Classical Mechanics - Moment Inertia

The period of physical pendulum is 2π√(I/mgd), where I is the moment of inertia about the pivot point and d is the distance from the pivot to the center of mass. A circular hoop hangs from a nail on a barn wall. The mass of the hoop is 3 kilograms and its radius is 20 centimeters. If it is displaced slightly by a passing breeze, what is the period of the resulting oscillations?

A. 0.63 s
B. 1.0 s
C. 1.3 s
D. 1.8 s
E. 2.1 s

(GR9677 #21)

Solution:

T = 2π√(I/mgd)
m = 3 kg
r = 20 cm = 0.2 m
In this case d = r

To find total I:
Parallel axis theorem: I = ml² + I_CM
I_CM  = I_loop = mr² 

In this case l = r
→ I = mr² + mr² = 2mr²

T = 2π√(2mr² / mgr) 
= 2π√(2r/g) 
= 2π√[2(0.2)/(10)] 
= 2π(0.2)
= 1.25 s

Answer: C

Optics - Thin Film

A soap film with index of refraction greater than air is formed on a circular wire frame that is held in a vertical plane. The film is viewed by reflected light from a white-light source. Bands of color are observed at the lower parts of the soap film, but the area near the top appears black. A correct explanation for this phenomenon would involve which of the following?

I. The top of the soap film absorbs all the light incident on it; none is transmitted.

II. The thickness of the top part of the soap film has become much less than a wavelength of visible light.

III. There a phase change of 180o for all wave-lengths of light reflected from the front surface of the soap film.

IV. There is no phase change for any wavelength of light reflected from the back surface of the soap film

A. I only
B. II and III only
C. III and IV only
D. I, II and III
E. II, III and IV

(GR9277 #21)

Solution:

I. FALSE. 

Soap film does not absorb light.

II. TRUE.

The gravity effect on water molecules decreases the thickness, t of the top of the soap film, become much less than a wavelength, λ of visible light.

When t ≪ λ, the path difference between the front and the rear surface reflections is small compared to λ. It causes an insignificant phase shift. The result is complete destructive interference of the 2 light waves, creating a black band.

III and IV. TRUE.

Thin film: 

n₁〈  n₂ → there is 180^o phase change
n₁ 〉n₂ → no phase change 

n = refractive index

From air to soap (front surface):

n_air〈  n_soap → 180^o phase change

From soap to air (back surface):
n_soap 〉n_air → no phase change 

Answer: E

Special Relativity - Space-Time Interval

Two observers O and O’ observe two events A and B. The observers have a constant relative speed of 0.8c. In units such that the speed of light is 1, observer O obtained the following coordinates:

Event A: x = 3, y = 3, z = 3, t = 3 

Event B: x = 5, y = 3, z = 1, t = 5

What is the length of the space-time interval between these two events, as measured by O’.

A. 1
B. √2
C. 2
D. 3
E. 2√3

(GR8677 #21)

Solution:

dS²  = dr² + (cdt)²  =  dx² + dy² + dz² − (cdt)² 
dS² = dS'² 

dx² = (5 − 3)² = 4
dy² = (3 − 3)² = 0
dz² = (1 − 3)² = 4
dt²  = (5 − 3)² = 4
c = 1

dS'² = 4 + 0 + 4 − 4 = 4
dS' = 2

Answer: C

Nuclear & Particle Physics - Hydrogen Spectrum

In the hydrogen spectrum, the ratio of the wavelength for Lyman-radiation  (n = 2  to  n = 1) to Balmer-radiation (n = 3  to  n = 2)  is

A. 5/48
B. 5/27
C. 1/3
D. 3
E. 27/5

(GR0177 #21)

Solution:

Rydberg formula:



λ_vac = the wavelength of the light emitted in vacuum
R_H = Rydberg constant for Hydrogen
n₁ and n₂ are integers such that

For Lyman-radiation (n = 2 → n = 1):



For Balmer-radiation (n = 3 → n = 2):



The Ratio:



Answer: B