1. Physics 218: Mechanics Instructor: Dr. Tatiana Erukhimova Lectures 39-41 Hw: Chapter 18 problems and exercises

2. >90>80>70>60>50>40 Average 65.4 Median 64 High 102 Low 2

3. m1 m2 R I The rope is assumed not to slip as the pulley turns. Given m 1, m 2, R, and I find the acceleration of mass m 1.

4. Two masses, and, are attached by a massless, unstretchable string which passes over a pulley with radius R and moment of inertia about its axis I. The horizontal surface is frictionless. The rope is assumed NOT to slip as the pulley turns. Find the acceleration of mass.

5. Rotational Kinetic Energy

6. A rigid body in motion about a moving axis Motion of a rigid body: combination of translation motion of the center of mass and rotation about an axis through the center of mass.

7. A primitive yo-yo is made by wrapping a string several times around a solid cylinder with mass M and radius R. You hold the end of the string stationary while releasing the cylinder with no initial motion. The string unwinds but does not slip or stretch as the cylinder drops and rotates. Use energy considerations to find the speed v cm of the center of mass of the solid cylinder after it has dropped a distance h.

8. Consider the speed of a yo-yo toy

9. Ch. 17 Statics

10. There is a ladder against a wall. There is no friction at the wall and the coefficient of friction at the floor is . At what angle can the ladder be placed so that it would slip if the angle were decreased infinitesimally?

11. The advantage of a lever GIVE ME A PLACE TO STAND AND I WILL MOVE THE EARTH

14. Precession under an applied external torque

16. What causes the seasons? 1.Seasons are NOT caused by varying distances from the Earth to the Sun 2.The primary cause of seasons is the 23.5 degree tilt of the Earth's rotation axis with respect to the plane of the ecliptic. Note: the Earth is actually closest to the Sun in January 4! The Seasons in the Northern Hemisphere Perihelion: 147.09 × 10 6 km; Aphelion: 152.10 × 10 6 km 23.5 o tilt

17. Myr ago Puzzle: Ice Ages! Occur with a period of ~ 250 million yr Cycles of glaciation within the ice age occur with a period of 40,000 yr Most recent ice age began ~ 3 million yr ago and is still going on!

18. Ice Age: Cause Theory: climate changes due to tiny variations in the Earth’s orbital parameters –Precession of the rotation axis (26,000 yr cycle) –Eccentricity (varies from 0.00 to 0.06 with 100,000 and 400,000 yr cycles) –Axis tilt (varies from 24.5 o to 21.5 o with 41,000 yr cycle Milutin Milankovitch 1920

19. 26,000 yr cycle Precession cycle

21. Varies from 0.00 to 0.06 (currently 0.017) Periodicity 100,000 and 400,000 yr Eccentricity cycle modulates the amplitude of the precession cycle

22. f1 f2 f3 f1+f2 f1+f2+f3 Adding oscillations with different phases and incommensurate frequencies f1 = sin[2 t + 1] f2 = 0.7 sin[3.1 t + 2.4] f3 = 1.3 sin[4.5 t + 0.3]

23. As a result, the flux of solar radiation received by the Earth oscillates with different periodicities and amplitudes This triggers changes in climate Why are these changes so irregular? Our Earth makes a complicated motion through space, like a crazy spaceship

24. Simple harmonic oscillator

25. Examples Lots of stuff Vibrates or Oscillates: –Radio Waves –Guitar Strings –Atoms –Clocks, etc…

26. Why do we care? Lots of engineering problems are oscillation problems –Buildings vibrating in the wind –Motors vibrating when running –Solids vibrating when struck –Earthquakes

27. Resonance

28. The forces No force Force in –x direction Force in +x direction

29. More Detail Time

30. Some Terms Amplitude: Max distance Period: Time it takes to get back to here

31. A block with mass m is attached to the end of a spring, with spring constant k. The spring is stretched a distance L and let go at t=0 Find the position of the mass at all times Where does the maximum speed occur? What is the maximum speed?

32. If the block is pulled a distance x 1 to the right and released from rest, how long will it take the block to return to its equilibrium position? How does this time change if the displacement is increased from x 1 to 2x 1 ? How fast will the block be moving at its equilibrium position for the x 1 displacement?

33. A bullet of mass m is fired with velocity of magnitude into a block of mass M. The block is connected to a spring constant k and rests on a frictionless surface. Find the velocity of the block as a function of time. (Assume the bullet comes to rest infinitely quickly in the block, i.e. during the collision the spring doesn’t get compressed.)