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

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.

21-cm line that comes from cool hydrogen

Note that the age of these surfaces is not necessarily the age of the planet as a whole. On geologically active objects (including Earth), vast outpourings of molten rock or the erosive effects of water and ice, which we call planet weathering, have erased evidence of earlier epochs

metallic iron-nickel core

Brahe didn’t have the ability to analyze them

Otherwise the pre-telescopic observations of Brahe would not have been sufficient for Kepler to deduce that its orbit had the shape of an ellipse rather than a circle.

original (unshifted) wavelength

It is the pattern of lines

clear that it was frozen water.

It discovered numerous sedimentary rocks

the type Ia supernova

in the direction of Sagittarius.

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.

philosophical tradition, going back to the Greeks and defended by the Catholic Church, held that pure human thought combined with divine revelation represented the path to truth

in those days, before the telescope, no one imagined testing these predictions.

Variation of Hubble’s Constant

rocks from the sky

Analyzing vectors using perpendicular components is very useful in many areas of physics,

The person taking the walk ends up at the tip of

The angles that vectors and make

Figure 6. To add vectors A and B

Calculate the momentum of a 2000-kg elephant charging a hunter at a speed of (b) Compare the elephant’s momentum with the momentum of a 0.0400-kg tranquilizer dart fired at a speed of (c) What is the momentum of the 90.0-kg hunter running at after missing the elephant?

Linear momentum is defined as

most broadly applicable form

State Newton’s second law of motion in terms of momentum.

What is the mass of a large ship that has a momentum of when the ship is moving at a speed of (b) Compare the ship’s momentum to the momentum of a 1100-kg artillery shell fired at a speed of 3

This quantity was the average force

A runaway train car that has a mass of 15,000 kg travels at a speed of down a track. Compute the time required for a force of 1500 N to bring the car to rest

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

Astrology, that unlikely and vague pseudoscience, makes much of the position of the planets at the moment of one’s birth. The only known force a planet exerts on Earth is gravitational. (a) Calculate the magnitude of the gravitational force exerted on a 4.20 kg baby by a 100 kg father 0.200 m away at birth (he is assisting, so he is close to the child). (b) Calculate the magnitude of the force on the baby due to Jupiter if it is at its closest distance to Earth, some away. How does the force of Jupiter on the baby compare to the force of the father on the baby? Other objects in the room and the hospital building also exert similar gravitational forces. (Of course, there could be an unknown force acting, but scientists first need to be convinced that there is even an effect, much less that an unknown force causes it.)

aching feet

Figure 2.

The Cavendish Experiment: Then and Now

gravity is able to supply the necessary centripetal force

Tides

The gravitational force is relatively simple. It is always attractive, and it depends only on the masses involved and the distance between them. Stated in modern language, Newton’s universal law of gravitation states that every particle in the universe attracts every other particle with a force along a line joining them. The force is directly proportional to the product of their masses and inversely proportional to the square of the distance between them.

”Weightlessness” and Microgravity

2: (a) Calculate the magnitude of the acceleration due to gravity on the surface of Earth due to the Moon. (b) Calculate the magnitude of the acceleration due to gravity at Earth due to the Sun. (c) Take the ratio of the Moon’s acceleration to the Sun’s and comment on why the tides are predominantly due to the Moon in spite of this number.

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

torque is analogous to force, angular acceleration is analogous to translational acceleration, and is analogous to mass (or inertia). The quantity is called the rotational inertia or moment of inertia of a point mass a distance from the center of rotation.

the sum of for all the point masses of which it is composed. That is,

Why is the moment of inertia of a hoop that has a mass and a radius greater than the moment of inertia of a disk that has the same mass and radius?

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.

radians (rad), defined

The arc length is the distance traveled along a circular path as shown in Figure 2 Note that is the radius of curvature of the circular path.

We define the rotation angle to be the ratio of the arc length to the radius of curvature:

We can loosely define energy as the ability to do work

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

The perpendicular lever arm is the shortest distance from the pivot point to the line along which acts;

Two children push on opposite sides of a door during play.

perpendicular lever arm

the second condition necessary to achieve equilibrium is that the net external torque on a system must be zero.

newtons times meters

Torque is the turning or twisting effectiveness of a 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?

