1 FOUNDATION STUDIES EXAMINATIONS June 2015 PHYSICS Semester One February Main Time allowed 2 hours for writing 10 minutes for reading This paper consists of 6 questions printed on 10 pages. PLEASE CHECK BEFORE COMMENCING. Candidates should submit answers to ALL QUESTIONS. Marks on this paper total 70 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 = i j k a x a y a z b x b y b z v dr a dv dt dt v = a dt r = v dt v = u + at a = gj x = ut + 1 2 at2 v = u gtj v 2 = u 2 + 2ax r = ut 1 2 gt2 j s = rθ v = rω a = ω 2 r = v2 r ω f = ω i + αt θ = ω i t + 1 2 αt2 ω 2 f = ω2 i + 2αθ p mv N1 : N2 : if F = 0 then δp = 0 F = ma N3 : F AB = F BA W = mg F r = µr τ r F Fx = 0 Fy = 0 τp = 0 W r 2 r 1 F dr W = F s KE = 1 2 mv2 P dw dt F = kx = F v P E = mgh P E = 1 2 kx2 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 9 10 9 Nm 2 C 2 ɛ 0 = 8.854 10 12 N 1 m 2 C 2 ( ) E lim δf δq 0 E = k q ˆr δq r 2 V W q E = dv dx V = k q r 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 v = E B r = m q df = i dl B τ = ni A B E BB 0 r = mv qb T = 2πm Bq KEmax = R2 B 2 q 2 2m db = µ 0 i dl ˆr 4π r 2 B ds = µ0 I φ = area B da ɛ = N dφ dt f = 1 T ω 2πf y = f(x vt) µ0 = 4π 10 7 NA 2 φ = B A ɛ = NABω sin(ωt) 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) p = p m cos(kx ωt) I = 1 2 ρvω2 s 2 m F µ n(db s) 10 log I 1 I 2 = 10 log I I 0
3 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 y = y 1 + y 2 y = [2a sin(kx)] cos(ωt) N : x = m( λ 2 ) AN : x = (m + 1 2 )( λ 2 ) (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 KEmax = 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 1 λ = ke2 2a 0 ( 1 n 2 f 1 n 2 i ev ) = R H ( 1 n 2 f (n = 1, 2, 3...) (k 1 4πε 0 ) E 2 = p 2 c 2 + (m 0 c 2 ) 2 1 n 2 i ) λ = h p x p x h π (p = m 0v (nonrelativistic)) E t h π 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 g = acceleration due to gravity = 10 ms 2 1 u = 1.660 10 27 kg = 931.50 MeV 1 ev = 1.602 10 19 J c = 3.00 10 8 ms 1 h = 6.626 10 34 Js e electron charge = 1.602 10 19 C R H = 1.09737 10 7 m 1 a 0 = Bohr radius = 0.0529 nm 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 E = m 0 c 2 E = pc
PHYSICS: Semester One. February Main 2015 4 Question 1 ( (4) + (3+2) + (1+2+2)= 14 marks): Part (a): The speed of sound in a gas, v, is expected to depend on the pressure p (SI unit: N/m 2 ) and density ρ of the gas, as well as the wavelength λ of the sound wave. Use dimensional analysis to determine the form of this dependence. Part (b): A string with linear mass density µ = 10 g/m is used to support a 5 kg mass as shown below, where the pulley is frictionless. y x 5 kg A transverse wave travels on the string in the +x direction with amplitude of 1 mm and frequency 100 Hz. (i) (ii) Write down a wave equation that describes this wave. Calculate the power transmitted by the wave along the string
PHYSICS: Semester One. February Main 2015 5 Part (c): The torque τ on an object around a pivot point A due to a force F acting at a point B is: τ = r AB F where r AB is the displacement vector between points A and B. A wooden stick AB is placed along the diagonal of a rectangular box as shown below: y B 1.5 m O x 2 m A z 2 m (i) Express the displacement vector r AB in terms of the unit vectors i, j and k. (ii) A force F acts on the stick at point B. The magnitude of this force is 5 N and it is in the same direction as the line OB. Express F in terms of the unit vectors i, j and k. (iii) Hence calculate the torque vector τ caused by F around the pivot A.
PHYSICS: Semester One. February Main 2015 6 Question 2 ( 4+4 = 8 marks): On the planet Zarq, a popular sport involves kicking a ball over a ditch. During a tournament of this exciting sport, a ball is kicked at a speed of 10 m/s at an angle α = tan 1 (3/4) above the horizontal, as seen below. The ball just makes it across an 8 m wide ditch. 10 m/s α 8 m (i) Calculate g Z, the acceleration due to gravity on Zarq. (ii) What is the maximum height (above the starting position) reached by the ball?
PHYSICS: Semester One. February Main 2015 7 Question 3 ( 4+4+2+2 = 12 marks): A light fitting is constructed using a uniform metal rod of length 3 m and mass 4 kg. The rod is attached to a wall at one end, using a hinge H. A lamp of mass 2 kg is attached to the other end A. A cable is connected from the wall to the rod at a point B, located 2 m from the hinge. as shown below: 30 H 60 B A (i) Draw a diagram clearly showing all of the forces acting on the metal rod. (ii) Use the conditions for static equilibrium to write down THREE equations involving these forces. (iii) Use these equations to calculate the tension T in the cable. (iv) Also calculate the components R x and R y of the reaction force of the wall acting on the rod at the hinge H.
PHYSICS: Semester One. February Main 2015 8 Question 4 ( 10 marks): A mass M rests on an inclined plane and is attached to an unstretched spring with spring constant k, as shown below: k M θ The mass is allowed to slide down the plane and momentarily comes to rest at a distance d from the starting point: d M The coefficient of friction between the block and the plane is µ. Use energy principles to obtain an expression for the distance d in terms of M,g,k,µ and θ.
PHYSICS: Semester One. February Main 2015 9 Question 5 ( 6 + 5 + 3 = 14 marks): Three blocks, with masses 4m, 2m and m are connected by two massless strings, running over massless and frictionless pulleys as shown. The coefficient of friction between blocks m and 2m and the surfaces is µ = 0.25. m 2m µ µ 4m θ = tan 1 (3/4) (i) Draw THREE diagrams (i.e. one for each block), clearly indicating the direction of motion and the forces acting on each block. (ii) Apply Newton s laws to each block to obtain FIVE equations of motion for the three blocks. (iii) Hence find an expression for the acceleration a of the 4m block.
PHYSICS: Semester One. February Main 2015 10 Question 6 ( 10+2 = 12 marks): A mass m travels in the x direction at a speed of 4 m/s before colliding 1-dimensionally with a mass of 3m, travelling at 1 m/s in the same direction. The collision between these masses is elastic. 4 m/s 1 m/s m 3m (i) Calculate the speeds of the two masses after the collision. Be careful to explain the physical principles you are using, as marks will be given for this. (ii) Draw a diagram clearly showing the directions of the motions of the two masses after the collision. END OF EXAM