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Understanding Simple Harmonic Motion in Physics

Explore the basics of an ideal spring-mass system including displacement, speed, acceleration, and mechanical energy. Learn about oscillations in various forms like pendulum and vibrating strings.

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Understanding Simple Harmonic Motion in Physics

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  1. Review: Oscillations of an Ideal Spring Recall the basics of an ideal spring-mass system: Its maximum displacement, A, is called its amplitude. It has maximum speed at minimum displacement; It has minimum speed at maximum displacement (A). It has maximum acceleration at maximum displacement; It has minimum acceleration at minimum displacement. If no external work is being done on it, its mechanical energy is a constant, comprised of some mix of kinetic and potential energy. OSU PH 212, Before Class #19

  2. A Universe Full of Oscillations But a spring-mass system isn’t the only form of Simple Harmonic Motion we can observe. Here’s a longer list: Spring-mass Pendulum Vibrating string Vibrating membrane Orbiting object (electron? planet?) OSU PH 212, Before Class #19

  3. Review: A mass is moving at constant speed in a circle of radius A meters. It completes each revolution in a time of T seconds. Find: (a) its speed, v, in m/s. (b) the magnitude of its acceleration, a, in m/s2. A.(a) 2A/T(b) 42A/T2 B.(a) 2/T(b) 42A/T2 C.(a) 2A/T(b) 0 D.(a) 2/T(b) 0 E. None of the above. OSU PH 212, Before Class #19

  4. Review: A mass is moving at constant speed in a circle of radius A meters. It completes each revolution in a time of T seconds. Find: (a) its speed, v, in m/s. (b) the magnitude of its acceleration, a, in m/s2. A.(a)2A/T(b)42A/T2 B.(a) 2/T(b) 42A/T2 C.(a) 2A/T(b) 0 D.(a) 2/T(b) 0 E. None of the above. OSU PH 212, Before Class #19

  5. Q: How would the previous answers look if we were to express 2/T as  instead? A:v = __________________ a = __________________ Q: How does this relate to amax and vmax of a mass oscillating on a spring? A: OSU PH 212, Before Class #19

  6. Q: How would the previous answers look if we were to express 2/T as  instead? A:v = A a = A2 Q: How does this relate to amax and vmax of a mass oscillating on a spring? A:Either axis of the 2D circular motion exhibits the above behavior—which matches that of a spring/mass system, OSU PH 212, Before Class #19

  7. OSU PH 212, Before Class #19

  8. Uniform Circular Motion and Simple Harmonic Motion • Notice how the projection of the motion of an object undergoing uniform circular motion onto one coordinate axis exactlydescribes simple harmonic motion. Look at the x-component of the position of the mass (we choose the x axis here since we’ve projected the circular motion onto a horizontal surface; either x or y would work this way): • x = A·cosθ • where A is the radius of our circle and is also the maximum amplitude of the shadow on the wall. • Remember that angular speed is ω = dθ/dt (rad/s). • At any time t, the angular displacement is simply θ = ωt. • Therefore, x(t) = A·cos(ωt). OSU PH 212, Before Class #19

  9. Displacement of an Object in Simple Harmonic Motion • x(t) = A·cos(ωt + f0) • xis the displacement (in ±meters) at a time t. x is a vector. • A is the maximum displacement (amplitude) magnitude of the cosine function, expressed in meters. Since cosine varies between –1 and +1, xwill oscillate between – A and +A. Those extrema occur whenever (ωt + f0) = nπ, where n = 0, 1, 2, 3, … • ω is the angular frequency, in radians/second. The angular frequency is related to frequency and period:ω = 2πf = 2π/T • Note that ω does not have units of hertz (cycles per second). • t is the time (in seconds). • f0 is the “angular position” of the oscillator in its cycle at t = 0. OSU PH 212, Before Class #19

  10. What is the amplitude of the Simple Harmonic Motion graphed above? • -10 cm • -5 cm • 5 cm • 10 cm • 20 cm • Not enough info. OSU PH 212, Before Class #19

  11. What is the amplitude of the Simple Harmonic Motion graphed above? • -10 cm • -5 cm • 5 cm • 10 cm • 20 cm • Not enough info. OSU PH 212, Before Class #19

  12. What is the frequency of the Simple Harmonic Motion graphed above? • 4 s • 2 s • 1 s • 1 Hz • 1/2 Hz • ¼ Hz OSU PH 212, Before Class #19

  13. What is the frequency of the Simple Harmonic Motion graphed above? • 4 s • 2 s • 1 s • 1 Hz • 1/2 Hz • ¼ Hz OSU PH 212, Before Class #19

  14. The initial phase angle of a sinusoidal function is the “angular position” (i.e. the point in its cycle) of the oscillator at t = 0. • Assuming this is a cosine function, how do you know the initial phase is not zero? What is the value of the initial phase angle? A: OSU PH 212, Before Class #19

  15. The initial phase angle of a sinusoidal function is the “angular position” (i.e. the point in its cycle) of the oscillator at t = 0. • Assuming this is a cosine function, how do you know the initial phase is not zero? What is the value of the initial phase angle? A: This graph is of the form x(t) = Acos(wt + f). If f were 0, then at t = 0, x would have a value of A (the positive maximum value). But looking at the graph, we can see that x(0) = –A/2. That is: Acos(0 + f) = –A/2 Or: cos(f) = –(1/2) So: f = 2p/3 rad orf = 4p/3 rad Which is correct? Notice that as t moves forward from 0, x(t) becomes –A before it next becomes 0. So: f = 2p/3 rad OSU PH 212, Before Class #19

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