For earthquakes, there are several types of disturbances, including disturbance of Earth’s surface and pressure disturbances under the surface.

Sound waves in air and water are longitudinal.

and energy

Sound in solids can be both longitudinal and transverse.

In equation form

Calculate the moment of inertia of a skater given the following information. (a) The 60.0-kg skater is approximated as a cylinder that has a 0.110-m radius. (b) The skater with arms extended is approximately a cylinder that is 52.5 kg, has a 0.110-m radius, and has two 0.900-m-long arms which are 3.75 kg each and extend straight out from the cylinder like rods rotated about their ends.

Starting with the formula for the moment of inertia of a rod rotated around an axis through one end perpendicular to its length prove that the moment of inertia of a rod rotated about an axis through its center perpendicular to its length is You will find the graphics in Figure 3 useful in visualizing these rotations.

MAKING CONNECTIONS

Some rotational inertias.

Example 1: Calculating the Effect of Mass Distribution on a Merry-Go-Round

Instead of memorizing these equations, it is helpful to be able to determine them from reason.

The brake cable on a bicycle carries the tension T from the handlebars to the brake

general result that if friction on an incline is negligible, then the acceleration down the incline is regardless of mass.

To do this, draw the right triangle

Flexible connectors are often used to transmit forces around corners

adding are perpendicular to each other and thus form a right triangle. This means that we can use the Pythagorean theorem to calculate the magnitude

there are no additional forces on the ball in the horizontal direction

static friction is a responsive force, being able to assume a value less than but no more than

1: (a) A 22.0 kg child is riding a playground merry-go-round that is rotating at 40.0 rev/min. What centripetal force must she exert to stay on if she is 1.25 m from its center?

What is the ideal banking angle for a gentle turn of 1.20 km radius on a highway with a 105 km/h speed limit (about 65 mi/h), assuming everyone travels at the limit?

Let us now consider banked curves,

Tire friction enables a vehicle to take the curve at significantly higher speeds.

In cases in which forces are not parallel, it is most convenient to consider components along perpendicular axes—in this case, the vertical and horizontal directions.

much, much less than the speed of light

his ideas were eventually accepted by the church and scientific communities.

Kinematics of Rotational Motion

four rotational kinematic equations (presented together with their translational counterparts):

the force (to bring the occupant to a stop) will be much less if it acts over a larger time.

average effective force

equivalent effective force

When opening a door, you push on it perpendicularly with a force of 55.0 N at a distance of 0.850m from the hinges. What torque are you exerting relative to the hinges?

The torque is always calculated with reference to some chosen pivot point. For the same applied force, a different choice for the location of the pivot will give you a different value for the torque, since both and depend on the location of the pivot. Any point in any object can be chosen to calculate the torque about that point. The object may not actually pivot about the chosen “pivot point.”

Torque plays the same role in rotational motion that force plays in linear motion.

Example 2: Calculating Force: Venus Williams’ Racquet

when the mass of the system is constant

Compare the player’s momentum with the momentum of a hard-thrown 0.410-kg football

The net external force equals the change in momentum

An object that has a small mass and an object that has a large mass have the same momentum. Which object has the largest kinetic energy? 2: An object that has a small mass and an object that has a large mass have the same kinetic energy. Which mass has the largest momentum?

The angles that vectors and make

Figure 6. To add vectors A and B

Extrapolating to a frictionless surface

find the masses of heavenly bodies

a small set of rules and a single underlying force

In the next image, her rate of spin increases greatly when she pulls in her arms, decreasing her moment of inertia.

Suppose a child walks from the outer edge of a rotating merry-go round to the inside. Does the angular veloc

Calculating the Angular Momentum

can be added using the familiar head-to-tail method or by trigonometric methods.

to show that the motion of heavenly bodies should be conic sections

Recall that the acceleration due to gravity is about on Earth.

an equation for

3: Draw a free body diagram for a satellite

both

In physics, the definition of time is simple

10.4

There is no simple, yet accurate, scientific definition for energy.

almost

Because gravitational potential energy depends on relative position, we need a reference level at which to set the potential energy equal to 0.

