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.