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Physics 6B Introduction to Physics II Winter 2003
Final Exam
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Problems Maximum Score Score 1–5 6 6–13 8 14–17 7 18–25 12 26–28 22 29 12 30 13 31 10 32 10 Total B 100
Closed book; no notes. A straightedge and pencil are needed for the ray diagram. You may assume that any numerical constant provided in this exam is accurate to two significant figures. Check that your exam includes 32 problems on 10 pages. Print your name and sign your exam.
The following constants and equations may or may not be needed:
- Acceleration of gravity: g =9.8 m/s^2
- Speed of sound in air: 350 m/s.
- Speed of light in vacuum: 3. 0 × 108 m/s.
- Gas constant: R=8.3 J/mol⋅K
- Stefan-Boltzmann constant: σ = 5. 7 × 10 −^8 W/(m^2 K^4 )
- Ratio of specific heat at constant pressure to that at constant volume, for air: γ=1.
- Young’s modulus of steel: Y=20× 1010 Pa
- Density of water: 1000 kg/m^3 Specific heat of water: 4200 J/kg/°C Specific heat of ice: 2100 J/kg/°C
- Heat of fusion of water: 330,000 J/kg
- Average power transmitted by a harmonic wave of amplitude A and angular frequency ω , on a string of
tension F : P = 21 μFω^2 A^2
- Intensity of a sound wave: B
p v
p I B A ρ ρ
ρ ω (^22)
2 max
2 2 2 max 2
=^1 = =
- Relation between pressure and displacement amplitudes in a longitudinal sound wave: p (^) max = BkA.
- Wave equation: 0
2
2 2 2
2 − = t
y x v
y ∂
- Doppler shift formula: (^) S S
L L (^) v v f
v v f ⋅
- Angle of a shock wave: sin α = vvs
- Malus’ law: I = I max cos^2 φ
- Equation for images formed by a single refracting surface: R
n n s
n s
n (^) a b b − a
′
Signature:
- Lateral magnification of a single refracting surface: ns
ns m b
= − a^ ′
- Thin lens equation: (^)
1 2
r r
n s s
- Angular magnification of a magnifying glass: f
M
25 cm
- Angular magnification of a telescope: M =− f 1 f 2
- Magnification of a compound microscope: 1 2
f f
M ∝
- Intensity for a single-slit diffraction/interference pattern:
[ ]
2 0 (sin )/
sin (sin )/
π θ λ
π θ λ a
a I I
- Intensity for a 2-slit interference pattern:
sin 2 2 0 cos^2 β
φ β I I
where θ λ
π φ sin 2 d = and θ λ
π β sin 2 a =
- Bragg condition for constructive interference of x -rays reflected from a crystal: 2 d sin θ = mλ.
- Angular radius of the first dark ring of a diffraction/interference pattern from a circular aperture:
D
λ sin θ (^) 1 = 1. 22.
- Bernoulli’s equation: (^22) 2
1 2 2
2 2 1
1 p 1 (^) + ρgy (^) 1 + ρv = p + ρgy + ρv.
- Poiseuille’s equation: (^)
L
R p p dt
dV^412 8 η
π
- Stoke’s law: (^) F = 6 πηrv
- (1 pnt) A toy boat of volume 0.020 m^3 weighing 15 kg is placed on the water in a bathtub filled to the brim. What is the mass of the water that flows over the edge of the bathtub? a) 5 kg b) 10 kg. c) 15 kg.
d) Not enough information is given to answer this question.
- (1 pnt) A highly viscous liquid is being pumped through a long cylindrical pipe. If the pressure difference between the two ends of the pipe is cut in half but at the same time the radius of the pipe is doubled, then the volume flow rate will increase by a factor of a) 2 b) 4 c) 8 d) 16
- (1 pnt) The flow of a fluid through a pipe is laminar. Which one of the following actions might result in the onset of turbulence, assuming all other properties are kept constant? a) Use a more viscous fluid. b) Increase the velocity of the fluid flow.
c) Decrease the diameter of the pipe. d) Use a less dense fluid.
- (1 pnt) In the case of a farsighted person (hyperopic) the image of a close (e.g. 25 cm) object
a) focuses in a different plane for horizontal versus vertical lines. b) forms between the lens and the retina. c) would form behind the retina, on the opposite side from the lens.
- (2 pnts) A perfectly exposed photograph is taken with a 1/500 s long exposure using a 50 mm focal length lens and an f -number of 2.0. The lens is then replaced by a 135 mm focal length lens with an f- number of 2.8. How long should the exposure be in order to photograph correctly the same subject with the new lens? a) 1/125 s. b) 1/250 s. c) 1/500 s. d) 1/1000 s.
- (1 pnt) To increase the depth of focus of your camera, you can
a) Increase the diameter of the aperture (lower f -stop setting). b) Reduce the diameter of the aperture (higher f -stop setting). c) Reduce the length in time of the exposure. d) Move the lens closer to the film.
- (1 pnt) The picture at the right shows waves from two sources interfering. The rings are wave crests, and the rays show lines of constructive interference. As the two sources are moved closer together, a) the lines of constructive interference will move further apart. b) the lines of constructive interference will move closer together. c) the lines of constructive interference shown will not change, but new lines will appear between them. d) the lines of constructive interference will not change position, but the interference will become more complete.
