I would like to solve only D and E thank you. 1.) A cube of lead (Pb) with dimen

Question

I would like to solve only D and E thank you.

1.) A cube of lead (Pb) with dimensions 1 cm on an edge is dropped into a large cubical (a) After steady-state is reached, will it float or stay at the bottom of the container? (c) If the cube is pressed down an infinitesimal distance further by a small externally container of mercury (Hg) which is of dimensions 1 m on an edge b) Determine the distance below the surface that the lead cube sinks into the Hg applied force and then let go, show that the cube undergoes simple harmonic motion as a result of this action, and in the absence of any viscous damping determine the period and the frequency of this simple harmonic motion. (ad) Imagine that this mechanically vibrating system, which produces acoustic waves in the fluid, is now attached to an electromagnetic resonator in order to produce resonance electromagnetic waves. In what region of the ENM spectrum would these EM waves lie? (e) Sketch and explain the design of a detection system for these EM waves

Explanation / Answer

1. dimensdions of cube of mercury, a = 1 cm on edge, Pb

dropeed in Hg, container 1m cube edge

a. rho Hg = 13.56 g/cc

rho Pb = 11.34 g/cc

hence the lead will float in mercury as its density is lower

b. distance below surfasce the cube sinks = d

then from force balance

rhoPb*a^3*g = rhoHg*a^2*d*g

d = (rhoPb/rhoHg)a = 0.83 a = 0.83628 cm

c. when cube is pressed by distance dy

extra buoyant force = dF

dF = -rhoHg*a^2*gdy

now, dF = rhoPb*y"*a^3

hence

y" = -(rhoHg*g/rhoPb*a)dy

this can be compared to equation of SHM

y" = -w^2*dy

hence

w = sqrt(rhoHg*g/rhoPb*a)

d. f = w/2*pi = 5.45102222 Hz

hence if this is frequency of emiited EM wave, then this belongs to radio waves part of EM spectrum

the waves are specificlally extremely low frequency

e. to detect presence of such waves we need a diffraction grating 1/2 the wavelenght of this EM wave

a = lambda/2

lambda*5.45 = c

lambda = 0.5504*10^8 m

a = 2.75*10^7 m

this is a very long distance and such a diffractoin grating can only be produced using celestial bodies indirectly

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