1 FOUNDATION STUDIES EXAMINATIONS June 2013 PHYSICS Semester One February Main Time allowed 2 hours for writing 10 minutes for reading This paper consists of 4 questions printed on 10 pages. PLEASE CHECK BEFORE COMMENCING. Candidates should submit answers to ALL QUESTIONS. Marks on this paper total 80 Marks, and count as 35% of the subject. Start each question at the top of a new page.
2 INFORMATION a b = ab cos θ a b = ab sin θ ĉ = v dr a dv dt v = u + at x = ut + 1 2 at2 v 2 = u 2 + 2ax i j k a x a y a z b x b y b z dt v = a dt r = v dt a = gj v = u gtj r = ut 1 2 gt2 j s = rθ v = rω a = ω 2 r = v2 r p mv N1 : N2 : if F = 0 then δp = 0 F = ma N3 : F AB = F BA W = mg F r = µr g = acceleration due to gravity = 10 m s 2 Φ = E da = q ɛ 0 C q V C = Aɛ d E = 1 q 2 = 1qV = 1CV 2 2 C 2 2 C = C 1 + C 2 1 C = 1 C 1 + 1 C 2 R = R 1 + R 2 1 R = 1 R 1 + 1 R 2 V = IR V = E IR P = V I = V 2 = R I2 R K1 : In = 0 K2 : (IR s) = (EMF s) F = q v B F = i l B df = i dl B τ = ni A B τ r F v = E B r = m q E BB 0 r = mv qb Fx = 0 Fy = 0 τp = 0 T = 2πm KE Bq max = R2 B 2 q 2 2m W r 2 r 1 F dr W = F s KE = 1 2 mv2 P E = mgh db = µ 0 i dl ˆr 4π r 2 B ds = µ0 I µ0 = 4π 10 7 NA 2 P dw dt = F v φ = area B da φ = B A F = kx P E = 1 2 kx2 ɛ = N dφ dt ɛ = NABω sin(ωt) dv v e = dm m v f v i = v e ln( m i m f ) F = v e dm dt F = k q 1q 2 r 2 k = 1 4πɛ 0 ɛ 0 = 8.854 10 12 N 1 m 2 C 2 E lim δq 0 ( δf δq ) 9 10 9 Nm 2 C 2 E = k q r 2 ˆr f = 1 T ω 2πf y = f(x vt) k 2π λ v = fλ y = a sin k(x vt) = a sin(kx ωt) = a sin 2π( x t ) λ T P = 1 2 µvω2 a 2 v = s = s m sin(kx ωt) F µ V W q E = dv dx V = k q r p = p m cos(kx ωt)
3 I = 1 2 ρvω2 s 2 m n(db s) 10 log I 1 I 2 = 10 log I I 0 where I 0 = 10 12 W m 2 ( ) v±v f r = f r s v v s where v speed of sound = 340 m s 1 1 λ = ke2 2a 0 ( 1 n 2 f 1 n 2 i ) = R H ( 1 n 2 f 1 n 2 i (a 0 = Bohr radius = 0.0529 nm) (R H = 1.09737 10 7 m 1 ) (n = 1, 2, 3...) (k 1 4πε 0 ) E 2 = p 2 c 2 + (m 0 c 2 ) 2 ) y = y 1 + y 2 E = m 0 c 2 E = pc y = [2a sin(kx)] cos(ωt) λ = h p (p = m 0v (nonrelativistic)) N : x = m( λ 2 ) AN : x = (m + 1 2 )( λ 2 ) x p x h π E t h π (m = 0, 1, 2, 3, 4,...) y = [2a cos( ω 1 ω 2 2 )t] sin( ω 1+ω 2 2 )t f B = f 1 f 2 y = [2a cos( k 2 )] sin(kx ωt + k 2 ) = d sin θ Max : = mλ Min : = (m + 1 2 )λ I = I 0 cos 2 ( k 2 ) E = hf c = fλ KE max = ev 0 = hf φ L r p = r mv L = rmv = n( h 2π ) δe = hf = E i E f r n = n 2 ( h 2 4π 2 mke 2 ) = n 2 a 0 E n = ke2 2a 0 ( 1 ) = 13.6 n 2 n 2 ev dn dt = λn N = N 0 e λt R dn dt T 1 2 MATH: = ln 2 = 0.693 λ λ ax 2 + bx + c = 0 x = b± b 2 4ac 2a y dy/dx ydx x n nx (n 1) 1 n+1 xn+1 e kx ke kx 1 k ekx sin(kx) k cos(kx) 1 cos kx k 1 cos(kx) k sin(kx) sin kx k where k = constant Sphere: A = 4πr 2 CONSTANTS: V = 4 3 πr3 1u = 1.660 10 27 kg = 931.50 MeV 1eV = 1.602 10 19 J c = 3.00 10 8 m s 1 h = 6.626 10 34 Js e electron charge = 1.602 10 19 C particle mass(u) mass(kg) e 5.485 799 031 10 4 9.109 390 10 31 p 1.007 276 470 1.672 623 10 27 n 1.008 664 904 1.674 928 10 27
PHYSICS: Semester One. February Main 2013 4 Question 1 ( 6 + (2+2+2+2+2) + (2+2)= 20 marks): Part (a): The pressure (i.e. force per area) P exerted on the bottom of a water tank is expected to depend on the density ρ of water, the acceleration due to gravity g and the height h of water in the tank. Use dimensional analysis to determine the form of this dependence. Part (b): The equation of a transverse wave on a string is given (in SI units) by: y = 0.05 cos 5π(x + 30t) Find the (i) (ii) (iii) (iv) (v) amplitude, wavelength, frequency and speed of the wave. If the tension in the string is 9 N, calculate µ, the mass per unit length of the string. Part (c): Two ants are sitting on a turntable rotating at a constant rate. Ant 1 is 20 cm from the centre of the turntable and its speed is measured at 5 m/s. Ant 2 s speed is measured to be 3 m/s. (i) How far from the centre of the turntable is Ant 2? (ii) Calculate the centripetal acceleration on Ant 1.
