Chapter 13 SHM?

1. Chapter 13 SHM? WOD are underlined

2. Remember Hooke’s Law • F = - k Δx • New Symbol: “k” • Spring constant. • “Stiffness” of the spring. • Depends on each spring’s dimensions and material. • In N/m

3. Question • If I let go, what will happen to the mass? Then what? Then what?

4. Simple Harmonic Motion • Repeating up and down motion, (like cos wave.) (Draw a picture.) • Motion that occurs when the net force obeys Hooke’s Law • The force is proportional to the displacement and always directed toward the equilibrium position • Show Example with Spring • The motion of a spring mass system is an example of Simple Harmonic Motion • Are springs the only type of SHM?

5. Simple Harmonic Motion • The motion of a spring mass system is an example of Simple Harmonic Motion • Are springs the only type of SHM: • No, • Jump Rope, Sound Waves, Pendulum, Swing, up and down motion of an engine piston

6. Motion of the Spring-Mass System • Initially, Δx is negative and the spring pulls it up. • The object’s inertia causes it to overshoot the equilibrium position. • Δx is positive now and the spring pushes it down. Again it will over shoot equilibrium.

7. Δx, v and a versus t graphs What type of curve is this? For Calculus, Derivative of sin is what? What happens if you bump the spring?

8. Δx, v and a

9. Δx, v and a • All three look like sinusoidal curves. • V is shifted backwards from Δx • a is shifted backwardwards from v.

10. Acceleration of an Object in Simple Harmonic Motion • Remember F = - k x & F = ma • Set them equal to each other: - k x = ma • Solve for a: • a = -kΔx / m • The acceleration is a function of position • Acceleration is not constant. • So non-inertial frame of reference. So, the kinematic equations are not valid here.

11. Amplitude: New Symbol “A” • Amplitude, A • The amplitude is the maximum position of the object relative to the equilibrium position: (Max Height) • In the absence of friction, an object in simple harmonic motion will oscillate between the positions x = ±A • What friction is there?

12. Amplitude: New Symbol “A” • Amplitude, A • The amplitude is the maximum position of the object relative to the equilibrium position: (Max Height) • In the absence of friction, an object in simple harmonic motion will oscillate between the positions x = ±A • What friction is there? • Air Resistance, Molecular Motion in Spring

13. Period: New Symbol “T” • Period: T • uppercase T stands for “period.” • Amount of time for the oscillator to go through 1 complete cycle. • (Time for 1 up and 1 down.) • Often measured from Max to Max, But can be measured from start to start, etc. • Measured in seconds.

14. Frequency: Another new symbol “ƒ” • “ƒ” is for frequency. • It is the number of cycles an oscillator goes through in one second. • It is measured in 1/seconds • 1/seconds => New unit “Hertz” or Hz. • What is the frequency of revolutions of a new M16 bullet?

15. Frequency: Another new symbol “ƒ” • “ƒ” is for frequency. • It is the number of cycles an oscillator goes through in one second. • It is measured in 1/seconds • 1/seconds => New unit “Hertz” • What is the frequency of revolutions of a new M16 bullet? • Ans:5100 Hz or Rev per Second.

16. Period and Frequency • The period, T, is the time per cycle. • The frequency, ƒ, is cycles per time. • Frequency is the reciprocal of the period • ƒ = 1 / T

17. Quick Recap(Pic for WOD) • A – maximum distance from rest postion. • T – time for one complete cycle • ƒ = 1 / T

18. In the table, label each +, -, or 0.

19. Question • When you compress (or stretch) a spring, you have to do work on it. You apply a force over some distance. • Can you get that energy back?

20. Elastic Potential Energy • (Energy stored in a spring. Ability of a spring to do work.) • Work done on a spring is stored as potential energy. • The potential energy of the spring can be transformed into kinetic energy of the mass on the end.

21. Energy Transformations • Suppose a block is moving on a frictionless surface. • Before it hits the spring, the total mechanical energy of the system is the kinetic energy of the block. What happens next?

22. Energy Transformations, 2 • The spring is partially compressed. • The mass has slowed down. • Σ ME = K.E. + P.E.

23. Energy Transformations, 3 • The spring is now fully compressed • The block momentarily stops • The total mechanical energy is stored as elastic potential energy of the spring

24. At all times, total Mechanical Energy is constant = KE + PE (Put into notes) Equations for SHM Energy: KE = ½ mv2 PE = ½ kx2

25. Keep in mind. • It takes the same energy to stretch a spring as compress it. • PE = ½ kx2 Is the same as = ½ k(-x)2 So PE is same at Max or Min A.

26. Back to Period and Frequency • Period • Frequency • What variable is not in these equations?

27. Back to Period and Frequency • Period • Frequency • What variable is not in these equations? A. T and f do not depend on Amplitude.

28. Problem • A 1 kg block is dropped from a height of 1 m onto a spring with k = 55 N/m. How far will the spring compress?

29. Problem (revisited) • A 1 kg block is dropped from a height of 1 m onto a spring with k = 55 N/m. What will it’s frequency and period of oscillation be?

30. Problem • A 1 kg block is dropped from a height of 1 m onto a spring with k = 55 N/m. • Q1. How far will the spring compress? • Q2. What will it’s frequency and period of oscillation be?