1. Physics 218: Mechanics Instructor: Dr. Tatiana Erukhimova Lectures 26, 27 Hw: Chapter 18 problems and exercises

2. 90 80 70 60 50 40 30 20 9080706050403020 High 100, Low 10 Average 67.3 Median 72 807 – 64.3 808 – 69.8 809 – 67.2

3. Ch. 17 Statics

4. 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?

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

8. Precession under an applied external torque

10. 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

11. 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!

12. 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

13. 26,000 yr cycle Precession cycle

15. 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

16. 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]

17. 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

18. Simple harmonic oscillator

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

20. 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

21. Resonance

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

23. More Detail Time

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

25. 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?

26. 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?

27. 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.)

28. Simple pendulum How long does it take to return to the equilibrium?

29. Have a great day! Hw: Chapter 18 problems and exercises