slope pring. otion, With sured alues nding , and each , and leastvalue ue for value dly to M as disk is disk is mbly is sition of the is initial value? (b) What If? How long does it take for the mechanical energy to drop to half its initial value? (c) Show that, in general, the fractional rate at which the amplitude decreases in a damped harmonic oscillator is PH the 125 fractional / LeClair rate at which the mechanical energy half decreases. Spring 2014 Problem Set 6 71. A block of mass m is connected to two springs of force constants kInstructions: 1 and k 2 as shown in Figures P15.71a and P15.71b. In each case, the block moves on a frictionless it credit. 1. Answer all questions below. Show yourtable work after for full is displaced from equilibrium released. Show that 2. All problems are due and by the end of the day on 14 in April 2014 the two cases the block exhibits simple harmonic motion 3. Late penalties will not be incurred until after spring break with periods 4. You may collaborate, but everyone must turn in their own work. √ k 2) h above the surface of a planet of mass M and radius R. 1. An object of mass m is droppedm(k from a height 1$ (a) T ! 2# Assume the planet has no atmosphere so k 1kthat 2 friction can be ignored. √ (a) What is the speed of the mass justmbefore it strikes the surface of the planet? Do not assume that h is (b) T ! 2# small compared with R. k1 $ k2 √ (b) Show that the expression from (a) reduces to v = 2gh for h R. (c) How long does it take for the object to fall to the surface for an arbitrary value of h? Use any means k1 k2 necessary to evaluate the integral required. Bonus points (10%) for code submissions. m 2. A block of mass m is connected to two springs of force constants k1 and k2 as shown below. The block moves on a frictionless table after it is displaced from equilibrium and released. Determine the period of simple harmonic motion. (Hint: (a) what is the total force on the block if it is displaced by an amount x? k1 k2 m 3. A mass m is connected (b) to two springs in series as shown below. What is the period of simple harmonic Figure motion if the mass is displacedP15.71 from equilibrium. Hint: what must be true of the displacement of each spring if the total displacement is ∆x? 72. A lobsterman’skbuoy is a solid k wooden cylinder of radius r and mass M. It is weighted at one end so that it floats upright in calm sea water, having density '. A passing shark tugs on the slack rope mooring the buoy to a lobster trap, pulling the buoy down a distance x from its equilibrium position and releasing it. Show that the buoy will execute simple harmonic if and thelength resistive effects atofone the 4. A horizontal plankmotion of mass m L is pivoted end. The plank’s other end is supported by water are neglected, and determine the period of the a spring of force constant k. The moment of inertia of the plank about the pivot is I = 13 mL2 . The plank oscillations. is displaced by a small angle θ from horizontal equilibrium and released. Find the angular frequency ω of m 73. simpleConsider bob on(Hint: a light stiff rod, forming a simple harmonic amotion. consider the torques about the pivot point.) pendulum of length L ! 1.20 m. It is displaced from the vertical by an angle "max and then released. Predict the subsequent angular positions if "max is small or if it is large. Proceed as follows: Set up and carry out a numerical method to integrate the equation of motion for the simple pendulum: d 2" g ! % sin " 2 dt L the mass is 5.00 kg and the spring has a force constant of 100 N/m. note that the mass of a segment of the is dm " (m/!)dx. Find (a) the kinetic energy o system when the block has a speed v and (b) the p of oscillation. Pivot v θ dx k x M Figure P15.61 P15.66 5. Assume the earth to be a solid sphere of uniform density. A hole is drilled throughFigure the earth, passing through its center, and a ball is dropped into the hole. Neglect friction. 