c. What are the amplitude, period, and angular frequency wo of the H atom's oscillation? (A) = P3 - Pz = P2-P, = 0.01nm am! = A = 0. 0Inm angular Wo=zt T= LS- L, Wo= er Frequency = = 1. 16 x10-175 1-16x10-14 = Na = 5.42x 10 14S d. The H atom has a mass of about 1.674 x 10-27 kg. What is the effective spring constant of the HCI molecule? WOL= K = wo'- ( 5 . 42 x1094) 2 (1. 674 x 10-27) K = 491.13 N/ M N . What is the vibrational energy E. of the HCI molecule? (Calculate this assuming the H atom is an ordinary mass on a spring.) 2 2491.13 x ( 0.01 x 10 - 9 ) 2 =2.456 x 10-20 f. What is the maximum speed of the H atom during its oscillation? At which of the times in the table above does the H atom have this speed? 1/z mul max = 2 muoz AL= EO Vmax is 5416. 91 m/s M g. On the graph below, draw a curve showing the H atom's velocity as a function of time. Be sure to use an appropriate scale for the vertical axis. 5416.91 mis t1 t2 t3 ta ts to -5916.9 h. At which times is the potential energy of the HCI molecule a maximum, and at which times is it zero? At which times is the kinetic energy of the oscillating H atom a maximum, and at which times is it zero? at t2 , ta , to EP = maximum at t,its, ts EP- is zero att2 ,ta , to - > EK is Zero at tilts , ts -> EP is max i. On the graph below, draw two curves indicating the potential energy and the kinetic energy of the H atom as functions of time. Be sure to use an appropriate scale for the vertical axis, and label the two curves. W Energy to DEAnalyze both qualitatively and quantitatively the properties and motion of systems that exhibit harmonic oscillations {with or without clamping] Apply the basic energy model to systems that exhibit harmonic oscillations and connect the mechanical energy to the motion Create and interpret graphs representing physical quantities in oscillations The vibration of diatomic molecules can be modeled as a harmonic oscillator. In this way. we can understand what it means to say that a molecule has "vibrational energy". Let's considerthe diatomic molecule HCL The potential energy of this d molecule is modeled by the Lennard-Jones potential shown below, wherer represents the distance between the two atoms and E0 is the ground state [minimum energy) ofthe molecule Let r1 : 0.12 nm, r2 = 0.13 nm, and r3 = 0.14 rim. Using the harmonic oscillator model, we can represent a diatomic molecule as two masses iatomic connected by a spring. in the case that one ofthe atoms is significantly larger than the other, we can treat the vibration as if only the smaller atom were moving while the larger atom is fixed in place. For the HE! molecule, this means we can assume the CE atom is stationary and only the H atom oscillates back and forth. This assumption is reasonable: even though the "spring\" exerts the same force on both atoms, the more massive atom will experience a much smaller acceleration. a. The table below shows diagrams for the position ofthe hydrogen atom (taking the Cl! atom to be at the origin) for certain times corresponding to critical points in the oscillation motion. At t1 = D, the two atoms are separated by their equilibrium distance r2; at :2 : 2.9 x 10''5 5, the two atoms are maximally separated by the distance 13. Fill in the remaining rows of the tablet corresponding to equal time intervals, with the appropriate diagrams and indicate the distance :r separating the two atoms. t2 = 2.9 x 10-15 s r3 = 5.8 x 10*15 s :. : 3.7 x 10-15 5 @- rlzo-I'an PI r5 = 1.15 x 10445 (gran-(=2? '5' Y;-.I'5f\\m r5 = 1.45 x 10445 W '7'.) {3:04an b. On the graph below, drawa curve showing the H atom's position 1' as a function of time using the information in the table from part a. Be sure to use an appropriate scale for the vertical axis (use numbers, not the symbols r1, 1'2, and r3). c. What are the amplitude, period, and angular frequency can of the H atom's oscillation? CH.) cFS'PZ *?1'P| = 0-0\ n~ amf= =- 0-\ u\\\\ We :11: 'T: X's1: Hui-aw --- I H : - b!- j r \\Nosf' SSH?! In"( .I i. On the graph below, draw two curves indicating the potential energy and the kinetic energy of the H atom as functions of time. Be sure to use an appropriate scale for the vertical axis, and label the two curves. W Energy Eo X PE ke t1 t2 t3 t4 ts to j. What is the period of oscillation for the kinetic and potential energy? How does this compare to the period of oscillation for the physical motion of the H atom? Tek= Perrod of oscillation OF KE= 13-t1= (3= 5.8x10-15 TEP = period of osc. or PE = E3- tv= tz = tz = 5.8 x14-0 T2 1. 16 x 10- 17 = 2 x 5.8 x 10 - 15 = T= 2 7EK = 2 TEP - Half of T bittAtom k. Calculate the energy of the HCI molecule assuming it's a quantum oscillator in its ground state. Is your result similar to, or very different from, your answer to part e? (h = 6.626 x 10-34 J . s and h = h/(2n)) E = (n+ yz) nwo atn= 0 E = VENwo - 6.626 * 10-24 x 5. 42x 101% = 2.8 6*10-40 Similar results, couple # of E. Suppose that there is damping present in the oscillations of the H atom (this is not physically realistic since damping would decrease the energy of the vibrations, but E is already the minimum allowed energy of the HCI molecule according to quantum mechanics). We'll take the damping coefficient to be 8 = 5.97 x 1013 s-1 (or y = 1.194 x 1014 s-1). 1. Does this damping coefficient result in underdamping, critical damping, or overdamping? Explain how you can tell. If the oscillations are underdamped, what is the new oscillation angular frequency w1? How does it compare to the undamped angular frequency wo? H results in underdumping be the damping Factor Ez = S = 5.9 9x 1013 1.194 x 1019 52: 0.5 01 New oscillation Argulan frog... WI= win N 1-EF Wn = 2TT un - 271 42. 5 7 52 x106 = wn = 207. 5076x106- 4 w,= 231. 6486 x 106 rad / sec. divert from us. m. What would be the amplitude of the H atom's oscillations at time to? On your graph in part b, sketch what the H atom's position as a function of time would look like with the damping. n. What would be the total mechanical energy of the H atom at time to (assume once again that it's an ordinary harmonic oscillator, but with the damping)? On your graph in part h, sketch what the total mechanical energy as a function of time would look like with the damping