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L A N C A S T E R U N I V E R S I T Y
2010 EXAMINATIONS
Part I
PHYSICS - Paper PH1.C ( 2 1/2 hours )
- Use a separate answer book for each section.
- Candidates should attempt all those sections identified with the modules for which they are registered.
- The time allocated is 30 minutes per section.
- An indication of mark weighting (30 marks per section) is given by the numbers in square brackets following each part.
- In each section attempted, candidates should answer all parts to the question.
PHYSICAL CONSTANTS
Planck’s constant h = 6. 63 × 10 −^34 J s h¯ = 1. 05 × 10 −^34 J s Boltzmann’s constant kB = 1. 38 × 10 −^23 J K−^1 Mass of electron me = 9. 11 × 10 −^31 kg Mass of proton mp = 1. 67 × 10 −^27 kg Electronic charge e = 1. 60 × 10 −^19 C Speed of light c = 3. 00 × 108 m s−^1 Avogadro’s number NA = 6. 02 × 1023 mol−^1 Permittivity of the vacuum ǫ 0 = 8. 85 × 10 −^12 F m−^1 Permeability of the vacuum μ 0 = 4 π × 10 −^7 H m−^1 Gravitational constant G = 6. 67 × 10 −^11 N m^2 kg−^2 Bohr magneton μB = 9. 27 × 10 −^24 J T−^1 (or A m^2 ) Bohr radius a 0 = 5. 29 × 10 −^11 m Gas constant R = 8 .31 J K−^1 mol−^1 Acceleration due to gravity g = 9.81 m s−^2 1 standard atmosphere = 1. 01 × 105 N m−^2 Mass of Earth = 5. 97 × 1024 kg Radius of Earth = 6. 38 × 106 m Density of iron = 7. 6 × 103 kg m−^3 Stefan-Boltzmann constant = 5. 67 × 10 −^8 Wm−^2 K−^4
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Section A: Module 131 - Vectors and Vector Algebra
A1. Let a = 3 i + j and b = 4 i − k. Here and in the following, i, j and k denote Cartesian basis vectors.
(a) Construct a unit vector n that is perpendicular to the plane spanned by a and b. [5] (b) Determine the number x such that the vector c = 2 i + x j + k lies in the plane spanned by a and b. [4]
A2. Let a = − i + k , b = 2 j + k , and c = − i + j. Calculate
(a) a ×
b × c
, [5]
(b)
a × b
× c. [5]
A3. The position vector of a particle is given, as a function of time t , by
r(t) = u t i + b cos(ωt) j
where u, b and ω are positive constants.
(a) Calculate the velocity v(t) and the acceleration a(t) of the particle. [4] (b) Sketch the path of the particle in the plane spanned by i and j. Show, on your diagram, the directions of the velocity and acceleration vectors at time t 0 = 2π/ω, and state their magnitudes at that time. [7]
Section C: Module 133 - Oscillations and Waves
C1. Draw diagrams to show the positions of the displacement nodes and anti-nodes for the following situations:
(a) the fundamental mode of a pipe open at both ends, (b) the second harmonic of a pipe closed at both ends, (c) the second overtone of a pipe closed at both ends, (d) the third harmonic of a pipe open at one end and closed at the other. [8]
C2. A loud speaker is located at point X and faces another loud speaker located at point Y 10 .0 m away from point X. The loud speakers each emit sinusoidal sound waves in phase at a constant frequency of 520 Hz with displacement amplitude 20. 0 μm. At time t = 0.00 s each speaker cone is maximally displaced. [Assume the speed of sound in air is 344 m s−^1 .]
(a) Write down a numerical expression for the resulting sound wave at the midpoint on the straight line XY as a function of the time t. (b) How many points of constructive interference would someone pass through if they walked from X to Y along the line XY? (c) Describe what a listener would hear at the midpoint of XY if the frequency of one of the speakers was changed to 524 Hz while the other speaker’s frequency remained unchanged. [13]
C3. Dr Beta has built a large pendulum consisting of a bob of mass 150 kg suspended at the end of a light, rigid rod of length 25.0 m. Dr Beta climbs a ladder and sets the pendulum in motion. Fascinated by observing the pendulum from his ladder Dr Beta carelessly grabs hold of the rod as it swings past him. If the new period of oscillation is 96.0% of the original period and Dr Beta weighs 68.0 kg, how far up the rod from the bob is Dr Beta clinging? [You can treat Dr Beta and the bob as point masses attached to the rod.] [9]
Section D: Module 134 - Electrical Circuits and Instruments
D1. Consider the following circuit containing a switch S, an ideal battery with e.m.f. ε = 12 V, resistors R 1 = 75 Ω and R 2 = 25 Ω, and capacitors C 1 = 1. 0 μF, C 2 = 0. 5 μF and C 3 = 0. 5 μF.
1
C 1 ε
−
C
C
2
3 R R 2
S
After the switch is closed:
(a) Calculate the initial current flowing through resistor R 1. (b) Calculate the combined capacitance of C 1 , C 2 and C 3 and hence determine the charge on C 1 when fully charged after current has stopped flowing. (c) Calculate the final energy stored in capacitor C 2. [10]
D2. An ideal inductor with inductance L = 0.1 H and a resistor of resistance R = 20 kΩ are connected in series to an ideal battery. After a long time the current in the circuit is 0.45 mA.
(a) Use Kirchoff’s Voltage Law to derive the differential equation that governs the current flowing in the circuit. Show that this differential equation is satisfied by a current of the form
i(t) = I 0
[
1 − exp(−
R
L
t)
]
(b) Considering the current from part (a), derive the voltage across the inductor as a function of time and use it to find the voltage of the battery. (c) After what time has the current increased to half of its final value? [10]
D3. Consider an inductor with L = 0.600 H, a capacitor with C = 50. 0 μF and a re- sistor with R = 50.0 Ω in series in an AC circuit with an external voltage source. The current follows the functional form I(t) = I 0 sin(ωt) with I 0 = 10.0 A and ω = 400 rad s−^1.
(a) Calculate the reactances XL and XC of the circuit. (b) Calculate the impedance of the circuit and the phase angle between current and voltage. What is the voltage amplitude across the capacitor? What is the voltage amplitude across the inductor? (c) What is the functional form of the voltage across the capacitor? Does the current lag or lead the voltage across the capacitor? Explain your answer. [10] please turn over
