Physics 218: Mechanics Instructor: Dr. Tatiana Erukhimova

Lectures 39-41

Hw: Chapter 18 problems and exercises

>90 >80 >70 >60 >50 >40

Average 65.4Median 64High 102Low 2

m1

m2

R I

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

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 .

1m 2m

1m

1m

2m

Rotational Kinetic Energy

2

2

1 IKE

22

2

1

2

1 cmcm ImvKE

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.

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 vcm of the center of mass of the solid cylinder after it has dropped a distance h.

Consider the speed of a yo-yo toy

Ch. 17 Statics

00 extextF

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?

The advantage of a lever

GIVE ME A PLACE TO STAND AND I WILL MOVE THE EARTH                                                  

Precession under an applied external torque

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 × 106 km; Aphelion: 152.10 × 106 km

23.5o tilt

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!

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 24.5o to 21.5o with 41,000 yr cycle

Milutin Milankovitch 1920

26,000 yr cycle

Precession cycle

http://en.wikipedia.org/wiki/Image:Earth_obliquity_range.svg

• 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

2 4 6 8 10

-1

-0.5

0.5

1

2 4 6 8 10

-1

-0.5

0.5

1

2 4 6 8 10

-1

-0.5

0.5

1

f1

f2

f3

2 4 6 8 10

-2

-1

1

2

f1+f2

2 4 6 8 10

-2

-1

1

2

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]

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

Simple harmonic oscillator

Examples

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

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

Resonance

The forcesNo force

Force in –xdirection

Force in +xdirection

More Detail

Time

Some Terms

Amplitude: Max distance

Period: Time it takes to get back to here

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?

If the block is pulled a distance x1 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 x1 to 2x1?

How fast will the block be moving at its equilibrium position for the x1 displacement?

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.)

mV

mV