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Reminder: Course Evaluation

Reminder: Course Evaluation. 5 bonus points (as detailed in syllabus) Window: Wed Nov 17 – Wed Dec 8 Go to Department of Physics homepage: www.as.uky.edu/physics Look for the link to the Online Course Evaluation under the “Courses” tab on the left-hand side of the webpage.

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Reminder: Course Evaluation

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  1. Reminder: Course Evaluation • 5 bonus points (as detailed in syllabus) • Window: Wed Nov 17 – Wed Dec 8 • Go to Department of Physics homepage: www.as.uky.edu/physics • Look for the link to the Online Course Evaluation under the “Courses” tab on the left-hand side of the webpage.

  2. Reminder: HW #10 due tonight, 11:59 p.m. • “Practice HW #11” posted on WebAssign (0 points, covers material after HW #10) • Last Time: Fluid Motion, Bernoulli’s Equation • Today: Hooke’s Law, Simple Harmonic Motion

  3. Periodic Motion: Vibrations & Waves water waves Sun cell phone tower sound waves PHY 213 atoms in solids

  4. Hooke’s Law x = 0 : Position of spring when not compressed/stretched (“equilibrium”) x Force exerted by spring : x = 0 : F = –kx k: “spring constant” compressed x < 0 If compressed, x < 0, so F > 0 ! F x “Restoring Force”

  5. Hooke’s Law x = 0 : Position of spring when not compressed/stretched (“equilibrium”) x Force exerted by spring : x = 0 : F = –kx stretched x > 0 x If stretched, x > 0, so F < 0 ! F “Restoring Force”

  6. Simple Harmonic Motion http://en.wikipedia.org/wiki/File:Muelle.gif Suppose mass released from x = +A on the right. Restoring force F = –kx < 0 accelerates mass towards x = 0. Magnitude of restoring force decreases as reaches x = 0. Mass reaches x = 0. Has reached maximum speed at x = 0, so “overshoots” equilibrium position, and compresses the spring. As mass moves to x < 0 values, restoring force F = –kx > 0, and increases in magnitude as mass moves to more negative values. Speed of object decreases, and finally briefly comes to rest at x = –A, before accelerating back towards x = 0. Ultimately returns to x = +A. Process repeats …

  7. Simple Harmonic Motion SHM occurs when : • The net force along the direction of motion obeys Hooke’s Law, F = –kx • Magnitude of net force proportional to displacement from equilibrium point, and always directed towards equilibrium point • Result is that object oscillates back and forth over the same path

  8. Object Suspended From Vertical Spring natural equilibrium length g Weight of mass stretches spring until equilibrium is reached. no mass m DEMO: If perturbed from this new equilibrium position, will undergo SHM about this new position !

  9. How to Determine a Spring Constant k natural equilibrium length y g y = 0 no mass m y = –d –ky = kd mg

  10. Example When a 4.25-kg object is placed on top of a vertical spring, the spring compresses a distance of 2.62 cm. What is the spring constant k?

  11. Simple Harmonic Motion: Definitions http://en.wikipedia.org/wiki/File:Muelle.gif The Amplitude A is the maximum displacement of the object from its equilibrium position. If no friction, will oscillate back and forth between x = +A and x = –A. The Period T is the time [seconds] for the object to move through one complete cycle of motion: x = +A to x = –A and back to x = +A (3) The Frequency f is the number of complete cycles (or vibrations) per unit of time. Note that f = 1/T. Units of [1/seconds] = Hertz = Hz

  12. Conceptual Question x = +A x = 0 x A block on the end of a horizontal spring is pulled from equilibrium at x = 0 to x = +A and then released. Through what total distance does it travel in one full cycle of its motion?

  13. Acceleration in SHM Acceleration is NOT constant !! Can’t use constant acceleration kinematic equations for SHM !! x = 0 : x = 0 : compressed x < 0 stretched x > 0 x F F x a > 0 : a < 0 : If moving to left, decelerating If moving to right, decelerating If moving to right, accelerating If moving to left, accelerating

  14. SHM: Velocity and Acceleration Vectors If moving left If moving left x = 0 : x = 0 : If moving right If moving right x = 0 : x = 0 :

  15. But Note ! When block has reached its amplitude, at x = +A or x = –A, its velocity is 0 (for a brief instant of time). At these points, the acceleration is a maximum : x = +A : a = –kx/m = –kA/m x = –A : a = –kx/m = +kA/m x = –A x = 0 x = +A x = 0

  16. Spring Elastic Potential Energy x = 0 x = 0 : compressed stretched x x If spring is compressed or stretched, it will exert a force, and so it has the potential to do work. Elastic potential energy associated with this spring force is : k : “spring constant” x : displacement of spring SI Unit: Joules

  17. Conservation of Energy Assuming only conservative forces (i.e., no non-conservative forces, such as friction), systems with springs will obey : Initial KE Initial Spring PE Initial Grav. PE Final KE Final Spring PE Final Grav. PE We would also need to add rotational KE, KEr, if the problem involved torques.

  18. Example A car with a mass of 1000 kg is driven into a brick wall in a safety test. The bumper behaves like a spring with spring constant k = 5 x 106 N/m, and is compressed 3.16 cm as the car is brought to rest. What was the speed of the car before impact, assuming no energy is lost in the collision with the wall?

  19. Velocity vs. Position x = 0 x x = +A x Total Mechanical Energy in the System : Recall: v = 0 at x = ±A + : moving to right Energy Conservation : Solve for v : – : moving to left

  20. Velocity vs. Position x = 0 x x = +A x Velocity : • If x = ±A, v = 0 (speed is 0 at maximum displacement) • If x = 0, (speed is maximum at x = 0)

  21. Example A 1.0-kg block connected to a massless spring with a spring constant of 20.0 N/m oscillates on a frictionless horizontal surface. (a) Calculate the total energy of the system and the maximum speed of the object if the amplitude of the motion is 3.0 cm. (b) What is the block’s speed when its displacement from equilibrium is 2.0 cm?

  22. Circular vs. Simple Harmonic Motion For object moving in a circle (radius A) at constant speed v : θ v0 θ C1 just a constant A Compare to SHM : C2 just a constant

  23. Period & Frequency Mathematically … Circular Motion & SHM are very similar ! So we can use this similarity to derive some analogies between Circular Motion and SHM. Circular : SHM : By conservation of energy By analogy with circular motion : Period for SHM: Time for one oscillation

  24. 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]

  25. Next Class • 13.4 – 13.7 : SHM Position, Velocity, Acceleration; Pendulum Motion

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