we have a lithium atom (atomic mass m = 6 amu) connected to two very much more massive atoms, as illustrated in the drawing below. We can pretend that the massive atoms don¡¯t move. The spring constants k1 and k2 are 73 N/m and 27 N/m, respectively. What is the angular frequency of the oscillation of the lithium atom? Units: THz (TeraHertz). 1 amu = 1.66 x 10-27 kg.
To calculate the angular frequency of the oscillation of the lithium atom, we need to calculate the effective spring constant and the reduced mass of the system.
1. Start by calculating the reduced mass (µ) of the system. The reduced mass represents the effective mass of a system of two masses connected by a spring. In this case, the lithium atom is much lighter compared to the two massive atoms.
The reduced mass (µ) is given by:
µ = (m1 * m2) / (m1 + m2)
Where m1 and m2 are the masses of the two atoms, and m1 is the mass of the lithium atom.
Using the given values:
m1 = 6 amu = 6 * 1.66 x 10^(-27) kg
m2 = m2 = infinite mass (due to being very massive)
µ = (6 * 1.66 x 10^(-27) kg * ∞) / (6 * 1.66 x 10^(-27) kg + ∞)
Since the mass of the second atom is infinite, the reduced mass becomes close to the mass of lithium atom itself:
µ ≈ m1 = 6 * 1.66 x 10^(-27) kg
2. Next, calculate the effective spring constant (k_eff) of the system. For a system with two springs connected in parallel, the effective spring constant is given by the sum of the reciprocals of the individual spring constants:
k_eff = (1 / k1) + (1 / k2)
Using the given values:
k1 = 73 N/m
k2 = 27 N/m
k_eff = (1 / 73 N/m) + (1 / 27 N/m)
3. Now, calculate the angular frequency (ω) of the oscillation using the formula:
ω = √(k_eff / µ)
Plugging in the values of k_eff and µ:
ω = √((1 / 73 N/m) + (1 / 27 N/m)) / (6 * 1.66 x 10^(-27) kg)
4. Finally, convert the angular frequency from radians per second to terahertz (THz):
1 THz = 10^12 Hz = 2π * 10^12 rad/s
The angular frequency in THz is:
ω_THz = ω / (2π * 10^12)
Calculate the above expression to find the final answer for the angular frequency in THz.