PHY 206 SPRING Problem #1. Problem #2 NAME: SIGNATURE: Problem #3 UM ID: Problem #4. Problem #5. Problem #6. Total. Prof. Massimiliano Galeazzi

Size: px

Start display at page:

Download "PHY 206 SPRING Problem #1. Problem #2 NAME: SIGNATURE: Problem #3 UM ID: Problem #4. Problem #5. Problem #6. Total. Prof. Massimiliano Galeazzi"

Rolf Knight
6 years ago
Views:

1 PHY 06 SPRING 006 Prof. Massimiliano Galeazzi Final May 3, 006 Problem #1 NAME: Problem # SIGNATURE: UM ID: Problem #3 Problem #4 Problem #5 Problem # Total Some useful relations: Pressure & depth: p = ptop + ρgh Continuity: A 1v1 = Av Bernoulli: 1 1 p1 + ρ gh1 + ρv1 = p + ρgh + ρv Specific heat capacity: dq = mcdt Latent heat: Q = ± ml Heat current: dq TH TC H = = ka dt L Average molecular energy per degree of freedom: ½ k B T 1 st lat of thermodynamic: U = Q - W Heat in an isobaric process: Q = n C P T Heat in an isochoric process: Q = n C V T Work: dw = p dv Ideal gas equation of state: p V = n R T 1

2 Engine efficiency: Entropy: Wave equation: Speed of propagation of a wave on a string: Doppler shift: Intensity: Light reflection: Light refraction: Mirrors and thin lenses: W e = Q H dq ds = T y( x, t) 1 = x v v = T µ v + v f L L = fs v + vs Power I = Area θ r =θ a na sin θa = nb sinθb y' s' + =, m = = s s' f y s y( x, t) with R = ( n 1) for lenses f R1 R Diffractive interface: na nb nb na y' na s' + =, m = = s s ' R y nbs Phase difference in interference: φ = π N, with N=difference in # of wavelengths For path difference: r 1 r N = λ Constructive interference: φ = nπ ; Destructive interference: φ = ( n + 1) π Constructive interference: r 1 r = nλ ; Destructive interference: r 1 1 r = ( n + ) λ Time dilation: T obs = γ T 0 with T 0 = proper time Length contraction: L obs = L 0 /γ with L 0 = proper length c + u Relativistic Doppler Shift: f c u Lorentz coordinate transformations: x' = γ ( x ut) y' = y z' = z ux t' = γ t c t obs = fo (for approaching source) Lorentz velocity transformations: vx u v' x = 1 ( vxu c ) v y v' y = γ[1 ( vxu c )] v v' = z z γ[1 ( vxu c )]

3 Problem #1 Sam is on Earth and sees two spaceships Alpha and Beta at a distance d from each other and moving toward each other along the x-axis. The spaceships are out of control and are headed to a head-on collision. Sam measures spaceship Alpha to be moving with speed v a and spaceship Beta with speed v b > v a. a) How long will it be, from Sam s point of view, before the two spaceships collide? b) How much time do the passengers of Alpha have to evacuate the spaceship before the collision? c) How fast is spaceship Beta traveling from the point of view of Alpha? d) Alpha sends a light signal to Beta to warn the passengers of that ship. However the signal is not seen by them because the frequency f with which the signal is received by Beta is not in the visible anymore. What was the original frequency of the signal? e) From Sam s point of view, both spaceships are identical. In reality, which one is longer? Why? NOTE: you can leave your answers in terms of γ, as long as you first explicitly write the expression for it.

4 PHY 06 SPRING 006 FINAL 5/3/006 4

5 PHY 06 SPRING 006 FINAL 5/3/006 Problem # Light with frequency f o is incident on a thin glass slab with index of refraction n. The light can be reflected by either the top or the bottom surface of the glass and the two reflected beams interfere constructively when the angle between the incident light beam and the normal to the glass is θ ο. Remember to write your answers only in terms of the input parameters f o, n, and θ ο. a) What is the wavelength of the light in the glass? b) What is the angle of refraction θ b inside the glass? c) What is the thickness d of the glass? d) For the same incident angle θ ο, which frequency will be transmitted constructively by the glass? e) If you put a second slab, this time made of plastic with index of refraction n <n, behind the first one and in contact with it, how do the conditions for interference in reflection and refraction change? 5

