By the end of this section

When a spinning figure skater

on a well-oiled swivel stool by starting yourself spinning slowly with your arms extended and then pulling your arms in.

Kepler’s second law is a consequence of the conservation of angular momentum.

Mass, Volume, and Density

This is the principle behind jet engines and rockets

This means that forces in nature do not occur alone

mass times its velocity

For every action there is an equal and opposite reaction

In the original Latin, the three laws contain only 59 words

Philosophiae Naturalis Principia Mathematica.

his friend Edmund Halley

born in Lincolnshire, England, in the year after Galileo’s death

beyond the orbit of Pluto.

Kepler’s third law

Note that the eccentricities of the planets’ orbits in our solar system are substantially less than shown here.

The universe could be a bit more complex than the Greek philosophers had wanted it to be.

Kepler found that Mars has an elliptical orbit, with the Sun at one focus (the other focus is empty). The eccentricity of the orbit of Mars is only about 0.1; its orbit, drawn to scale, would be practically indistinguishable from a circle, but the difference turned out to be critical for understanding planetary motions.

If the foci (or tacks) are moved to the same location, then the distance between the foci would be zero. This means that the eccentricity is zero and the ellipse is just a circle; thus, a circle can be called an ellipse of zero eccentricity. In a circle, the semimajor axis would be the radius.

The ratio of the distance between the foci to the length of the major axis is called the eccentricity of the ellipse.

We wrap the ends of a loop of string

sum of the distance from two special points inside the ellipse to any point on the ellipse is always the same

observations

behavior of planets based on their paths through space.

occupied most of Kepler’s time for more than 20 years

Tubingen

Brahe didn’t have the ability to analyze them

exploding star

Three years after the publication of Copernicus’ De Revolutionibus, Tycho Brahe was born

The square of a planet’s orbital period is directly proportional to the cube of the semimajor axis of its orbit.

This animation (http://tiny.cc/88cyqy) shows the phases of Venus. You can also see its distance from Earth as it orbits the Sun. The Astronomy Picture of the Day has an animation of Venus  https://apod.nasa.gov/apod/ap060110.html (http://tiny.cc/vadyqy) and a  set of images of Venus as viewed from Earth.  https://apod.nasa.gov/apod/ap170317.html (http://tiny.cc/ebdyqy)

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

predictions

still taught at Harvard University

simple experiment dropping two balls of different weights

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

clumsy and lacking the beauty and symmetry of its successor.

the nearer a planet is to the Sun, the greater its orbital speed.

We have all experienced seeing an adjacent train, bus, or ship appear to move, only to discover that it is we who are moving.

heavenly bodies must be made up of combinations of uniform circular motions.

His great contribution to science was a critical reappraisal of the existing theories of planetary motion and the development of a new Sun-centered, or heliocentric, model of the solar system.

We can now take a brief introductory tour

In this simplified model of a hydrogen atom

We can use Kepler’s law (see Orbits and Gravity) and our knowledge of the visible star to measure the mass of the invisible member of the pair.

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

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.

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

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

pressure has units of energy per unit volume, too.

Bernoulli’s equation is a form of the conservation of energy principle.

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.

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

Sound in solids can be both longitudinal and transverse.

In equation form

and energy

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

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

PERIOD OF SIMPLE HARMONIC OSCILLATOR

most fishermen feel no obligation to truthfully report the mass

potential energy stored in a spring

elastic potential energy

solve

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.

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

dissipative forces

superposition and interference

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

beating of hearts

guitar, atoms

four types of waves in this picture

10.4

Rotational Kinetic Energy

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?

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

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

Conceptual Questions

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

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

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

The equation is very fundamental and broadly applicable.

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.

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.

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?

MAKING CONNECTIONS

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.

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

Some rotational inertias.

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

Kinematics of Rotational Motion

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

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?

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.

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.

Forces and Torques in Muscles and Joints

Simple Machines

Applications

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?

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.

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

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.”

newtons times meters

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

perpendicular lever arm

Introduction to Rocket Propulsion

It can be shown that

Collisions of Point Masses in Two Dimensions

Inelastic Collisions in One Dimension

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)?

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

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

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?

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?

Example 2: Calculating Force: Venus Williams’ Racquet

World Energy Use

Work, Energy, and Power in Humans

Power

Conservation of Energy

Nonconservative Forces

Potential Energy of a Spring

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

A 100-g toy car is propelled

Suppose a 350-g kookaburra

9-megaton fusion bomb

What is the final speed of the roller coaster

this point is arbitrary;

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

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

we refer to this as the

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.

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?

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.

lawn mower.

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

Example 4: Work and Energy Can Reveal Distance, Too

rotational kinetic energy

This expression

Figure 1. (a) A graph

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

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.

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

a small set of rules and a single underlying force

find the masses of heavenly bodies

Solving for yields

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.)

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.

3: Draw a free body diagram for a satellite

The Cavendish Experiment: Then and Now

”Weightlessness” and Microgravity

Tides

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

Figure 2.

Example 1:

equivalent effective force

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

Tidal friction exerts torque that is slowing Earth’s rotation

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

We can loosely define energy as the ability to do work

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

almost

This quantity was the average force

most broadly applicable form

The net external force equals the change in momentum

Linear momentum is defined as

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

an equation for

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.

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

both

gravity is able to supply the necessary centripetal force

aching feet

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

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?

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?

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

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.

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

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

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

radians (rad), defined

5.3 Elasticity: Stress and Strain

Problems & Exercises

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.

TAKE-HOME EXPERIMENT

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.

“Aerodynamic” shaping

more generalized fashion as

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.

Problems & Exercises

Extended Topic:

Calculate the tension in the wire

can exert pulls only parallel to its length