Conservation of Energy

To the mouse and any smaller animal, [gravity] presents practically no dangers. You can drop a mouse down a thousand-yard mine shaft; and, on arriving at the bottom, it gets a slight shock and walks away, provided that the ground is fairly soft. A rat is killed, a man is broken, and a horse splashes. For the resistance presented to movement by the air is proportional to the surface of the moving object. Divide an animal’s length, breadth, and height each by ten; its weight is reduced to a thousandth, but its surface only to a hundredth. So the resistance to falling in the case of the small animal is relatively ten times greater than the driving force.

more generalized fashion as

This means a skydiver with a mass of 75 kg achieves a maximum terminal velocity of about 350 km/h while traveling in a pike (head first) position, minimizing the area and his drag. In a spread-eagle position, that terminal velocity may decrease to about 200 km/h as the area increases. This terminal velocity becomes much smaller after the parachute opens.

For most large objects such as bicyclists, cars, and baseballs not moving too slowly, the magnitude of the drag force is found to be proportional to the square of the speed of the object.

as the person’s velocity increases, the magnitude of the drag force increases until the magnitude of the drag force is equal to the gravitational force, thus producing a net force of zero.

TAKE-HOME EXPERIMENT

Problems & Exercises

that is, a push or a pull

internal forces within the body

of all external forces

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

Any force or combination of forces can cause a centripetal or radial acceleration. Just a few examples are the tension in the rope on a tether ball, the force of Earth’s gravity on the Moon, friction between roller skates and a rink floor, a banked roadway’s force on a car, and forces on the tube of a spinning centrifuge

As a skater forms a circle, what force is responsible for making her turn? Use a free body diagram in your answer

frictionless banked curve

Centripetal force is always perpendicular to the path and pointing to the center of curvature,

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

Linear or tangential acceleration refers to changes in the magnitude of velocity but not its direction. We know from Chapter 6 Uniform Circular Motion and Gravitation that in circular motion centripetal acceleration, refers to changes in the direction of the velocity but not its magnitude.

At its peak, a tornado is 60.0 m in diameter and carries 500 km/h winds. What is its angular velocity in revolutions per second?

These equations mean that linear acceleration and angular acceleration are directly proportional.

Centripetal acceleration ac occurs as the direction of velocity changes; it is perpendicular to the circular motion. Centripetal and tangential acceleration are thus perpendicular to each other.

Explain why centripetal acceleration changes the direction of velocity in circular motion but not its magnitude. 3: In circular motion, a tangential acceleration can change the magnitude of the velocity but not its direction. Explain your answer.

Analogies exist between rotational and translational physical quantities. Identify the rotational term analogous to each of the following: acceleration, force, mass, work, translational kinetic energy, linear momentum, impulse.

superposition and interference

four types of waves in this picture

beating of hearts

Inelastic Collisions in One Dimension

solve

dissipative forces

The simplest oscillations occur when the restoring force is directly proportional to displacement.

The force constant is related to the rigidity (or stiffness)

elastic potential energy

potential energy stored in a spring

restoring force.

At a staggering 2.5 million light years from the Earth, this galaxy is the nearest one to our own galaxy

A 100-g toy car is propelled

Suppose a 350-g kookaburra

An object’s gravitational potential is due to its position relative to the surroundings within the Earth-object system.

What is the final speed of the roller coaster

this point is arbitrary;

we refer to this as the

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

boldface symbol, such as , stands for a vector. Its magnitude is represented by the symbol in italics, and its direction by

this involves determining the perpendicular components of a single vector, for example the x– and y-components,

Resolving a Vector into Components

Applications

The airplane wing is a beautiful example of Bernoulli’s principle in action.

pressure has units of energy per unit volume, too.

The net work done increases the fluid’s kinetic energy. As a result, the pressure will drop in a rapidly-moving fluid, whether or not the fluid is confined to a tube.

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

Compare the kinetic energy of a 20,000-kg truck moving at 110 km/h with that of an 80.0-kg astronaut in orbit moving at 27,500 km/h.

