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To determine the velocity and acceleration of the oscillating mass at a specific time, we'll need to use the equations for simple harmonic motion (SHM). The position of an object undergoing SHM can be described by the equation:

x(t) = A * cos(ωt + φ)

where:

  • x(t) is the position of the object at time t,
  • A is the amplitude of the oscillation,
  • ω is the angular frequency (2π times the frequency), and
  • φ is the phase angle.

To find the velocity and acceleration, we need to differentiate the position equation with respect to time.

Given: Amplitude (A) = 4.00 m Frequency (f) = 0.5 Hz Phase angle (φ) = π/4

We can calculate the angular frequency (ω) using the formula: ω = 2πf

Substituting the given values: ω = 2π * 0.5 = π rad/s

Now, let's find the position (x) at time t = 5s:

x(t) = A * cos(ωt + φ) x(5) = 4.00 * cos(π * 5 + π/4) x(5) = 4.00 * cos(6π + π/4) x(5) = 4.00 * cos(25π/4)

To calculate the velocity, we differentiate the position equation with respect to time:

v(t) = dx(t)/dt = -A * ω * sin(ωt + φ)

Substituting the known values: v(5) = -4.00 * π * sin(π * 5 + π/4) v(5) = -4.00 * π * sin(6π + π/4) v(5) = -4.00 * π * sin(25π/4)

Finally, to calculate the acceleration, we differentiate the velocity equation with respect to time:

a(t) = dv(t)/dt = -A * ω^2 * cos(ωt + φ)

Substituting the known values: a(5) = -4.00 * π^2 * cos(π * 5 + π/4) a(5) = -4.00 * π^2 * cos(6π + π/4) a(5) = -4.00 * π^2 * cos(25π/4)

Calculating these values will give us the velocity and acceleration of the object at time t = 5s.

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