62. Review problem. A particle of mass 4.00 kg is attached to 67. A ball of mass m is connected to two rubber ba a spring with a force constant of the 100ball N/m. It is oscillating (a) Calculate the time for to return to the release point. Hint: what sort of motion results? length L, each under tension T, as in Figure P15.6 on a horizontal frictionless surface an time amplitude (b) Compare the result of partwith a to the requiredoffor the ball to complete a circular orbit of radius RE ball is displaced by a small distance y perpendicular 2.00 m. Aabout 6.00-kg theobject earth is dropped vertically on top of the length of the rubber bands. Assuming that the t 4.00-kg object as it passes through its equilibrium point. does not showonthat (a) the restoring 6. A satellite in a circular orbitmuch of radius r. the The area A enclosed by thechange, orbit depends r2 because The two objects stick istogether. (a)Earth By how does is $ (2T/L)y and (b) the system exhibits simple har 2 = πr Determine system how thechange following amplitudeA of the. vibrating as properties a result ofdepend the on r: (a) period, (b) kinetic energy, (c) angular motion with an angular frequency # " √2T/mL . and (d) speed. collision?momentum, (b) By how much does the period change? (c) By how much does the energy change? (d) Account for 7. Here are some functions: the change in energy. y 2 A 63. A simple pendulum length of 2.23 m and a mass fof2 (x, t) = f1 (x, t)with = Aea−b(x−vt) 2 b (x − vt) + 1 6.74 kg is given an initial speed of 2.06 m/s at its equilibL L 2 3 −b bx +vt ( ) rium position.f3 (x, Assume t) = Ae it undergoes simple harmonic f4 (x, t) = A sin (bx) cos (bvt) motion, and determine its (a) period, (b) total energy, and Figure P15.67 (c) maximum angular displacement. (a) Which ones satisfy the wave equation? Justify your answer with explicit calculations. 64. Review problem. One end of a light spring the withwave force con- write down the corresponding functions g(x, t) (b) For those functions that satisfy equation, stant 100 representing N/m is attached vertical wall. A light string a waveto of athe same shape traveling in theisopposite 68.direction. When a block of mass M, connected to the en tied to the other end of the horizontal spring. The string spring of mass ms " 7.40 g and force constant k, is s 8. The equation for driven damped oscillator changes from horizontal to avertical as it passes over isa solid simple harmonic motion, the period of its motion is pulley of diameter 4.00 cm. The pulley is free to turn on a d2 x dx q +The 2γωo vertical + ωo2 xsection = E(t) fixed smooth axle. of the string sup2 M ' (m s(1) /3) dt dt m % " 2 & ports a 200-g object. The string does not slip at its contact k with the (a) pulley. Find frequency of oscillation of the Explain thethe significance of each term. object if the (b) E250 g, A two-part experiment is conducted with the u iωt (b) mass Let Eof=the Eo epulley and isx (a) = xonegligible, ei(ωt−α) where o and xo are real quantities. Substitute into the above and (c) 750 g. blocks of various masses suspended vertically fro expression and show that √ xo = q qEo /m 2 (2) 2 (ωo2 − ω 2 ) + (2γωωo ) (c) Derive an expression for the phase lag α, and sketch it as a function of ω, indicating ωo on the sketch. 9. (a) A diatomic molecule has only one mode of vibration, and we may treat it as a pair of masses connected by a spring (figure (a) below). Find the vibrational frequency, assuming that the masses of A and B are different. Call them ma and mb , and let the spring have constant k. (b) A diatomic molecule adsorbed on a solid surface (figure (b) below) has more possible modes of vibration. Presuming the two springs and masses to be equivalent this time, find their frequencies. Figure 1: From http: // prb. aps. org/ abstract/ PRB/ v19/ i10/ p5355_ 1 . 10. Energetics of diatomic systems An approximate expression for the potential energy of two ions as a function of their separation is PE = − b ke2 + 9 r r (3) The first term is the Coulomb interaction representing the electrical attraction of the two ions, while the second term is introduced to account for the repulsive effect of the two ions at small distances. (a) Find b as a function of the equilibrium spacing ro . (What must be true at equilibrium?) (b) For KCl, with an equilibrium spacing of ro = 0.279 nm, calculate the frequency of small oscillations about r = ro . Hint: do a Taylor expansion of the potential energy to make it look like a harmonic oscillator for small r = ro .