FOUNDATION STUDIES EXAMINATIONS September 2009

Transcription

1 1 FOUNDATION STUDIES EXAINATIONS September 2009 PHYSICS First Paper July Fast Track Time allowed 1.5 hour for writing 10 minutes for reading This paper consists of 4 questions printed on 7 pages. PLEASE CHECK BEFORE COENCING. Candidates should submit answers to ALL QUESTIONS. arks on this paper total 60 arks, and count as 15% of the subject. Start each question at the top of a new page.

2 2 INFORATION a b = ab cos θ a b = ab sin θ ĉ = v dr a dv dt v = u + at x = ut 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 C 2 R = R 1 + R 2 1 R = 1 R R 2 V = IR V = E IR P = V I = V 2 = R I2 R K1 : In = 0 K2 : (IR s) = (EF 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 = N 1 m 2 C 2 E lim δq 0 ( δf δq ) 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 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 = 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 hc ( 1 n 2 f 1 n 2 i ) = R H ( 1 n 2 f 1 n 2 i (a 0 = Bohr radius = nm) (R H = 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 )( 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 ax : = mλ in : = (m )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 ATH: = ln 2 = 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 = kg = ev 1eV = J c = m s 1 h = Js e electron charge = C particle mass(u) mass(kg) e p n

4 PHYSICS: First Paper. July Fast Track k m Question 1 ( 15 marks): The figure above shows a block of mass, m (kilogram) suspended by a spring, of spring constant, k (N ewton/metre), from a fixed beam. When the block is pulled down and released, it oscillates vertically, with a period of P (second). You are given, that P may depend on m, k, and the acceleration due to gravity, g. Use dimensions to derive an expression for P, in terms of m, k, and g.

5 PHYSICS: First Paper. July Fast Track Question 2 ( ( ) = 15 marks): a m µ T The Figure above shows two blocks, of masses m and, connected by a string and pulley system. The strings and the pulleys are of negligible mass, and the pulleys have negligible friction. The coefficient of friction between block m, and the horizontal surface on which it slides is µ. The acceleration of block m is a, and the tension in the string attached to it is T, as labeled. The system is released from rest. (i) Draw a diagram of each block, labeling all particular forces that act upon it. Label also, the acceleration of each block. (ii) Write down Newton s equation of motion for each block, in both the vertical, and horizontal directions (three equations). (iii) Hence derive an expression for the horizontal acceleration, a, of block m, in terms of the parameters labeled in the Figure, and the acceleration of gravity, g.

6 PHYSICS: First Paper. July Fast Track A wheel(µ = 0) ladder wall 8 m 10 m 6 m B floor µ ladder about to slip Question 3 ( ( ) = 15 marks): The above Figure shows a uniform ladder, AB, of length 10 m, and mass, which leans between a vertical wall, and the floor. There is a frictionless wheel at end A of the ladder, so that there is zero friction between the wall and the ladder at that end. At end B, however, the coefficient of friction between the ladder and the floor is µ. The ladder is at the point of slipping, when end A of the ladder is 8 m above the floor, and end B of the ladder is 6 m horizontally from the wall. (i) Draw a diagram of the ladder, and label all the forces that act upon it. (ii) Write down the equations for equilibrium of the ladder (three equations). (iii) floor. Hence find the value of the coefficient of friction, µ, between the ladder and the

7 PHYSICS: First Paper. July Fast Track Q y 10 N r P 5 N 8 N 3 m x z 4 m 3 m Question 4 ( = 15 marks): The figure above shows a rectangular box of labeled dimensions, aligned along the x, y, and z axes. Forces of 5 N, 8 N and 10 N, act at corner P of the box, in the directions indicated. (i) Write expressions for each of the three forces, in terms of the ijk unit vectors, and hence find their vector sum, F. (ii) Express the position vector, r, of point P, in terms of the ijk unit vectors. (iii) Calculate the torque, τ of total force F, about the origin, O, given that - τ = r F (cross product) ANSWERS: END OF EXA Q1. P = C m, where k dimensionless const. k Q2. (ii) R mg = 0, T µr = ma, g 4T = a 4 ; (iii) a = 4g( 4µm +16m ). Q3. (ii) N µr = 0, R mg = 0, 3mg 8N = 0; (iii) µ = 3 8. Q4. (i) F = 8i 8j 11k N; (ii) r = 4i + 3j + 3k m; (iii) τ = 9i + 20j 8k Nm.

8 1 FOUNDATION STUDIES EXAINATIONS December 2009 PHYSICS Second Paper July Fast Track Time allowed 1.5 hour for writing 10 minutes for reading This paper consists of 4 questions printed on 7 pages. PLEASE CHECK BEFORE COENCING. Candidates should submit answers to ALL QUESTIONS. arks on this paper total 60 arks, and count as 15% of the subject. Start each question at the top of a new page.

9 PHYSICS: Second Paper. July Fast Track y y v m m rest (a) 2R O rest x (b) O x v Figure 1: Question 1 ( 15 marks): Figure 1(a) shows two balls, of masses, m and ( > m), connected at the ends of a straight rod, of negligible mass, and length, 2R. Initially, the rod is horizontal, aligned with the x-axis, and pivoted at its centre, O. When the rod is released from rest, falls, while m rises, until the rod is vertical, and aligned along the y-axis, as depicted in Figure 1(b). At this stage, each of the two balls is moving with a velocity, v. There is negligible friction at the pivot. Use energy principles to derive an expression for the velocity, v, in terms of, m, R, and the acceleration due to gravity, g.

