1. Week # 5 Simple harmonic motion/oscillation and Dr. Mohammad Mezaael International Foundation Year (IFY) China- Kaplan

2. Circular Motion Vocabulary r = radius m= mass v = velocity F c = centripetal force F T = tension force (sometimes written as T, not to be confused with the T for period) F f = friction force t = time T = period = “sec / rev” linear (tangential) velocity = 2πr/T “m / s” rotational (angular) velocity (ώ) = “rev / sec” = 1 / T a c = centripetal acceleration

3. Circular Motion Around Us

4. The angle  is the angular position. How do we describe the position of the particle? Again  is defined to be positive in the counter-clock- wise direction. Angles are usually measured in radians. s is arc length. r is the radius of the circle.

5. Radians For a full circle.

6. Part I-Intro to Circular Motion Tangential Speed and Velocity Frequency and Period Centripetal Force Centripetal Acceleration

7. Frequency, f : #revolutions per unit time f = # rev / time Units: (1/sec)=sec -1 =Hertz (Hz) rpm (#rev/min) rps (#rev/sec) r

8. Period Period T : time for 1 revolution Unit: sec, min, h Relating Frequency and period f= 1 T

9. Arc Length Arc Length s (unit: meter) Distance traveled along a circular path. s

10. -not a real force -feeling due to inertia

11. Rotation → axis inside object (Earth rotates around its axis) Revolution → axis outside object (Earth revolves around the Sun) Rotational speed → rot. speed = # of rev / time (rev/s) Period → T = time / # of rev (s/rev) Linear velocity → v = 2πr / T (m/s) = 2  rf T = 2 p/w

12. Circular Motion Around Us

14. Centripetal Acceleration, a c = v 2 r Has same direction as centripetal force.(Always points to the center). Is perpendicular to the tangential velocity at any given instant.

15. Centripetal Force F c =ma c

16. Problem 1 A biker travels once around a circular track of radius 20.0m in 3s. Calculate: a) the average tangential speed b) the frequency c) the period Answers: 41.9m/s, f=0.33Hz, T=3s

17. Problem 2 A coin sits 0.10m from the center of a record player spinning at 45rpm. a) What is the frequency in Hertz? b) What is the period? c) What is the linear speed? Answer: 0.75Hz, 1.33s, 0.47m/s

18. Problem 3 A child on a merry-go-round sits 1.5m from the center. They spin 3 times in one min. The mass of the child is 40kg. Find the friction(centripetal force) acting on the child. Answer: 5.9N

19. Part II More Centripetal Force Problems Car Rounding a Curve Loop-the- loop Rotor

20. What force plays the role of the centripetal force when a car rounds a curve?

21. Car Rounding a Bend

22. Example 1: Car Rounding a Curve A car is travelling with a speed of 45km/h on a circular horizontal track of radius 50m. What is the minimum coefficient of friction, so that the car stays on the track? Answer: 0.3

23. What force plays the role of the centripetal force when a ball is on the top of a loop-the-loop?

24. Example 2: Loop-the-loop What is the critical velocity of a ball at the top of the loop of radius.3m so that it completes the loop? Answer: 1.7m/s

25. Example 3: Rotor A brave student rides in a rotor of radius 5m whose floor drops when it reaches a speed of 20mi/h. What is the coefficient of friction between the student and the wall of the rotor, so that the student does not fall? Answer: 0.6

26. Rotor Ride

27. Centripetal: (“c”) “center seeking” F c and a c always act toward center Velocity is tangent to the curve. Magnitude is constant, direction is not. V ┴ F net-c, a c Radius is the ┴ distance from the axis of rotation to the object Never put F c on the F.B.D.. Ie. F c = F f, F T, F N Projectile Motion is not true Circular motion, Why? In circular motion the force is always ┴ to the velocity; the magnitude of velocity is not effected by the force

28. 1. What does centripetal mean? A ball is attached to a string and whirled around at a constant speed in a circle with a radius 3 meters. (Questions 2-6) 2. What is the direction of the net force on the ball? 3. What is the direction of the accel. of the ball? Is it constant? 4. What is the direction of the velocity of the ball? Is it constant? 5. What are the 3 important equations? Does rotational speed depend upon the radius? Does linear speed depend upon the radius? 6. If it takes 50 seconds for the ball to make 20 revolutions around a 3 m radius, calculate the rotational speed and linear velocity of the ball. 7. What were the relationships we found in the lab?

