = 0.250/s, M = 2.0 g

Determine the following:

a) Time for one complete oscillation

b) Displacement from the mass from equilibrium when time = 0.5 seconds

c) Speed of mass (M) when time = 1.5 seconds

d) Acceleration of mass (M) when time = 2.5 seconds

e) Total Energy of the System (Spring and Mass) at any time

Explanation / Answer

GIven Amplitude A = 0.1 m frequency f = 0.25 0 / s mass M = 2 g a)   time for one complete oscillation                 T = 1 / f                      = 1 / 0.25                      = 4 s b) Dispacement from the mass from equilibrium when t = 0.5 s            x = A cos t                = A cos ( 2f t )                = 0.1 cos (2*0.25* 0.5)                = 0.099 m c)   speed v = dx / dt                      = A * - sin t                     = - (2*0.25) ( 0.1 ) sin ( 2*0.25*0.5 )                      = - 0.0021 m/s   d) acceleration a = dv / dt                               = - 2 A cost                               =  - (2*0.25)2 ( 0.1 ) sin ( 2*0.25*0.5 )                               = -0.0033 m/s 2 e)    total energy of the system is                T = ( 1/2) k A 2

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