To calculate the total energy of a frictionless oscillator attached to a spring, you need to consider both the potential energy and the kinetic energy associated with the oscillation.
- Potential Energy: The potential energy of a spring is given by the equation:
PE = (1/2)kx²
where PE is the potential energy, k is the spring constant, and x is the displacement from the equilibrium position.
Given that the spring constant (k) is 5.5 N/m and the amplitude (A) is 1.5 cm (or 0.015 m), the maximum displacement (x) can be calculated as half of the amplitude:
x = A/2 = 0.015 m / 2 = 0.0075 m
Now we can calculate the potential energy:
PE = (1/2) * (5.5 N/m) * (0.0075 m)²
- Kinetic Energy: The kinetic energy of an oscillator is given by the equation:
KE = (1/2)mv²
where KE is the kinetic energy, m is the mass of the oscillator, and v is the velocity.
Given that the mass (m) is 250 grams (or 0.25 kg), we need to find the maximum velocity (v). The maximum velocity can be calculated using the equation:
v = ωA
where ω is the angular frequency.
The angular frequency (ω) can be determined using the equation:
ω = √(k/m)
Substituting the values:
ω = √(5.5 N/m / 0.25 kg) = √22 rad/s ≈ 4.69 rad/s
Now we can calculate the maximum velocity:
v = (4.69 rad/s) * (0.015 m)
Finally, we can calculate the kinetic energy:
KE = (1/2) * (0.25 kg) * (v)²
- Total Energy: The total energy (E) of the oscillator is the sum of the potential energy (PE) and the kinetic energy (KE):
E = PE + KE
Now you can substitute the calculated values into the equation to find the total energy.