The complete question is;
A 1.15-kg mass oscillates according to the equation x = 0.650 cos(8.40t) where x is in meters and t in seconds. Determine (a) the amplitude, (b) the frequency, (c) the total energy, and (d) the kinetic energy and potential energy when x = 0.360 m.
Answer:
A) Amplitude; A = 0.650 m
B) Frequency; f = 1.337 Hz
C) total energy = 17.142 J
D) Kinetic energy = 11.884 J
Potential Energy = 5.258 J
Explanation:
We are given;
Mass;m = 1.15 kg
Equation; x = 0.650 cos (8.40t)
(a) The standard form of a wave function is in the form y(x,t) = Asin(kx−ωt+ϕ)
So, comparing terms in our equation in the question to this, the amplitude is;
A = 0.650 m
(b) we know that formula for frequency is;
f = ω/2π
Again, comparing terms in the standard equation and our question, we can see that ω = 8.4
Thus;
f = 8.4/(2π)
f = 1.337 Hz
(c) Formula for the total energy is given by;
E = m•ω²•A²/2
Plugging in the relevant values, we have;
E = (1.15)(8.40)²(0.650)²/2
E = 17.142 J
(d) we want to find the kinetic energy and potential energy when x = 0.360 m.
The formula for kinetic energy in this case is given by;
K = (1/2)•m•ω²•(A² - x²)
Thus;
K = (1/2) × (1.15) × (8.40)² × ((0.650)² - (0.360)²)
K = 11.884 J
Also, the formula for the potential energy in this case is given by;
U = (1/2)•m•ω²•x²
Thus;
U = (1/2) × (1.15) × (8.40)² × (0.360)²
U = 5.258 J
