A 200 g mass attached to a horizontal spring oscillates at a frequency of 2.0 Hz. At *t *= 0 s, the mass is at *x *= 5.0 cm and has *v**x *= -30 cm/s.

Write down an equation that describes the position of the oscillating mass as a function of time ( ie. what is **x(t)? ****).**

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First, we have to solve for the phase constant and the amplitude in order to get the position at t = 0.4s. Consider the following expressions:

`x(t) = A cos(omega*t + phi)`

`v(t) = dx/dt = -omega A sin(omega*t + phi)`

where:

x = length/distance = 5.0 cm

v = velocity = -30 cm/s

f = frequency = 2.0Hz

t = time = 0 s

A = amplitude

`phi ` = phase constant

`omega ` = angular frequency

`omega = 2*pi*f`

`omega = 2*pi*2.0`

`omega = 4pi`

**Phase Constant**

`x(t) = A cos(omega*t + phi)`

`5.0 = A cos (4pi*0 + phi)`

`5.0 = A cos (0 + phi)`

`5.0 = A cos(phi)` equation 1

`v(t) = dx/dt = -omega A sin(omega*t + phi)`

`-30 = -4pi A sin(4pi*0 + phi)`

`-30 = -4pi A sin(0 + phi)`

`30 = 4pi A sin(phi)` equation 2

Now divide the expression 2 by the expression 1 (2/1):

`30/5.0 = (4pi A sin(phi))/(A cos(phi))`

`6 = 4pi (sin phi)/(cos phi)`

Remember from trigonometric identities: tanx = sinx/cosx

`6 = 4pi*tan phi`

`6/(4pi) = tan phi`

`phi = tan^-1 (6/(4pi))`

`phi = 25.52 ` -> phase constant

** **

**Amplitude**

`A = (x)/(cos phi)`

`A = 5.0/(cos 25.52)`

`A = 5.5407`

Now we can solve the position at t = 0.4 s

Finally,

`x(t) = A cos(omega*t + phi)`

`x(t) = 5.5407 cos (4pi*0.4 + 25.52)`

**x(t) = 4.8 cm**

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