A 2.4m в ку (20 points total) An 800 gram mass is attached to a light spring of force constant K (N/m) and is free to oscillate on a horizontal, frictionless surface as shown. The mass is displaced 40 cm in the negative direction from the equilibrium position (i.e. to the left of the equilibrium position) and released from rest. The angular frequency of motion of the mass is 3 radians per second. Find (in SI units!) the following: X-0 T a) (3 points) The period of the block's motion rad/s = 2π f Fe HA 2TT 2 ог 1.2575 F='/> T = ½ = Period = 21 b) (3 points) The spring constant K. О c) (2 points) The maximum speed of the mass Derive x(t) V(t) = 0.5 sm (5t+1) Va dx at zo. 5sm (51+0) max speed N 5 m/s Page 2 of 8 d) (2 points)The maximum acceleration of the mass Derive v(t) (t) = -0.5 sm (5t 0) e) azd 2-2.5c03 (51+0) max acceleration = 2 fm 15. (4 points)The phase constant. Show your calculations! x(t) = 0.4m ( .++ Q 0. um cos(0) = 0 -1 205 (0) = Q 5% +2 (3 points) An expression for the displacement as a function of time (NOTE: include numerical values for all constants e.g. A, and p as well as units) f) Find x(t) A = 0.4m W3 8 = 15/2 Rad dx dr x(t)= 0.4m cos ( 0 ·t + + 11/2₂ rad) 2 (3 points) An expression for the velocity as a function of time (NOTE: include numerical values for all constants e.g. A, wand p as well as units) Find V(t). = √(x) = -0.3% sm (5 r. + π/2 Rad) 12 3 0.1m Problem 3. Standing Waves (20 points total) Score: /20 A standing wave is set up in a 4-meter long string that is fixed at both ends. The wave function for this standing wave is y(x,t) =[0.1 sin(x)] cos(80л t) where x and y are in meters and t is in seconds. a. (6 points) Neatly sketch the complete standing wave pattern on the axes below. Put the appropriate numerical scale on the y-axis. Mark all of the nodes with an O and mark all of the antinodes with an X on your sketch. y (m) 0.1mL 0.05 om -0.05mt 2 ° λ = 4m 11 = 2.4 0 x (m) 0 -o-im F x b. (2 points) Which harmonic is this? (e.g., first, second, third, fourth, ...) Ans. Second 2 c. (7 points) Write an equation for the velocity ("transverse velocity") of a particle of the string located at x-0.5 m. Your answer should be a function of time, not a nuptber. x = 0.5 sm [2 ad.251.33 21.33 zad/st] 1. 0.5 sm (1½ rad - 251.83 zad, t +1 d. (5 points) Two traveling waves interfere to form this standing wave. Write the wave functions of both of these two traveling waves. Be sure to include numerical values for all known quantities. = 12 A smk/x) cos wit y = 10.2 sim π/ π/cos 251.33) О Page 5 of 8 Charge, Coulomb's Electric Force, Electric Field (20 points total) Consider the two charges shown below. q-7 μC and q2--5 μC (one is + and one is -), and they are 0.6 m apart. - 3×106 Sim 60° 0.6m Cos 60' B 30° x origi snift (a) (4 points) If the charge qi consist of a sphere of protons, calculate the number of the protons that make up the charge q₁. The proton charge is 1.6x10-19C. 7×10 °C 1.6 x 1019/ Protim 13 Co 0 protens (b) (6 points) Calculate the magnitude of the force on q: caused by q1. b 13x1067 F21 = 12191 = (1.6×10 12 (3×10 C)(3×10 C) = Y 10.6m)² 5 F2 = (c) (7 points) Write the vector expression (F, i +Fy j) for the force in part (b). m2 F1 = (1.6 × 10 6 m²)(3×10) 13x10°C) 2.1 F2, (0.63)2 0.01 +0.01 Ng -19 루 COS 301+ (1.6×10 5 (0.6 (d) (3 points) If charge q is removed, write down the vector expression for the Electric Field at the position formerly occupied by q. (HINT: Think! Don't over calculate here - the answer is straightforward) are removed →→ Direction if Charges 2 -0.05N9-0.08771 3×10 Page 6 of 8. of field chane +2 Fields. A uniform electric field is produced between two parallel metal plates by connecting the plates to a battery as shown. The distance between the plates is 0.5 cm. 30V Battery V bat B 30V 0.015m + OV (a) (3pts) If a proton is placed at point B and released from rest, it moves to the left. Based on this, label the plates + and - and sketch the electric field lines between the plates on the above diagram. (Ignore points A, B and C when you make your sketch.) +2 (b) (4pts) If the battery voltage is Vbat 30 V, what is the magnitude of the electric field between the plates in SI units? Show all your work and write your answer in the spaces provided. AV = 30V find E AV = E.d 30 V 2 E. 0.005m E=50,000/m v/m m = 50,000 N/C Magnitude of Electric Field = 30/05 50,0 +2 SI units N/C (c) (1pts) If a charge of 0.05 C (a positive charge) is moved from point A to point B (circle one response), a the charge is moving to a region of higher potential the charge is moving to a region of lower potential c. the potential of points A and B is the same d. insufficient information is given Page 7 of 8 (d) (1pts) If a charge of 0.05 C (a positive charge) is moved from point A to point B, its potential energy (circle one response), increases b decreases c. remains the same d. insufficient information is given (e) (4pts) A charge of 0.05 C (a positive charge) is released from rest at point C. Calculate the change in electric potential energy of this charge when it moves from point C to point A. Show all your work and write your answer in the spaces provided. Recall that Vat=30 V. AU-Ed Nm = J DUE = QAV AUE = q Ed AUE=(0.05C) (50,000 N/c) (0.605m) = 5 Nm = 5J +3 Change in electric P.E.= SI units J (f) (3pts) What is the kinetic energy of this 0.05 C charge just before it strikes the plate 30 at point A? Show all your work and write your answer in the spaces provided. AU 2 - AUE (0.050) (303) = 1/2 (1.67 × 10 7 kg) vz k=12m²K=8.1728× 10½ kg. m/s² = 4.09 N Kinetic Energy- SI units Jor N (g) (4pts) If the mass of this 0.05 C charge is 3.5×10 kg, what is its speed just before it strikes the plate at point A? Show all your work and write your answer in the spaces provided. key. m/s = 2 V = ½ 13.5x10¹ leg) (v²). Speed of Particle= +2 SI units/s