Physics 218: Mechanics Instructor: Dr. Tatiana Erukhimova

Lectures 35,36

An ant of mass m is standing at the center of a massless rod of length l. The rod is pivoted at one end so that it can rotate in a horizontal plane. The ant and the rod are given an initial angular velocity 0. If the ant crawls out towards the end of the rod so that his distance from the pivot is given by , find the angular velocity of the rod as a function of time, angular momentum, force exerted on the bug by the rod, torque about the origin.

2

2bt

l

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

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

Suppose there were an axle at the origin with a rigid, but massless rod attached to it with bearings so that the rod could freely rotate. At the end of the rod, of length b, there is a block of mass M as shown below:

b

x0

v0, mrod

axle

A bullet is fired at the block. If the bullet strikes the block and sticks, what will be the angular velocity of the block about the axle? Neglect gravity.

A block of mass M is cemented to a circular platform at a distance b from its center. The platform can rotate, without friction, about a vertical axle through its center with a moment of inertia, Ip. If a bullet of mass m, moving horizontally with velocity of magnitude vB as shown, strikes and embeds itself in the block, find the angular velocity of the platform after the collision.

b

top view

vB

axle

What is the moment of inertia of a disk of thickness h, radius R and total mass M about an axis through its center?