How Do You Solve Complex Oscillation Problems in Physics?

[johnq2k7]

Physics Advanced Oscillation problem, Help Needed!

Consider an atom of mass m bonded to the surface of a much larger immobile body by electromagnetic forces. The force binding the atom to the surface has the expression:

F= exp((a*cos(z))+(b*sin(z))) +d*tan(z)

where a,b, and d are constants and z is positive upwards. The equilibrium point is defined to be the origin, so z=0 there. Ignore an motion except in the vertical direction. THe whole assembly is subject to normal Earth gravity.

a.) For small oscillations, give an approx. expression for the binding force on the atom.
b.) What restriction are there on the values of a,b, and d so that the force on the atom actually is a restoring force and the atom can reach a stationary equilibrium?
c.) What is the angular frequency w_0 and frequency v_0 of oscillation of this undamped system?
d.) Would the oscillation frequency change if there were no gravity? Why?
e.) Subject to the considerations above, if the atom has a mass of 1 atomic mass unit, b= -1.658 x10^30 and d= -3.2361 x10^4 N, what is the frequency v_0 of he oscillation?
f.) Suppose that a photon with this frequency that is incident on this atom would absorbed. What wavelength does this correspond to? What part of the spectrum does it fall in?


I need a lot of help with these problems, help would be appreciated!

 

[CaptainEvil]

Because we're dealing with small oscillations, we can expand your function F as a taylor series and ignore higher powers of z.

F(z) = F₀ + z(\deltaF/\deltaz)₀

for part b, at equilibrium, your electric force must balance gravity. then F₀ = mg and you can solve for a in terms of mg.

You also know K < 0 for a restoring force, and you can find limits on b and d. (you can find K in your F(z) taylor series. Remember F = F₀ - kx (or in this case kz))

Try it and see how you do,

cheers

 

[johnq2k7]

ok for part a.) would it be F(z) = (z) (((exp^(acos(z)+ bsin(z))* (asin(z)+bcos(z)))dsec^2(z))for part b.) I'm confused if F(0)= mg do u state,

mg= exp(acos(z)+bsin(z)) +dtan(z)

and then find equivalent expressions for a,b,d?

how do u compute part c. and d.?

 

[CaptainEvil]

No,

F at 0 is e^a and
The first derivative of F at 0 is be^a + d

so F (z) = e^a + (be^a +d)z

for part c, you should have equations in your textbook, or google them

 

[johnq2k7]

then would part b be equivalent to the following,

since F(0)= mg

would it be mg= e^a + (be^a +d)z

and then the expressions for a,b, and d would be determined in terms of mg, b,d,etc

for part c.) angular frequency is measured as the sqrt (k/m) since

k= dF/dz at 0

would u substitute F'(z) at 0 expression into the equation.. how do u relate mass though

addtionally, for frequency, the expression is equivalent to 1/2pi multipled by the sqrt (k/m)


?

 

[CaptainEvil]

You are correct for part (b)

For part (c), those equations are all you need. Don't plug in values just yet, you do that in part (e) when mass is given to you.

 

[johnq2k7]

for part d.) when there's no gravity

would F(0)= 0

therefore would the frequency not change considering the value of k is determined by dF/dz at 0

and since the frequency is given by 1/2pi times the sqrt (k/m)

would the oscillation frequency not change?

for part f.)

do u determine the value of the oscillation frequency from part e.. and use the wave spectra to determine the wavelength and the spectra it falls under


please help

 

[CaptainEvil]

because a is dependent on g, and a is in the expression for oscillation frequency, that it will be affected if gravity should be neglected.

When you find the frequency, use the relationship between frequency, speed of light and wavelength to calculate wavelength of the photon

 

[johnq2k7]

additionally,

how do i evaluate the restrictions on a,b, and d for the part b.

i know that the expression

will mg= e^a + (b*(e^a)+ d)

how do i simplify the expression in terms of a
and b, and d

if i apply logarithm rules.. it still won't work
plus it says to state the restrictions

sorry for the trouble

 

[CaptainEvil]

mg = e^a only since gravity balances electric force ONLY AT EQUILIBRIUM

then you know a = ln(mg)

additionally, since spring constant k must be a restoring force, that k < 0

you know from F(z) that be^a + d = k, so if that expression needs to be < 0 , solve for b and d

 

[johnq2k7]

since F is approx equal to -kx since were assuming it's a restoring force

shouldn't the expression be k>0 and NOT k<0 ?

confusion

 

[CaptainEvil]

yes so in order for k to be negative, it needs to be < 0