A spring and mass system are oscillating with an amplitude of 8.4 cm. The spring constant is 204 N/m and the mass is 540 g. Find the mechanical energy of the system to 2 sf.
x = .084 sin wt
max x = .084 when sin w t = 1
then
Pe = (1/2) k x^2 = (1/2) 204)(.084)^2
now what is Ke when sin wt = 1?
v = .084 w cos wt
if sin wt = 1, then cos wt = 0 because
sin^2 + cos^2 = 1 always
so
v = 0 when sin wt = 1
so
TOTAL energy = PE at sin wt = 1
= (1/2) (204)(.084)^2 Joules still
To find the mechanical energy of the system, we need to consider both the potential energy stored in the spring and the kinetic energy of the mass.
The potential energy (PE) of the spring is given by the equation:
PE = (1/2) * k * x^2
Where k is the spring constant and x is the displacement from the equilibrium position (amplitude in this case).
In this scenario, the displacement (x) is given as 8.4 cm, which can be converted to meters (m) by dividing by 100:
x = 8.4 cm / 100 = 0.084 m
The spring constant (k) is given as 204 N/m.
Now we can calculate the potential energy stored in the spring:
PE = (1/2) * 204 N/m * (0.084 m)^2
PE ≈ 0.713 N·m or J (Joules)
The kinetic energy (KE) of the mass can be calculated using the equation:
KE = (1/2) * m * v^2
Where m is the mass and v is the velocity of the mass.
The mass (m) is given as 540 g, which can be converted to kilograms (kg) by dividing by 1000:
m = 540 g / 1000 = 0.54 kg
Since the mass is oscillating, the maximum velocity (v) occurs when the displacement is zero. At this point, all the potential energy is converted to kinetic energy.
The maximum velocity (v) can be determined from the properties of simple harmonic motion. The maximum velocity is related to the amplitude (A) and the angular frequency (ω) of the system by the equation:
v = A * ω
The angular frequency (ω) is related to the spring constant (k) and the mass (m) by the equation:
ω = √(k / m)
Plugging in the values:
ω = √(204 N/m / 0.54 kg)
ω ≈ √(377.78)
ω ≈ 19.44 rad/s
Now, we can calculate the kinetic energy:
KE = (1/2) * 0.54 kg * (A * ω)^2
KE = (1/2) * 0.54 kg * (0.084 m * 19.44 rad/s)^2
KE ≈ 0.494 J or N·m
Finally, to determine the mechanical energy (E) of the system, we sum the potential and kinetic energies:
E = PE + KE
E ≈ 0.713 J + 0.494 J
E ≈ 1.21 J (rounded to 2 significant figures)
Therefore, the mechanical energy of the system is approximately 1.21 J.