Here, we are required to obtain information about the 2.75 kg particle and generate equations for it's velocity and acceleration, potential and kinetic energy, and find the it's total energy.
By convention, a particle moving as a function of time as follows; X(t) = Acos(ωt + Ф).
(a). By comparison with the equation of the 2.75kg particle, the amplitude, A = 5m.
Also, ω = 1.25.
ω = 2πf =1.25
Therefore, frequency, f = 1.25 / 2× 3.142..
f = 0.199Hz ≈ 0.2Hz.
And the angular frequency, ω = 1.25rad/sec.
The period, T = 1/f = 1/0.2 = 5seconds.
(b). The equation of the velocity of the particle can be obtained by taking the differential of the displacement function, x(t).
Therefore, velocity v = dx/dt
Therefore, v = 1.25 × 5 × -sin(1.25t + π/4).
Therefore, v = -6.25 sin(1.25t + π/4)
(c). The equation of the particle's acceleration can be obtained by further differentiating the equation of the particle's velocity.
Therefore, acceleration a = dv/dt.
Therefore, a = -6.25 × 1.25 cos(1.25t + π/4).
Therefore, a = -7.8125 cos(1.25t + π/4).
(d). v = fλ = ωk....................where k = spring constant.
Therefore, k = fλ/ω.
(e). The equation for the potential energy is given as P.E = 1/2 × kx² = 1/2 × mω²x².
Therefore, P.E = 1/2 × m × ω² × 25 cos²(1.25t + π/4).
The equation for the kinetic energy is given as K.E = 1/2 × m × v²
Therefore, K.E = 1/2 × m × 39.0625 × sin²(1.25t + π/4).
Therefore, K.E = 19.53125 × m × sin²(1.25t + π/4).
(f). The total energy = The kinetic energy + The potential energy
The total energy = 12.5 × m × ω² × cos²(1.25t + π/4) + 19.53125 × m × sin²(1.25t + π/4).
Read more:
https://brainly.com/question/16036080
