The total energy of the system can be calculated by considering both the potential energy and the kinetic energy of the mass-spring system.
The potential energy of the spring is given by the formula:
Potential Energy = (1/2) * k * x^2
where k is the spring constant and x is the amplitude of the oscillation.
In this case, the spring constant (k) is 4.5 N/m and the amplitude (x) is 0.045 m. Plugging these values into the formula, we have:
Potential Energy = (1/2) * 4.5 N/m * (0.045 m)^2
Potential Energy = (1/2) * 4.5 N/m * 0.002025 m^2
Potential Energy = 0.0045375 J
The kinetic energy of the mass is given by the formula:
Kinetic Energy = (1/2) * m * v^2
where m is the mass and v is the velocity of the mass.
In this case, the mass (m) is 0.35 kg. To determine the velocity (v), we need to consider the relationship between the amplitude and the maximum velocity in simple harmonic motion. The maximum velocity (vmax) is related to the angular frequency (ω) and the amplitude (x) as follows:
vmax = ω * x
The angular frequency (ω) can be calculated using the formula:
ω = sqrt(k / m)
Plugging in the values for k and m, we have:
ω = sqrt(4.5 N/m / 0.35 kg)
ω = sqrt(12.8571 rad/s^2)
Using the calculated value of ω and the given amplitude (x), we can now find the maximum velocity (vmax):
vmax = (12.8571 rad/s^2) * 0.045 m
vmax = 0.57857 m/s
Finally, we can calculate the kinetic energy using the mass and the maximum velocity:
Kinetic Energy = (1/2) * 0.35 kg * (0.57857 m/s)^2
Kinetic Energy = 0.0608 J
To find the total energy of the system, we simply add the potential energy and the kinetic energy:
Total Energy = Potential Energy + Kinetic Energy
Total Energy = 0.0045375 J + 0.0608 J
Total Energy = 0.0653375 J
Therefore, the total energy of the system is approximately 0.0653 J.