A car’s bumper is designed to withstand a 4.0-km/h (1.1-m/s) collision with an immovable object without damage to the body of the car. The bumper cushions the shock by absorbing the force over a distance. Calculate the magnitude of the average force on a bumper that collapses 0.200 m while bringing a 900-kg car to rest from an initial speed of 1.1 m/s.

lawn mower.

How fast must a 3000-kg elephant move to have the same kinetic energy as a 65.0-kg sprinter running at 10.0 m/s?

can exert pulls only parallel to its length

Once you have determined the tension in one location, you have determined the tension at all locations along the rope.

resolve the tension vectors into their horizontal and vertical components.

Calculate the tension in the wire

the acceleration parallel to the incline when there is 45.0 N of opposing friction.

Extended Topic:

forces parallel to the slope

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.

Solving for yields

they are actually valid for all bodies satisfying the two previously stated conditions.

2: Calculate the mass of the Sun based on data for Earth’s orbit and compare the value obtained with the Sun’s actual mass.

Examples of gravitational orbits abound.

find the masses of heavenly bodies

Tidal friction exerts torque that is slowing Earth’s rotation

Three children are riding on the edge of a merry-go-round that is 100 kg, has a 1.60-m radius, and is spinning at 20.0 rpm. The children have masses of 22.0, 28.0, and 33.0 kg. If the child who has a mass of 28.0 kg moves to the center of the merry-go-round, what is the new angular velocity in rpm?

The equation is very fundamental and broadly applicable.

It seems quite reasonable, then, to define angular momentum as

Conceptual Questions

Calculate the angular momentum of the Earth in its orbit around the Sun.

An ice skater is spinning

Compare this angular momentum with the angular momentum of Earth on its axis.

These expressions are the law of conservation of angular momentum. Conservation laws are as scarce as they are important.

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

The two-dimensional curved path of the horizontally thrown ball is composed of two independent one-dimensional motions (horizontal and vertical).

The two-dimensional curved path of the horizontally thrown ball is composed of two independent one-dimensional motions (horizontal and vertical).

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

Problems & Exercises

Power

A bullet is accelerated down the barrel of a gun by hot gases produced in the combustion of gun powder. What is the average force exerted on a 0.0300-kg bullet to accelerate it to a speed of 600 m/s in a time of 2.00 ms (milliseconds)?

The quantity is given the name impulse.

Example 1:

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

average effective force

chickens

A system is said to be in stable equilibrium if, when displaced from equilibrium, it experiences a net force or torque in a direction opposite to the direction of the displacement.

some systems in stable equilibrium are more stable than others.

unless some effort is made to keep it moving.

Extrapolating to a frictionless surface

graphically, using the head-to-tail method, or analytically, using components. The techniques are the same as for the addition of other vectors, and are covered in Chapter 3 Two-Dimensional Kinematics.)

we assume the vertical forces cancel

For completeness, the vertical forces are also shown; they are assumed to cancel since there is no acceleration in the vertical direction. The vertical forces are the weight and the support of the ground

It can help us make predictions.

The energy moves forward through the water, but the water stays in one place.

The waves on the strings of musical instruments are transverse

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

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.

Laminar flow is characterized by the smooth flow of the fluid in layers that do not mix. Turbulent flow, or turbulence, is characterized by eddies and swirls that mix layers of fluid together.

objects are larger than the size of most molecules

much, much less than the speed of light

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

Forces and Torques in Muscles and Joints

The displacement as a function of time t in any simple harmonic motion

PERIOD OF SIMPLE HARMONIC OSCILLATOR

5.3 Elasticity: Stress and Strain

Momentum, like energy, is important because it is conserved.

This equation means that the total kinetic and potential energy is constant for any process involving only conservative forces.

Potential Energy of a Spring

Nonconservative Forces

Work, Energy, and Power in Humans

work done on a system by a constant force is defined to be the product of the component of the force in the direction of motion times the distance through which the force acts.

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

Rotational Kinetic Energy

10.4

6.4 Fictitious Forces and Non-inertial Frames: The Coriolis Force

World Energy Use

First, the equation for conservation of momentum for two objects in a one-dimensional collision is

Kinematics of Rotational Motion

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

In physics, the definition of time is simple

Collisions of Point Masses in Two Dimensions

Introduction to Rocket Propulsion

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.)