9-megaton fusion bomb

Potential Energy of a Spring

Collisions of Point Masses in Two Dimensions

It can help us make predictions.

the force exerted by the person pushing the mower must be greater than the friction opposing the motion

Introduction to Rocket Propulsion

It can be shown that

restoring force.

most fishermen feel no obligation to truthfully report the mass

Here, is the restoring force, is the displacement from equilibrium or deformation, and is a constant related to the difficulty in deforming the system. The minus sign indicates the restoring force is in the direction opposite to the displacement.

If we follow a small volume of fluid along its path,

This equation tells us that, in static fluids, pressure increases with depth

When a fluid flows into a narrower channel, its speed increases.

2: A 75.0-kg person climbs stairs, gaining 2.50 meters in height. Find the work done to accomplish this task.

How much work is done by the boy pulling his sister 30.0 m in a wagon as shown in Figure 3? Assume no friction acts on the wagon

2: Give an example of a situation in which there is a force and a displacement, but the force does no work. Explain why it does no work.

The directions of the velocity of an object at two different points are shown, and the change in velocity Δv is seen to point directly toward the center of curvature. (See small inset.) Because ac = Δv/Δt, the acceleration is also toward the center; ac is called centripetal acceleration. (Because Δθ is very small, the arc length Δs is equal to the chord length Δr for small time differences.)

Example 1: How Does the Centripetal Acceleration of a Car Around a Curve Compare with That Due to Gravity?

Example 2: How Big Is The Centripetal Acceleration in an Ultracentrifuge?

the resultant is independent of the order in which the vectors are added.

the negative of any vector has the same magnitude but the opposite direction,

The topic of chaos has become quite popular over the last few decades.

Figure 1. (a) A graph

rotational kinetic energy

On the whole, solutions involving energy are generally shorter and easier than those using kinematics and dynamics alone.

This expression

Example 4: Work and Energy Can Reveal Distance, Too

momentum, which will affect the outcome of their collisions

Many systems oscillate, and they have certain characteristics in common.

guitar, atoms

The forces due to pressure have well-defined directions: they are always exerted perpendicular to any surface.

“Aerodynamic” shaping

Suppose you attach the object with mass to a vertical spring originally at rest, and let it bounce up and down. You release the object from rest at the spring’s original rest length. (a) Show that the spring exerts an upward force of on the object at its lowest point. (b) If the spring has a force constant of and a 0.25-kg-mass object is set in motion as described, find the amplitude of the oscillations. (c) Find the maximum velocity.

A system is in unstable equilibrium if, when displaced, it experiences a net force or torque in the same direction as the displacement from equilibrium

Two children of mass 20.0 kg and 30.0 kg sit balanced on a seesaw with the pivot point located at the center of the seesaw. If the children are separated by a distance of 3.00 m, at what distance from the pivot point is the small child sitting in order to maintain the balance?

air resistance is negligible

Simple Machines

Models for an Expanding Universe

Variation of Hubble’s Constant

Hubble’s Law

f we then put this speed and the Hubble constant into Hubble’s law equation, we can solve for the distance.

Basically, if we can obtain a spectrum of a galaxy, we can immediately tell how far away it is.

just as Lemaître had suggested.

When Hubble laid his own distance estimates next to measurements of the recession velocities (the speed with which the galaxies were moving away), he found something stunning: there was a relationship between distance and velocity for galaxies. The more distant the galaxy, the faster it was receding from us.

Humason was collaborating with Hubble by photographing the spectra of faint galaxies

No one at the time quite knew what to make of this discovery.

the spectral lines of most galaxies showed an astounding redshift.

spectra of galaxies contained a wealth of information about the composition of the galaxy and the velocities of these great star systems.

Other Measuring Techniques

(credit: NASA, ESA, A. Riess (STScI))

Observations show that supernovae of this type all reach nearly the same luminosity (about 4.5 × 109LSun) at maximum light.

make out an individual cepheid variable star in the galaxy M100 and measure its distance to be 56 million light-years.

they could measure distances using certain kinds of intrinsically luminous variable stars, such as cepheids

we know the precise way light is dimmed by distance

If we know the distance to a galaxy, we can convert how bright the galaxy appears to us in the sky into its true luminosity

The Formation of the Galaxy

Stellar Populations

First, an asteroid might have ventured too close to the black hole and been heated to a very high temperature

One of the stars has been observed for its full orbit of 15.6 years.

