# SHM: atom in a lattice

1. Feb 7, 2008

### catkin

[SOLVED] SHM: atom in a lattice

1. The problem statement, all variables and given/known data

An atom of mass $$5 \times 10^{-26}$$ kg is in a cubic lattice with all bonds between adjacent pairs of atoms having a spring constant about 100 N/m. Estimate the natural frequency of oscillation of the atom. If it is able to absorb radiation at this frequency, what part of the electromagnetic spectrum does it absorb?

2. Relevant equations
$$F = ma$$ (Newton 2)
$$F = -kx$$ (Hooke)
$$a = -\omega^{2}x$$ (SHM)

3. The attempt at a solution
From Newton 2 and Hooke
$$a = -kx / m$$
Substituting in SHM
$$-kx / m = -\omega^{2}x$$
$$\omega = \sqrt{k/m}$$
$$= \sqrt{\frac {100} {5 \times 10^{-26}}}$$
$$= 4 \times 10^{13}$$ Hz ct1sf

My difficulty is that the answer given in the book is $$7.12 \times 10^{12}$$ Hz. What have I done wrong? Using $$\omega = \sqrt{k/m}$$ and the given value of m, k would have to be ~2.55 N/m. ???

2. Feb 7, 2008

### Shooting Star

The ans given is correct. Check your arithmetic.

3. Feb 7, 2008

### catkin

Arithmetic check:
$$\sqrt{\frac {100} {5 \times 10^{-26}}}$$
$$= \sqrt{\frac {100} {5}} \times 10^{13}}$$
$$= \sqrt{20}\times 10^{13}}$$
$$= 4.472\times 10^{13}}$$

I still can't see my mistake

4. Feb 7, 2008

### catkin

Ah! f = ω / 2π