Physics 325 HW 5A due 1:00 pm Thursday March 5, 2015 ___________________name1. 15 points A pendulum, with a massless rod of length L and apoint bob of mass m, swings in a gravitational field as pictured.It has energy E, not necessarily small. ( we choose our referenceposition at which potential energy = 0 at θ = 0 )In terms of E, L, g, and m, what is its maximum angularspeed dθ/dt?How large must the energy E be such that the pendulumwhirls around rather than just oscillates?In terms of E, L, m and g, what is the amplitude θmax of itsoscillations?2. 15 points A mass M hangs from the ceiling by a linear spring of stiffness k. The spring has anatural length L.At static equilibrium, under the influence of the weightMg, the spring has a new length, what is it?Write an expression for the system's potential energy PEin terms of the length X of the spring.Find the value of X for which total potential energy PEis minimum.What is the second derivative d2PE/dX2 ?What is the period of the system's vibrations?Phys 325 HW 5B due 1:00 pm Thursday March 5, 2015 Name:________________1. 15 points A compound pendulum is formed of a rigidrod of length 2a, one mass m at its tip and another mass 2m atits midpoint. What is the period of its small oscillations θ(t)?One approach: Start by constructing formulas for its PE as afunction of θ and its KE as a function of dθ/dt.2. 15 points The pictured mass/spring system has m = 3 kg, k=75 N/m; and c = 18 Nsec/m. x(t) represents rightwarddeviation away from equilibrium. It has initial conditionsxo = 2 mm vo = 10 mm/sec. What is its x(t) subsequent totime zero, in mm ?3. 15 points The displacement history of a mass-spring-viscous resistance element Single Degreeof Freedom system is given by the plot. The information from t < 1 was not recorded. From theplot, and using a fine scale ruler for accuracy, accurately• estimate the log decrement δ of this system and• show that δ is not varying in time.• Also estimate the damped frequency ωd.• From these, determine the natural frequency ωn.• Given that the mass is 1 kg, make a good estimate for the stiffness k of the
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