These stars have now been observed for almost two decades,

There are also thousands of stars within a parsec of Sagittarius A*.

temperature of 10 million K.

Very Large Array (VLA) of radio telescopes in Socorro, New Mexico.

supernova remnants

supermassive black holes by astronomers, to indicate that the mass they contain is far greater than that of the typical black hole created by the death of a single star.

less than the diameter of Mercury’s orbit.

the mass closer to us can bend the light from farther away. With just the right alignment, the image of the more distant object also becomes significantly brighter.

these should produce dark features in the ultraviolet spectra of objects lying beyond the Galaxy,

least 2 × 1012MSun, which is about twenty times greater than the amount of luminous matter.

forced to the reluctant conclusion that this matter is invisible

Rotation Curve of the Galaxy

indicating a great deal of mass beyond the Sun’s orbit.

heir orbital velocities are even greater than the Sun’s

should

objects orbiting at large distances from a massive object will move more slowly than objects that are closer to that central mass.

In science, what seems to be a reasonable assumption can later turn out to be wrong

100 billion times the mass of the Sun

concentrated at a point in the center of the sphere

the Galaxy is roughly spherical

Kepler Helps Weigh the Galaxy

rotation curve

The concentration of matter in the arms exerts sufficient gravitational force to keep the arms together over long periods of time

Scientists have used supercomputer calculations to model the formation and evolution of the arms.

might

21-cm line that comes from cool hydrogen

Once the light pollution subsides to negligible levels, the Milky Way can be readily spotted arching over the sky on clear, moonless nights.

our Glaxy is not unique in its characteristics. There are many other flat, spiral-shaped islands of stars, gas, and dust in the universe.

Heavier-element abundance

We know that this extensive dark matter halo exists because of its effects on the orbits of distant star clusters and other dwarf galaxies that are associated with the Galaxy.

shaped rather like a peanut

Mount Wilson Observatory,

In 1785

band standing in formation during halftime at a football game

William Herschel was a German musician who emigrated to England and took up astronomy in his spare time.

You could do a much better job from a helicopter flying over the city than you could if you were standing in Times Square.

The variable v is counted as positive if the velocity is one of recession, and negative if it is one of approach. Solving this equation for the velocity, we find

fossil life on Mars

no life was detected on Mars.

Climate Change on Mars

The rovers (Spirit, Opportunity, and Curiosity)

Some local source of heating must have released this water

The channel is about 2.5 kilometers across.

They are about the same to the nearest power of 10.

polar cap area

together with a great deal of water ice.

Several types of clouds can form in the martian atmosphere.

At a pressure of less than 0.006 bar, the boiling point is as low or lower than the freezing point, and water changes directly from solid to vapor without an intermediate liquid state (as does “dry ice,” carbon dioxide, on Earth).

0.007 bar, less than 1% that of Earth.

We know that the one requirement shared by all life on Earth is liquid water.

Mars seems to be the most promising place to look for life

The most remarkable thing about these organic molecules is that they include equal numbers with right-handed and left-handed molecular symmetry.

complex organic molecules in them—chemicals based on carbon, which on Earth are the chemical building blocks of life.

with the different materials sorted according to density.

This value (which we round off to 4.5 billion years in this book) is taken to represent the age of the solar system

Some Striking Meteorites

do not appear to come from the comets

Such a fall occurs when a single larger object breaks up during its violent passage through the atmosphere

It is the pattern of lines

Check Your Learning

656.6 nm

difference in wavelength

For particular absorption or emission lines

original (unshifted) wavelength

The wavelengths of the absorption lines can be measured accurately, however, and their Doppler shift is relatively simple to detect.

Solving this equation for the velocity

the wavelength emitted by the source

describe changes in the wavelengths of radio waves or X-rays

Color Shifts

Volcanoes

This is the way many, but not all, of the mountain ranges on Earth were formed.

astronomy exams

It is a cooling system for the planet.

Heat escaping from the interior provides energy for the formation of our planet’s mountains, valleys, volcanoes, and even the continents and ocean basins themselves.

white

To find primitive rock, we must look to smaller objects such as comets, asteroids, and small planetary moons

Metamorphic rocks

common sandstones, shales, and limestones

Jupiter’s moon Io

rework the surface of our planet constantly