Final answer:
The difference in energy between allowed oscillator states is calculated using the equation for the period of a simple pendulum. The value of n for a state with an energy of 5.00 J is calculated using the formula for quantum energy. The quantization of energy in this system can be observed.
Step-by-step explanation:
To determine the difference in energy between allowed oscillator states, we can use the equation for the period of a simple pendulum:
T = 2 * pi * sqrt(l/g)
Where T is the period, l is the length of the pendulum, and g is the acceleration due to gravity.
Given that each oscillation of the orangutan takes 3.00 seconds, we can calculate the period as:
T = 3.00 s / 1 oscillation
T = 6.00 s
Now we can use the equation to solve for the length of the pendulum:
6.00 s = 2 * pi * sqrt(l/9.8 m/s^2)
Simplifying the equation:
3.00 s = pi * sqrt(l/9.8 m/s^2)
Squaring both sides:
9.00 s^2 = pi^2 * (l/9.8 m/s^2)
Solving for l:
l = (9.00 s^2 * 9.8 m/s^2) / pi^2
Calculating the value of l:
l ≈ 91.16 m
Now, we can calculate the energy of a single quantum using the formula:
E = (n + 1/2) * hf
Where E is the energy, n is the quantum number, h is Planck's constant (6.626 × 10^-34 J*s), and f is the frequency of oscillations.
Given that the frequency is 1/3 Hz (since each oscillation takes 3.00 seconds), we can solve for the energy:
E = (n + 1/2) * (6.626 × 10^-34 J*s) * (1/3 Hz)
Now, we can plug in the energy value of 5.00 J and solve for the quantum number:
5.00 J = (n + 1/2) * (6.626 × 10^-34 J*s) * (1/3 Hz)
Simplifying the equation:
15.00 J*Hz = (n + 1/2) * 6.626 × 10^-34 J*s
Solving for n:
n + 1/2 = (15.00 J*Hz / 6.626 × 10^-34 J*s)
n + 1/2 ≈ 2.27 × 10^34
n ≈ 2.27 × 10^34 - 1/2
n ≈ 2.27 × 10^34
The quantization of energy in this system can be observed, as the energy is not continuous, but rather quantized into distinct states.