You have fabricated an array of rectangular double$-$clamped silicon beams for detecting small

airborne droplets. The beams measure $100$ nm $($thick$)$ by $200$ nm $($wide$)$ by $5\backslash$mu m $($long$)$ and have

the elasticity modulus and density values stated below.

E M $($Si$)=110$ GPa $(1$ GPa $=109$ N$/$m $2$

$)$

$\backslash$rho $($Si$)=2330$ kg$/$m $3$

$($a$)$ Assuming no damping, calculate the natural resonance frequency $($in MHz$)$ and quality $($Q$)$

factor $($assuming their resonance has a bandwidth of $30$ kHz$)$ of the beams.

$($b$)$ How much would you expect the resonance to shift $($in kHz$)$ if the beams capture a $50$ fg $(1$ fg $=$

$10-15$ g$)$ droplet?