PHYSICS: Semester One. February Main 2013 5 Question 2 ( (3+5+2) + (2+4+4) = 20 marks): Part (a): A bird enclosure at a zoo has the dimensions and position shown in Figure 1. A parrot is hanging on to the edge of the cage at point X, before flying across the cage to point Y. (i) Find the displacement vector s of the bird s journey from X to Y. Express your answer in terms of the unit vectors i, j and k. (ii) A wind blows through the cage with a force F = 5 N, directed parallel to the vector AB. Express this force in terms of the same unit vectors. (iii) Calculate the work done by the wind on the bird. y 3 m 1 m 1 m Y B F A 2 m O x 1 m X 4 m z Figure 1:
PHYSICS: Semester One. February Main 2013 6 Part (b): A new invention consists of a uniform beam AB of length 4 m and mass m = 10 kg which is hinged to a support H as shown in Figure 2. A mass M = 100 kg is attached to end A via a massless rope and a massless and frictionless pulley. A mass M is attached to the other end B. (i) (ii) (iii) State the conditions for static equilibrium of the beam. Draw a diagram clearly showing all of the forces acting on the beam. Calculate the largest mass M that can be placed at B before the mass M lifts off the ground and the beam moves. A α α M H 01 01 01 01 01 0011 0011 0011 0011 0011 000111 000111 AH = 1 m AB = 4 m M = 100 kg α = 30 degrees B M Figure 2:
PHYSICS: Semester One. February Main 2013 7 Question 3 ( (4+4+2) + (2+4+4) = 20 marks): Part (a): A student throws a basketball at an angle of 45 as shown in Figure 3. The ball hits a nearby wall normally (i.e. at 90 to the wall) at a height 3.2 m above the point of release. Assume g = 10 m/s 2. (i) Calculate the speed V that the ball was initially thrown with. (ii) How far away from the wall was the ball when it was thrown? (iii) If the ball loses 36% of its kinetic energy in the collision with the wall, how far from the wall would the student need to stand to catch the ball without it bouncing on the ground? Assume the catching height is the same as the height from which the ball was thrown. V =? 0011 000111 0011 0011 θ = 45 01 01 01 3.2 m Figure 3:
PHYSICS: Semester One. February Main 2013 8 Part (b): Another student is playing pool. The white ball is propelled at a speed of 4 m/s toward a stationary red ball, as shown in Figure 4. Both balls have the same mass. After the collision, the red ball is travelling at 3 m/s at an angle θ to the x-axis, where tan θ = 3/4. (i) Draw a clear, labelled diagram to show the approximate paths of the two balls after the collision. (ii) Calculate the magnitude and direction of the white ball s velocity after the collision. (iii) Is the collision between the balls elastic? Be sure to explain your answer. y v = 3 m/s u = 4 m/s rest x θ (a) before collision (b) after collision Figure 4:
PHYSICS: Semester One. February Main 2013 9 Question 4 ( (4+6) + 10 = 20 marks): Part (a): Two blocks of masses M and m (where M > m) are connected by a massless string, running over a massless and frictionless pulley as shown in Figure 5. The coefficient of friction between block m and the level surface is µ, while the contact between block M and the incline is frictionless. (i) Draw two diagrams clearly indicating the direction of motion and the forces acting on each block. (ii) Hence use Newton s laws to find an expression for the acceleration a of block M in terms of the parameters shown in the diagram. g M µ m µ = 0 θ Figure 5:
PHYSICS: Semester One. February Main 2013 10 Part (b): A block of mass m = 2 kg starts at rest and then slides down an inclined plane towards a spring as shown in Figure 6. The upper section AB of the plane is frictionless, and the lower section BC has a coefficient of friction µ = 0.2. The block compresses the spring and is pushed back up the plane where it momentarily comes to rest at a point X. Use energy principles to calculate the distance AX. Take g = 10 m/s 2 in your calculations. rest A µ = 0 2 m B 2 m µ = 0.2 2 m k C Figure 6: END OF EXAM