10 PHYSICS: Second Paper. July Fast Track y y 1 m/s 4 m/s Bef ore 2 2 rest x rest 1 m/s Af ter 4 m α θ 2 3 m 2 v x Figure 2: Question 2 ( (11 + 4) = 15 marks): One ball, of mass,, moving with a velocity of 4 m/s in the +x-direction, strikes two stationary balls, both of mass 2, as shown in Figure 2. The x-y plane is horizontal. After the collision. the ball rebounds back along the x-axis with a velocity of 1 m/s, while one of the 2 balls moves with a velocity of 1 m/s at an angle of α (where tan α = 3 ) above the x-axis, as illustrated. 4 (i) Using momentum principles, find the velocity of the other ball, of mass 2, after the collision. Find both magnitude (v), and direction (θ). (ii) Is this collision elastic? Show your reasoning.

11 1 FOUNDATION STUDIES EXAINATIONS January 2010 PHYSICS Final Paper July Fast Track Time allowed 3 hours for writing 10 minutes for reading This paper consists of 6 questions printed on 13 pages. PLEASE CHECK BEFORE COENCING. Candidates should submit answers to ALL QUESTIONS. arks on this paper total 120 arks, and count as 45% of the subject. Start each question at the top of a new page.

12 PHYSICS: Final Paper. July Fast Track y 4 m A 3 m r O B x F = 10 N z 2 m Figure 1: Question 1 ( ( ) + ( ) = 20 marks): Part (a): Figure 1 shows a rectangular box, with a corner at the origin, O, and with its sides aligned along the x-,y- and z-axes. Dimensions of the box are labeled. A force of 10 N acts at corner, A, of the box, in the direction of the diagonal, AB. (i) Write down an expression for the position vector, r, of point A, in terms of unit vectors ijk. (ii) Express force, F, in terms of unit vectors ijk. (iii) Find the torque, τ, of force, F, on the box, about the origin, O as pivot. Give your answer in terms of ijk unit vectors. You are given that - τ = r F (cross product)

13 PHYSICS: Final Paper. July Fast Track E 20 kg 50 kg H C 10 m F 6 m 8 m Figure 3: Question 2 ( (2 + 8) + ( ) = 20 marks): Part (a): Figure 3 shows a lever, HE, of mass 20 kg, and length, 10 m, being used to lift a block of mass, = 50 kg. A massless cable runs horizontally from the centre, C, of the lever, over a frictionless pulley, and vertically down to the block. The lever is hinged at end H. A force, F, is exerted at end E, of the lever, via a vertical cable. Dimensions are labeled. Take the acceleration due to gravity g = 10 ms 2. (i) Draw a labeled diagram of the lever, showing all forces that act upon it, as the block is just lifted. (ii) Use conditions for equilibrium, to determine the value of the force, F, required to just lift the block. Determine also, the reaction of the hinge at H on the lever, at this stage.

14 PHYSICS: Final Paper. July Fast Track y r r 3 µ 2 µ ω rotating disc Figure 4: Part (b): Figure 4 shows three blocks, of masses 3, and 2, connected by a string, of negligible mass, over a pulley, of negligible mass and friction. Blocks and 2 rest on the horizontal surface of a disc, at distances r and 2r along the same radius from its centre. Block 3 rests on a horizontal surface, at the bottom of a hole in the centre of the disc. The coefficient of static friction between each of blocks and 2, and the surface of the disc is µ. The disc is rotated with a slowly increasing angular velocity, ω, with the vertical y-axis as its spin axis. If ω continues to increase, blocks and 2 will eventually slip, on the surface of the disc. (i) Draw a diagram of each block, labeling all particular forces that act on each, just before blocks and 2 slip. Label also, any acceleration of each block. (ii) Write down Newton s equation of motion for each block, in both the vertical, and horizontal directions, just before blocks and 2 slip (five equations). (iii) Hence find the maximum angular velocity, ω m, at which the disc can spin, before the blocks slip on the disc surface. Express ω m in terms of µ, r, and the acceleration of gravity, g.

15 PHYSICS: Final Paper. July Fast Track k m µ rest rest rest d Figure 5: Question 3 ( (10) + (10) = 20 marks): Part (a): Figure 5 shows two blocks, of masses m and, and a spring, of spring constant, k, connected by a massless string, which passes over a massless, frictionless pulley between the two blocks. The system is released from rest. At this stage the spring is unstretched. Block,, falls a vertical distance, d, before coming momentarily to rest, dragging block m along the horizontal surface, with which it has a kinetic friction coefficient, µ, and extending the spring. Use energy principles to derive an expression for d, in terms of the parameters labeled in Figure 5, and the acceleration due to gravity, g.

16 PHYSICS: Final Paper. July Fast Track m/s rest Figure 6: Part (b): Figure 6 shows two balls, of masses and 2, that hang side-by-side just touching. Ball of mass is pulled aside and then released, so that it has a velocity of 2 m/s just before it makes an inline elastic collision with the other stationary ball. Use momentum and energy principles to determine the velocities of both balls, immediately after the collision.

17 PHYSICS: Final Paper. July Fast Track Question 5 ( ( ) + ( ) = 20 marks): Part (a): A vibrator source, set to a frequency, f = 500 Hz, and an amplitude of a = 1.00 mm, sends a transverse wave along a string, of mass per unit length, µ = kg/m, which is stretched to a tension of T = 10 N. (i) Calculate the speed, v, with which the wave travels along the string. (ii) Calculate the wavelength of the wave. (iii) Write down a possible particular wave function for the wave. (iv) What is the power, P, of the source of this wave? Part (b): The Paschen series of spectral lines for atomic hydrogen, are formed by electron transitions terminating on the n = 3 energy level. Use the Bohr theory for the hydrogen atom to answer the following questions. (i) Calculate the longest and shortest wavelengths in this spectral series. (ii) What is the total energy of an electron in the n = 3 level? (iii) What is the radius of the orbit of an electron in the n = 3 level?