29. An object moves with simple harmonic motion whenever its acceleration is proportional to its displacement from some equilibrium position and is oppositely directed. Simple harmonic motion/oscillation Solution to this 2 nd order differential equation is: or:

30. Properties of simple harmonic motion  …Angular frequency A…amplitude  …phase constant, phase angle Position of particle at time t: T…period, time to complete one full cycle  t  …phase

31. Properties of simple harmonic motion Displacement: Period T: Frequency: Angular frequency: Units: 1/s = 1 Hz Velocity: Acceleration:

32. Properties of simple harmonic motion Phase of velocity differs by  /2 or 90° from phase of displacement. Phase of acceleration differs by  or 180° from phase of displacement. Displacement, velocity and acceleration vary sinusoidally. Acceleration of particle is proportional to the displacement, but is in the opposite direction (a = -  2· x). The frequency and period of the motion are independent of the amplitude. (demo).

33. The block-spring system The frequency depends only on: - the mass of the block - the force constant of the spring. Note: The frequency does not depend on the amplitude.

34. A spring stretches by 3.90 cm when a 10.0 g mass is hung from it. A 25.0 g mass attached to this spring oscillates in simple harmonic motion. (a)Calculate the period of the motion. (b)Calculate frequency of the motion. A) 1.00 HzB) 1.21 HzC) 1.43 HzD) 1.50 HzE) 1.59 Hz example

35. Energy of a harmonic oscillator Kinetic energy: Potential energy: Total energy:

36. A 0.200 kg mass is attached to a spring and undergoes simple harmonic motion with a period of 0.250 s. The total energy of the system is 2.00 J. Black board example 15.3 (a)What is the force constant of the spring? (b)What is the amplitude of the motion? (c)What is the velocity of the mass when the displacement is 8.00 cm? (d)What is the kinetic and potential energy of the system when the displacement is 8.00 cm?

38. The physical pendulum For small angles (less than about 10°) Small angle approximation: sin  ~  I… moment of interia m… mass of object g… acceleration due to gravity d… distance from pivot point to center of mass

39. The (mathematical) pendulum (point mass) For small motion (less than about 10°).

40. In this case, it can be shown that the period depends on the moment of inertia: Substituting the moment of inertia of a point mass a distance l from the axis of rotation gives, as expected,

41. Black board example 15.4 Find the period of a 14.7 inch (0.37 m) long stick that is pivoted about one end and is oscillating in a vertical plane. A) 1.00 sB) 1.22 sC) 2.00 s D) 2.44 sE) 3.00 s

42. Damped Oscillations The previous image shows a system that is underdamped – it goes through multiple oscillations before coming to rest. A critically damped system is one that relaxes back to the equilibrium position without oscillating and in minimum time; an overdamped system will also not oscillate but is damped so heavily that it takes longer to reach equilibrium.

43. Simple harmonic motion and uniform circular motion

45. Damped, simple harmonic motion b is damping constant Spring constant, k

46. Forced (Driven) Oscillations and Resonance A damped, harmonic oscillator (ang. frequency  ) is driven by an outside, sinusoidal force with ang. frequency  d  Resonance when  d =  (can get very large amplitudes) b is damping constant

47. Summary Period: time required for a motion to go through a complete cycle Frequency: number of oscillations per unit time Angular frequency: Simple harmonic motion occurs when the restoring force is proportional to the displacement from equilibrium.

48. The amplitude is the maximum displacement from equilibrium. Position as a function of time: Velocity as a function of time:

49. Acceleration as a function of time: Period of a mass on a spring: Total energy in simple harmonic motion:

50. Potential energy as a function of time: Kinetic energy as a function of time: A simple pendulum with small amplitude exhibits simple harmonic motion

51. Period of a simple pendulum: Period of a physical pendulum:

52. Oscillations where there is a nonconservative force are called damped. Underdamped: the amplitude decreases exponentially with time: Critically damped: no oscillations; system relaxes back to equilibrium in minimum time Overdamped: also no oscillations, but slower than critical damping

53. An oscillating system may be driven by an external force This force may replace energy lost to friction, or may cause the amplitude to increase greatly at resonance Resonance occurs when the driving frequency is equal to the natural frequency of the system