Download Angle Between Vectors - Vectors and Vector Algebra - Exam and more Exams Algebra in PDF only on Docsity!
L A N C A S T E R U N I V E R S I T Y
2011 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 ϵ 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
please turn over
Section A: Module 131 - Vectors and Vector Algebra
A1. In the following questions ˆi, ˆj and ˆk are the Cartesian basis vectors. Leta⃗ = ˆi + ˆj − ˆk and ⃗b = 2ˆi − ˆj + 2ˆk. [10]
(a) Calculate the angle between the vectorsa⃗ and ⃗b. (b) Construct two unit vectors that are perpendicular to the plane containinga⃗ and ⃗b.
A2. For the vectors
a⃗ = ˆi + ˆj + ˆk,⃗ b = ˆi + 2ˆj + ˆk,⃗ c = 3ˆi + ˆj + ˆk,
calculate [10]
(a)a⃗ ×
⃗b ×c⃗
(b)a⃗ ·
⃗b ×c⃗
(c) Do these vectors form a left- or right-handed basis? Explain your answer.
A3. As a function of time the position of a particle is given by
r⃗ (t) = a cos(ωt)ˆi + b sin(ωt)ˆj
where ω, a and b are positive constants. [10]
(a) Calculate the velocityv⃗ (t) and the accelerationa⃗ (t) of the particle. (b) Sketch the path of the particle and indicate on your sketch the velocity vector and the acceleration vector at some chosen instant of time.
Section C: Module 133 - Oscillations and Waves
C1. A string of length 1.12 m is clamped tightly at both ends. The transverse displace- ment of the string as a function of distance x along the string and time t is given by y(x, t) = A cos(ωt) sin(kx)
(a) What kind of wave does this function represent? (b) Define the symbols A, ω and k. (c) Find an expression for the allowed values of k and explicitly calculate the three smallest allowed values of k. (d) Sketch the third harmonic of the string and label the nodes and anti-nodes. [10]
C2. A mass of 5.00 kg is attached to an ideal spring with spring constant 4.50 N m−^1 sus- pended from a hook. The mass undergoes simple harmonic motion when displaced from equilibrium.
(a) Calculate the period of oscillation of the mass on the spring. (b) On a single graph, sketch the potential energy, kinetic energy and total energy of the mass-spring system as a function of displacement from equilibrium. (c) During an oscillation the mass is measured to be displaced 2.50 cm from equi- librium and to be travelling at a speed of 0.200 m s−^1. What is the amplitude of the oscillation? [10]
C3. Dr Beta is in the basket of a stationary hot air balloon excitedly recording the sounds of shockwaves from passing supersonic jet planes. At the exact moment that a jet flies directly beneath the balloon Dr Beta overbalances and falls out of the basket. Dr Beta falls directly downwards while the jet continues to fly in a straight line at constant altitude. After 0.500 s Dr Beta hears the sonic boom from the jet and remembers to deploy his parachute. If the jet is flying at Mach 1.80, how high above the jet was the balloon when Dr Beta fell out? [You can neglect air resistance. The speed of sound in air is 344 m s−^1 .] [10]
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 = 20 V, resistors R 1 = 8.0 Ω, R 2 = 20.0 Ω and R 3 = 30.0 Ω, and capacitors C 1 = 40. 0 μF and C 2 = 60. 0 μF.
ε=20V (^) + C 1 =40 μF C (^) 2 =60 F μ −
2
3
R = 20Ω
R = 30Ω S
R = 1 8Ω
(a) Calculate the initial current through resistor R 1 just after the switch is closed. (b) Calculate the power dissipated by resistor R 3 just after the switch is closed. (c) Calculate the combined capacitance of C 1 and C 2 and hence determine the voltage drop over C 1 when fully charged after current has stopped flowing. [10]
D2. An ideal inductor with inductance L = 10.0 mH and a resistor with R = 60 kΩ are connected with a switch in series to an ideal battery with e.m.f ε = 12.0 V.
(a) Determine the current flowing through the resistor (i) just after the switch has been closed, (ii) after the switch has been closed for a long time. (b) (i) State and sketch the functional form of the current through the inductor after closing the switch. (ii) After what time has the current reached 80% of its final value? Determine the voltage drop across the inductor at this time. (c) At the time calculated in part (b), at what rate is energy dissipated by the resistor and at what rate is energy being stored in the inductor? Why are both rates in general not the same? [10]
Section E: Module 135 - Optics and Optical Instruments
E1. Explain what is meant by refractive index and chromatic dispersion. Why are dispersion effects important in optical imaging using lenses? What are the benefits of replacing the objective lens with a mirror in an astronomical telescope? [10]
E2. A spherical convex mirror has a radius of curvature of 10cm. An object is placed in front of it at a distance of 5 times its focal length.
(a) Draw a principal-ray diagram showing the formation of the image. (b) State whether the image is real or virtual, erect or inverted. (c) Calculate the distance of the image from the mirror and the magnification of the image. (d) Explain why a convex mirror is used as a rear view mirror in a car. [10]
E3. (a) Explain what is meant by interference. Describe how this effect is used to make
(i) highly reflective coatings and (ii) non-reflective coatings for optical devices. (b) An optical medium with a refractive index of 1.42 is applied to a glass which has a refractive index of 1.52. (i) Determine the minimum thickness of the film coating which should be used to make the glass non-reflective at 650 nm. (ii) Explain the changes needed to make the glass highly reflective while keep- ing the same thickness. (^) [10]
End of Paper
