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# Simple Harmonic Motion

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Describes simple harmonic motion of weight- spring systems and pendulums.
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### Simple Harmonic Motion

2. 2. 200 grams Vibrating Tuning fork A weight on a spring A boy on a swing 2
3. 3. Simple Harmonic Motion • Simple harmonic motion (SHM) is a repeated motion of a particular frequency and period. • The force causing the motion is in direct relationship to the displacement of the body. (Hooke’s Law) • The displacement, velocity, acceleration and force characteristics are specific a various points in the cycle for SHM. • SHM can be understood in terms of the displacement, velocity, acceleration and force vectors related to circular motion. • Recall that the displacement vector for circular motion is the radius of the circular path. The velocity vector is tangent to the circular path and the acceleration vector always points towards the center of the circle. 3
4. 4. 400 grams 200 grams F O R C E (N) ELONGATION (M) Slope = spring constant 600 grams Elongation of spring 4
5. 5. Simple Harmonic Motion • SHM motion can be represented as a vertical view of circular motion. Using this concept, we can see the variations in the vector lengths and directions for displacement, velocity and accelerations as those values for SHM. • Use the following slide showing a mass on a spring, vibrating in SHM to examine the variations in these three vectors as the reference circle rotates. • See if you can decide which trig functions (sine, cosine or tangent) govern each to the three vectors in SHM 5
6. 6. Displacement = +max Velocity = 0 Acceleration = - max Kinetic Energy = 0 Net Force = - max Displacement = 0 Velocity = max Acceleration = 0 Kinetic Energy = max Net Force = 0 Top of cycle Mid cycle Bottom of cycle Displacement = - max Velocity = 0 Acceleration = + max Kinetic Energy = 0 Net Force = + max CLICK HERE 6
7. 7. The velocity vector (black) is always directed tangentially to the circular path. The acceleration vector (red) is always directed toward the center of the circular path 7
8. 8. Displacement Vector of Circular motion & Displacement in SHM Acceleration Vector 8
9. 9. Displacement vector on Reference Circle 200 grams 200 grams 200 grams Vertical View Simple Harmonic Position 200 grams y = +max y = 0 y = -max y = 0 9
10. 10. Displacement vector on Reference Circle Vertical View Note that the vertical view of the displacement vector is 0 at 00, 100 % upward at 900, 0 at 1800, 100 % downward at 2700 and finally 0 again at 3600 What trig function is 0 at 00, 1.0 (100%) at 900, 0 at 1800, -1.0 (100% and pointing down) at 2700, and 0 again at 3600 y = 0 00 y = -max 2700 y = 0 1800 y = +max 900 The SINE y = Amp x sin θ 10
11. 11. The velocity vector is always tangent to the circular path 11
12. 12. The reference circle is turned sideways and viewed vertically. This shows the velocity vector of a body in Simple Harmonic Motion. 12
13. 13. Velocity Vector of Circular motion & Velocity in SHM 13
14. 14. Velocity vector on Reference Circle 200 grams 200 grams 200 grams Vertical View Simple Harmonic Position 200 grams V = 0 V = + max V = 0 V = -max 14
15. 15. Note that the vertical view of the velocity vector is 100 % upward at 00, 0 at 900, 100% downward at 1800, 0 at 2700 and finally 100% again at 3600 What trig function is 1.0 (100%) at 00, 0 at 900, 1.0 (100%) at 1800, 0 at 2700, and 1.0 again at 3600 Velocity vector on Reference Circle Vertical View V = 0 900 V = + max 00 V = 0 2700 V = -max 1800 The COSINE V = Vmax x cos θ 15
16. 16. Acceleration Vector of Circular motion & Acceleration in SHM 16
17. 17. Acceleration vector on Reference Circle 200 grams 200 grams 200 grams Vertical View Simple Harmonic Position 200 grams a = -max a = 0 a = +max a = 0 17
18. 18. Acceleration vector on Reference Circle Vertical View a = -max, 900 a = 0 00 a = +max, 2700 a = 0 1800 Note that the vertical view of the acceleration vector is 0 at 00, 100 % downward at 900, 0 at 1800, 100 % upward at 2700 and finally 0 again at 3600 What trig function is 0 at 00, -1.0 (100%) at 900, 0 at 1800, +1.0 (100% and pointing down) at 2700, and 0 again at 3600 The - SINE a = amax x ( -sin θ) 18
19. 19. 200 grams 200 grams 200 grams 200 grams 200 grams 0 o 90 o 180 o 270 o 360 o V = Velocity max x Cos  Acc. = Acc. max x (-Sin ) Y = Amplitude x Sin  19
20. 20. y t Y = Amp x Sin  Amp dy/dt t dy/dt = Vmax Cos  d2y/dt2 t d2y/dt2 = Amax (-Sin ) Displacement Velocity Acceleration 20