HCI 08 Oscillations Solutions (Discussion Questions)

1 Oscillation Tutorial Hints & Suggested Solutions to Discussion Questions D1 Amplitude of oscillation = 50 mm Period of oscillation = 2 s x = 50 sin (2π /2) t (x in mm) [Note: Taking to the right of 650 mm mark as positive direction] when x = +25 mm 25 = 50 sin (2π /2) t Time duration for which shutter remained open, t = 1/6 s = 0.17 s Key Concepts: Characteristics of S.H.M? Able to write down the displacement-time function for a S.H.M. from initial condition (i.e. when t = 0 s). Note: For SHM, displacement is NOT proportional to time. D2 Hints: Check at equilibrium x = 0, is there tension? How is tension related to the extension of spring? B. Key Concepts: Link to topic of Dynamics & Forces. Note: Tension in spring, T is NOT proportional to the displacement, x. Qn D2, D10 & D11 is about oscillation of a vertical spring mass system. D3 Hints: Sketch the displacement, x vs phase angle, ϕ graph for the scenario. Note that at time t = 0 s, Mass A is released from its maximum displacement position (2 cm below equilibrium position). Mass B is released from it maximum displacement position (1 cm below equilibrium position) only when Mass A is at displacement of 1 cm b elow equilibrium position. B. 1 2cos 3    x Mass A Mass B

2 Key Concepts: Understand phase angle and phase difference for oscillations. D4 Hints: i) Can you write down the function that relates the acceleration, a to time, t? ii) Can you write down the function that relates the displacement, x to time, t? iii) Can you write down the function that relates the velocity, v to time, t? A. From a – t graph, a = ao cos ωt We can then infer that displacement, x = - xo cos ωt Hence, velocity, v = vo sin ωt Sketch the x – t and v – t graphs on the a – t graph (same time axis), and determine that at P when a is positive, x is negative and v is positive. Key Concepts: See the relationship between a-t, x-t & v-t graphs of SHM. Know the kinematics of SHM. D5 Hints: i) How do you know that the oscillation is SHM? ii) Where is the equilibrium position of the SHM? iii) What is the amplitude of the SHM? iv) What is the displacement of SHM when the depth of water is 1.5 m? v) Can you write down the appropriate function that relates displacement, x to the time, t? 2.0 m is the equilibrium position. Amplitude, x0 = 1.0 m T = 12 hrs Depth of water, x = - 1.0 cos (2π /12) t + 2.0 1.5 = - 1.0 cos (2π /12) t + 2.0 Time duration that the boat will have to wait before entering, t = 2.0 hrs Key Concepts: Characteristics of S.H.M? Able to write down the displacement-time function for a S.H.M. from initial condition (i.e. when t = 0 s). D6 (b) (i)