- (1 pnt) During your visit to the International Space Station you take along your 90 mm refracting telescope, which has high-quality, nearly perfect lenses. You point it at a star known to be a binary system (two stars close together), but the image that you see looks like a single star. You double the magnification, but it still looks the same. To resolve the two stars into separate images, you would have to a) replace the objective lens with one having a shorter focal length. b) reduce the aperture diameter by covering the objective lens with black paper with a relatively small hole cut in it. c) get a telescope with a larger diameter objective lens. d) use instead a reflecting telescope with a 90 mm diameter objective mirror.
- (2 pnts) The graph below shows the intensity pattern of interference from laser light shining through several narrow closely spaced slits and onto a screen. How many slits are there? The angle δ is the phase difference, in radians, between the light from two adjacent slits (so that the big peaks are separated by 2π radians in phase).
10 5 0 5 10
0
10
20
2530
0
I( δ, 4 )
-10 δ 10 a) Three slits. b) Four slits. c) Five slits. d) Six slits.
S1 (^) S
- (1 pnts) Suppose that you are trying to resolve two closely spaced spectral lines using a spectrometer with a diffraction grating. You are looking at the 3rd-order lines of the diffraction/interference pattern, but unfortunately, the two lines are blurred together and cannot be resolved. Which of the following changes would increase the spectrometer resolving power and improve your ability to resolve the two lines? a) Increase the number of diffraction-grating lines that are illuminated by the light from the source being studied, but keep the same spacing between the lines. b) Decrease the distance between the diffraction grating and the telescope used to view the lines. c) Keep the number of illuminated diffraction-grating lines constant, but decrease the spacing between the lines. d) Look instead at the 1st-order lines.
- (1 pnt) A 1 cm diameter hole is drilled in the center of a 10 cm square plate of aluminum. When the aluminum is cooled down afterwards, the diameter of the hole will a) decrease b) increase c) remain unchanged
- (1 pnt) When a block of metal is heated up, the length, height, and width each increase by 1%. By how much does the volume of the block increase? a) 0.0001% b) 1% c) 2% d) 3%
- (1 pnt) Several ice cubes are floating in a bucket of fresh water. When the ice melts, the water level in the bucket a) decreases. b) remains unchanged. c) increases.
- (2 pnts) A mechanic using a manual hydraulic jack to lift a car of weight 27,000 N pushes on a reservoir of oil using a piston of radius 1 cm. The car is lifted by the same oil pushing on a piston of radius 30 cm. With how much force F must the mechanic push in order to lift the car slowly and steadily?
a) 30 N b) 300 N c) 700 N d) 13,500 N
- (2 pnts) In the example of the previous problem, what distance must the mechanic push the small piston in order to lift the car 1 cm? a) 1 cm b) 30 cm c) 900 cm d) 2700 cm
- (2 pnts) Two parallel steel plates d =5 mm apart have some high-viscosity oil between them. They are kept sliding past each other at constant speed by the application of a force to the top plate (the bottom plate is fixed in place to the floor). By what factor must the force be changed to keep the same speed if both the plate separation and the viscosity of the oil are doubled? a) 0. b) 1.0 (no change) c) 2. d) 4.
- (2 pnts) A telescope lens is coated with a thin film to reduce the reflection of visible light of wavelength λ. If the index of refraction n of the coating material is greater than the index of refraction of the lens glass, then the thickness of the coating should be a) λ/(8n) b) 3 λ/(4n) c) λ/(4n) d) λ/(2n)
- (12 pnts) Laser light is shining through 4 equally spaced slits, with distance d=0.2 mm between adjacent slits. The slit width is much less than d, and the wavelength of the light is λ=500 nm. a) Draw two phasor diagrams that illustrate how the equal amplitudes from the four slits add together at the locations of the first minimum (picture on left) and second minimum (picture on right) away from the central maximum in the interference pattern.
b) What is the phase angle between phasors from adjacent slits in the case of the first minimum?
c) If a screen is placed 2 m away behind the slits, what is the distance in meters from the central maximum (m=0) on the screen to the brightest point on the second principal maximum (m=1)? You may use small-angle approximations: θ ≈ sin θ ≈tan θ.
d) What is the distance on the screen from the central maximum to the first minimum adjacent to the central peak?
- (13 pnts) An architectural engineer desires a fountain in front of her latest building. She wants the water to squirt 20 m high from the narrow opening in the pipe illustrated here. Use g =10 m/s^2 in the following for the acceleration of gravity. Assume that friction and viscosity effects are negligible. The radius of the pipe is 10 cm , except at the nozzle (Point 2), where it necks down to a radius of 7.1 cm.
a) What must be the speed of the water just at Point 2, the exit to the pipe, in order to reach the desired altitude?
b) What then is the speed of the water at Point 1, in the pipe 2 meters lower?
c) What must be the gauge pressure of the water at Point 1? (Gauge pressure is the difference between the absolute pressure and atmospheric pressure.)
d) What is the flow rate of water from the fountain, in cubic meters per second?
- (10 pnts) A 0.015 kg string 1.5 m long that is fixed at both ends and under a tension of 100 N is vibrating in its third harmonic. The maximum displacement of any segment of the string is 2.0 mm. a) Graph the shape of the string when it is at its maximum amplitude of oscillation. (You do not need to draw a precise sine function, but do put the nodes and antinodes in the correct locations with the correct amplitude.)
x (meters)
Amplitude (mm)
b) What is the wavelength λ of this standing wave?
c) What is the frequency f of the wave oscillation?
d) Write a wave function y ( x , t )appropriate for this standing wave.