6 PHY 06 SPRING 006 FINAL 5/3/006 6

7 PHY 06 SPRING 006 FINAL 5/3/006 Problem #3 An expandable balloon is filled with a volume V of an ideal gas (much denser than air) and thrown into the water (density ρ w ). The mass of the gas is M, the mass of the balloon is m. Neglect any elastic tension of the balloon. a) What is the density of the gas inside the balloon? b) What is the fraction of the balloon underneath the water level? c) The balloon is forced into the water down to a depth L, much bigger than the diameter of the balloon. Assuming that the temperature is the same everywhere, what is the new volume of the air inside the balloon? d) If the balloon is suddenly released, what is its initial acceleration? Assume that the density of the water does not change with depth. e) The balloon will then reach the surface and jump out of the water. What is its acceleration right after it gets completely out of the water? Neglect any friction. 7

8 PHY 06 SPRING 006 FINAL 5/3/006 8

$PHY 06 SPRING 006 FINAL 5/3/006 Problem #4 A glass rod, with index of refraction n, has one end shaped as a half sphere with radius R (see figure).$

9 PHY 06 SPRING 006 FINAL 5/3/006 Problem #4 A glass rod, with index of refraction n, has one end shaped as a half sphere with radius R (see figure). An object with height y is placed at a distance R to the left of the glass. a) Draw on the figure at least two principal rays and the image of the object due to the first surface of the glass. b) What is the distance between the vertex V and the image? c) What is the height of that image? d) The rod is now cut along the dashed line in the figure, leaving the shape of a hemisphere. What is the distance between the center of the hemisphere O and the final image? 9

10 PHY 06 SPRING 006 FINAL 5/3/006 10

11 PHY 06 SPRING 006 FINAL 5/3/006 Problem #5 A monoatomic ideal gas is initially at temperature T o, pressure p o, and volume V o (state a in the figure). The pressure in the gas is increased isochorically to a value 4p o (state b). The gas then expands isothermally to a volume V o (state c) and finally returns to its original state following a straight line on a p-v diagram (see figure). a) What are the values of volume, temperature, and pressure is states b and c? Remember to write your answers in terms of only the input parameters T o, p o, and V o. b) What are the values of work W, change in internal energy U, and heat transfer Q in process a-b? c) What are the values of work W, change in internal energy U, and heat transfer Q in process b-c? d) What are the values of work W, change in internal energy U, and heat transfer Q in process c-a? e) What is the efficiency of an engine using this cycle? 11

12 PHY 06 SPRING 006 FINAL 5/3/006 1

13 PHY 06 SPRING 006 FINAL 5/3/006 Problem #6 In the figure, string 1 and string have the same length L from the wall to the wheel, however, in that length, string 1 has mass m 1 and string has mass m. Both strings are under tension owing to the hanging block of mass M [figure (a)]. Remember to write your answers in terms of the input parameters L, m 1, m, and M. a) What is the frequency of the fundamental harmonic produced by standing waves on string 1? b) The block is now divided into two blocks (with M 1 + M = M) and the apparatus rearranged as shown in Figure (b). Find the values of M 1 and M such that the two strings produce the same sound. c) What is the frequency of the sound produced with the arrangement described in part (c)? d) What is the average energy transported by the standing waves generated in each string? 13

14 PHY 06 SPRING 006 FINAL 5/3/006 14

PHY 206 SPRING Problem #1 NAME: SIGNATURE: UM ID: Problem #2. Problem #3. Total. Prof. Massimiliano Galeazzi. Midterm #2 March 8, 2006

PHY 206 SPRING Problem #1 NAME: SIGNATURE: UM ID: Problem #2. Problem #3. Total. Prof. Massimiliano Galeazzi. Midterm #2 March 8, 2006 PHY 06 SPRING 006 Prof. Massimiliano Galeazzi Midterm # March 8, 006 NAME: Problem # SIGNAURE: UM ID: Problem # Problem # otal Some useful relations: st lat of thermodynamic: U Q - W Heat in an isobaric