Respuesta :
This question involves the concept of simple harmonic motion.
a) amplitude = 5m
frequency = 0.199 /s
angular frequency = 1.25 rad/s
period = 5.0277 s
b) Equation of velocity; v = -6.25sin(1.25t + (π/4))
c) equation of the acceleration; a = 7.8125 sin(1.25t + (π/4))
d) spring constant = 4.2969 N/m
e) Potential energy equation; U = 53.711cos²(1.25t + [tex]\frac{\pi}{4}[/tex])
kinetic energy equation; K = 53.711sin²(1.25t + (π/4))
f) total energy = 53.711 J
The general expression of the phase form of simple harmonic motion is;
x(t) = Acos(ωt + φ)
where;
A is amplitude
ω is angular frequency
t is time
φ is phase constant
We are given;
Mass of particle; m = 2.75 kg
Motion of particle; x(t) = 5cos(1.25t + (π/4))
- a) Comparing this general expression to the one we are given in the question, we can see that;
A = 5 m
ω = 1.25 rad/s
φ = π/4
formula for frequency is;
f = ω/2π
f = 1.25/2π
f = 0.199 /s
Period is given by the formula;
T = 1/f
T = 1/0.1989
T = 5.0277 s
- b) From first derivative of the position equation, the velocity expressed with respect to time is given as;
v = −ωAsin(ωt + φ)
Plugging in the relevant values, we have;
v = -1.25 × 5 sin(1.25t + (π/4))
v = -6.25sin(1.25t + (π/4))
- c) From second derivative of the position equation, the acceleration with respect to time is given as;
a = −ω²Acos(ωt + φ)
Plugging in the relevant values, we have;
a = [tex]1.25^{2}[/tex] × 5 sin(1.25t + (π/4))
a = 7.8125 sin(1.25t + (π/4))
- d) Formula for the spring constant is;
k = ω²m
k = 1.25² × 2.75
k = 4.2969 N/m
- e) The potential energy is given by the formula;
U = ½Kx²
U = ½(4.2969)[4cos(1.25t + π/4)]²
U = 53.711cos²(1.25t + [tex]\frac{\pi}{4}[/tex])
The kinetic energy is given by the formula;
K = ½m[tex]v^{2}[/tex]
K = ½ × 2.75 × [-6.25sin(1.25t + (π/4))]²
K = 53.711sin²(1.25t + (π/4))
- F) The total energy is the sum of the potential and kinetic energy. Thus;
E = 53.711cos²(1.25t + [tex]\frac{\pi}{4}[/tex]) + 53.711sin²(1.25t + (π/4))
E = 53.711[cos²(1.25t + [tex]\frac{\pi}{4}[/tex]) + sin²(1.25t + (π/4))]
From trigonometry, we know that;
sin²x + cos²x = 1
Thus;
E = 53.711 J
Read more at; brainly.com/question/12722639
