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MSTC Physics C

MSTC Physics C. Study Guide Chapter 12 Sections 2 and 3. Simple Harmonic Motion.

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MSTC Physics C

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  1. MSTC Physics C Study Guide Chapter 12 Sections 2 and 3

  2. Simple Harmonic Motion System, when moved away from its central position, that experiences a restoring force proportional to the displacement from this position and oscillates for an indefinite period of time with no loss of mechanical energy

  3. Simple Harmonic Oscillator If displacement vs time is plotted for a SHO motion along the x axis can be written as x = A cos(ωt + δ) where A = amplitude, ω = angular frequency, and δ = phase constant (how long after t=0 the max is reached)

  4. Simple Harmonic Oscillator Motion of SHO is periodic and repeats every time ωt increases by 2 π radians ex- here δ = π at ωt = 0 x = A cos π = -A at ωt = 2 π x = Acos(2 π + π) = Acos 3 π = -A

  5. Simple Harmonic Oscillator (ω t + δ) is called the phase of the motion Recall that d2x/dt2 = - ω2x for SHM Since x = = A cos(ωt + δ) v = dx/dt = - ωAsin(ωt + δ) a = d2x/dt2 = -A ω2cos(ωt + δ) = - ω2x

  6. Simple Harmonic Oscillator Since sine and cosine oscillate between +/- 1 vmax = ωA amax = ω2A

  7. Sample Problem A mass of 200 g is connected to a light spring of force constant 5 N/m and is free to oscillate on a horizontal, frictionless surface. If the mass is displaced 5 cm from equilibrium and released from rest, (a) find the period of its motion. (b) Determine the maximum speed of the mass. (c) What is the maximum acceleration of the mass? (d) Express the displacement, speed, and acceleration as functions of time. (e) At what time does the maximum speed and the maximum acceleration occur?

  8. Sample Problem A spring stretches 0.15 m when a 0.3 kg mass is hung from it. The spring is then stretched an additional 0.1 m from this equilibrium point and released. Determine (a) the spring constant, (b) the amplitude of the oscillation, (c) the maximum velocity, (d) the maximum acceleration, (e) the period and frequency, (f) the displacement as a function of time, and (g) the velocity at t=0.15 s.

  9. Energy of a SHO For a SHO the energy oscillates between KE and U Consider a mass spring oscillator on a frictionless table Total energy at any time E = KE + Us E = ½ mv2 + ½ kx2 Know that x = A cosωt and v = -ωAsinωt Can show that E = ½ kA2 The total energy of a SHO is constant and proportional to A2

  10. Sample Problem A spring with constant k = 19.6 N/m has an amplitude of 0.1 m. Attached to the spring is a 0.3 kg mass. The equations for displacement and velocity as fns of time are x = -0.1cos(8.1t) and v = 0.81sin(8.1t). If the spring is vibrating horizontally, find the total energy, the kinetic energy and potential energy as fns of time, the velocity when the mass is 0.05 m from equilibrium, the kinetic and potential energies at ½ amplitude.

  11. Springs In Combination F will distribute K1 F itself among the 3 K2 springs so K3 F = F1 + F2 + F3 all springs will stretch the same amount so F = K1 x + K2 x + K3 x = (K1 + K2 + K3) x In parallel Keq = K1 + K2 + K3 + …..

  12. Springs In Combination K1 K2 K3 Each spring will stretch a different amount given by x1 = F1 / K1 So total x = F1/K1 + F2/K2 + F3/K3 The force will be transmitted equally so x = F (1/K1 + 1/K2 + 1/K3) In series 1/Keq = 1/K1 + 1/K2 + 1/K3

  13. Sample Problem Three springs with force constants K1 = 10 N/m, K2 = 12.5 N/m, and K3 = 15 N/m are connected in parallel to a mass of 0.5 kg. The mass is then pulled to the right and released. Find the period of motion. The same three springs are now connected in series. Find the period of motion.

  14. Sample Problem An object suspended from a spring exhibits oscillations of period T. Now the spring is cut in half and the 2 halves are used to support the same object as shown. Show that the new period of oscillation is T/2.

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