A water molecule is sometimes modeled as an oxygen connected by \"springs\" to t

Question

A water molecule is sometimes modeled as an oxygen connected by "springs" to two hydrogen atoms. Suppose that one of the H atoms in a water molecule vibrates with a frequency of 3 GHz (1 GHz = 1 billion Hz). Given that the mass of a hydrogen atom is about 1.67 times 10^-27 kg, a. Estimate the "spring constant" for this system. b. What is the period of the vibration? c. If the amplitude of the vibration is on the order of 1 nm, estimate the energy of the atom's vibrations, remembering that there are two hydrogen atoms and therefore two "springs".

Explanation / Answer

a)we know that, the angular frequency in SHM is given by

w = sqrt (k/m)

w^2 = k/m

k = w^2 m

k = (2 pi f)^2 m

k = 4 x 3.14^2 x (3 x 10^9)^2 x 1.67 x 10^-27 = 5.93 x 10^-7 N/m

Hence, k = 5.93 x 10^-7 N/m

b)Period is:

T = 1/f

T = 1/3 x 10^9 = 3.33 x 10^-10 s

Hence, T = 3.33 x 10^-10 s

c)We know that when two springs are in series

1/Keq = 1/k1 + 1/k2 = 2/k

Keq = k/2 = 5.93 x 10^-7/2 = 2.97 x 10^-7 N/m

U = 1/2 k x^2

U = 0.5 x 2.97 x 10^-7 x (10^-9)^2 = 1.485 x 10^-25 J

Hence, U = 1.485 x 10^-25 J

Related Questions

Navigate

• Browse (All)
• Browse A
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.