Remember: “Practice HW #11” posted on WebAssign ( 0 points, covers material after HW #10 )

1. Remember: “Practice HW #11” posted on WebAssign (0 points, covers material after HW #10) Solutions will be posted on Friday afternoon • Last Time: Hooke’s Law, Simple Harmonic Motion • Today: SHM Position, Velocity, Acceleration; Pendulum Motion

2. Review: SHM Period & Frequency For SHM, relations between period, frequency, and angular frequency for SHM : time for one oscillation [seconds] number of oscillations per second [Hz] If think of one oscillation as corresponding to 2π radians, ω = number of radians/second [rad/s]

3. Conceptual An object of mass m is attached to a horizontal spring, stretched to a displacement of A from equilibrium, and then released. It then undergoes harmonic oscillations on a frictionless surface with period T0 . This is then repeated with a new object of mass 4m. What is its new period of oscillation? (a) 2T0 (b) T0 (c) T0/2 (d) T0/4

4. Example A 0.326-kg object is attached to a spring and executes SHM with a period of 0.25 s. If the total energy of the system is 5.83 J, find: the maximum speed of the object, the force constant of the spring, the amplitude of the motion.

5. Position, Velocity, Acceleration vs. Time Again, we will use the close mathematical relationship between Circular Motion and SHM : At some time, the x-position is: ω Suppose at t = 0, θ = 0. Then we have θ = ωt, so : θ But: ω = 2π/T = 2πf A

6. Position, Velocity, Acceleration vs. Time Displacement x : Velocity v : Velocity v :

7. Warning !! Displacement x : In these formulas, ωt is in radians. So you need to make sure your calculator is set to RADIANS, NOTdegrees when calculating sines and cosines !! Velocity v : Velocity v :

8. Demo: SHM Motion IS Sinusoidal !

9. Conceptual Question If the amplitude A of a system undergoing SHM is doubled, which of the following quantities does NOT change? Total energy Maximum speed Maximum acceleration Period

10. Conceptual Question Suppose the position of an object moving with SHM is given by: x = 4 cos (6πt), where x is in meters, and t is in seconds. What is the period of this oscillating system? 4 s 1/6 s 1/3 s 6π s not enough information

11. Example: 13.28 (modified) The position of an object connected to a spring varies with time according to x = A sin (Bt), where: A = 0.052 m, B = 8π 1/s • What is the period and frequency of this motion? • What is the amplitude of this motion? • Find the first time after t = 0 that the object reaches x = 0.026 m. • At this time, what is the object’s velocity and acceleration?

12. Motion of a Pendulum When a pendulum swings back and forth, is the motion SHM? To answer this, we need to examine the restoring force, the force of gravity, that acts along the circular arc. Key Point: Mass on a spring moves in 1-D only. Here, we have motion in 2-D. But we will consider “small oscillations”.

13. Motion of a Pendulum Let’s see if we can find a “Hooke’s Law” for a pendulum … End results …

14. Comments So we found : This implies : • Period of a pendulum does not depend on its mass • Period of a pendulum does not depend on its amplitude (provided we are considering “small oscillations”) • What is the length of the pendulum in our lecture hall ?

15. Conceptual Question If a pendulum clock is tuned (i.e., its length is set) so that it keeps perfect time at the base of a very tall mountain, will it also keep perfect time when it is moved to the top of this mountain?

16. Example: 13.39 Earth/Mars Comparison ~15% of Earth’s volume ~11% of Earth’s mass The free-fall acceleration on Mars is 3.7 m/s2. What length of pendulum has a period of 1 s on Earth? On Mars? An object is suspended from a spring with k = 10 N/m. What mass would result in a period of 1 s on Earth? On Mars?

17. Next Class • 13.7 – 13.8 : Intro to Properties of Waves (PHY 213), Review for Final