When we first introduced gravitational potential energy in Section 7.1, we assumed that the earth’s gravitational force on a body of mass m doesn’t depend on the body’s height. This led to the expression U = mgy

## Planetary Motions and the Center of Mass

We have assumed that as a planet or comet orbits the sun, the sun remains absolutely stationary. This can’t be correct; because the sun exerts a gravitational force on the planet, the planet exerts a gravitational force on the sun of the same magnitude but opposite direction. In fact, both the su...

The name planet comes from a Greek word meaning “wanderer,” and indeed the planets continuously change their positions in the sky relative to the background of stars. One of the great intellectual accomplishments of the 16th and 17th centuries was the threefold realization that the earth is also ...

## Conditions for equilibrium

In this chapter we’ll apply the first and second conditions for equilibrium to situations in which a rigid body is at rest (no translation or rotation). Such a body is said to be in static equilibrium

## The Continuity equation

The mass of a moving fluid doesn’t change as it flows. This leads to an important relationship called the continuity equation

## Center of gravity

In most equilibrium problems, one of the forces acting on the body is its weight. We need to be able to calculate the torque of this force. The weight doesn’t act at a single point; it is distributed over the entire body. But we can always calculate the torque due to the body’s weight by assum...

## Gravitation

Some of the earliest investigations in physical science started with questions that people asked about the night sky. Why doesn’t the moon fall to earth? Why do the planets move across the sky? Why doesn’t the earth fly off into space rather than remaining in orbit around the sun? The study of gr...

## The Motion of satellites

Artificial satellites orbiting the earth are a familiar part of technology But how do they stay in orbit, and what determines the properties of their orbits? We can use Newton’s laws and the law of gravitation to provide the answers. In the next section we’ll analyze the motion of planets in the...

## Nature of physics

Introduce the systems of units used to describe physical quantities and discuss ways to describe the accuracy of a number.

## Determining the value of G

To determine the value of the gravitational constant G, we have to measure the gravitational force between two bodies of known masses m1 and m2 at a known distance r. The force is extremely small for bodies that are small enough to be brought into the laboratory, but it can be measured with an in...

## Uncertainty and significant figures

Measurements always have uncertainties. If you measure the thickness of the cover of a hardbound version of this book using an ordinary ruler, your measurement is reliable to only the nearest millimeter, and your result will be 3 mm. It would be wrong to state this result as 3.00 mm; given the limit...

## The simple Pendulum

A simple pendulum is an idealized model consisting of a point mass suspended by a massless, unstretchable string. When the point mass is pulled to one side of its straight-down equilibrium position and released, it oscillates about the equilibrium position.

## eLastiCitY and pLastiCitY

Hooke’s law—the proportionality of stress and strain in elastic deformations— has a limited range of validity. In the preceding section we used phrases such as “if the forces are small enough that Hooke’s law is obeyed.” Just what are the limitations of Hooke’s law? What’s more, if you pull, sque...

## Using and Converting Units

An equation must always be dimensional consistent. You can’t add apples and automobiles; two terms may be added or equated only if they have the same units.

## viscosity

Viscosity is internal friction in a fluid. Viscous forces oppose the motion of one portion of a fluid relative to another. Viscosity is the reason it takes effort to paddle a canoe through calm water, but it is also the reason the paddle works. Viscous effects are important in the flow of fluids ...

Any spherically symmetric mass distribution can be thought of as a combination of concentric spherical shells. Because of the principle of superposition of forces, what is true of one shell is also true of the combination. So we have proved half of what we set out to prove: that the gravitational...

## Bulk stress and strain

When a scuba diver plunges deep into the ocean, the water exerts nearly uniform pressure everywhere on his surface and squeezes him to a slightly smaller volume. This is a different situation from the tensile and compressive stresses and strains we have discussed.

## Fluid Mechanics

Fluids play a vital role in many aspects of everyday life. We drink them, breathe them, swim in them. They circulate through our bodies and control our weather. The physics of fluids is therefore crucial to our understanding of both nature and technology

## Estimates and order of magnitudes

We have stressed the importance of knowing the accuracy of numbers that represent physical quantities. But even a very crude estimate of a quantity often gives us useful information. Sometimes we know how to calculate a certain quantity, but we have to guess at the data we need for the calculation. ...

## Why Gravitational forces are important

gravitational forces are negligible between ordinary household-sized objects but very substantial between objects that are the size of stars. Indeed, gravitation is the most important force on the scale of planets, stars, and galaxies

## Vibrations of molecules

The following discussion of the vibrations of molecules uses the binomial theorem. If you aren’t familiar with this theorem, you should read about it in the appropriate section of a math textbook.

## Pressure in a fLuid

A fluid exerts a force perpendicular to any surface in contact with it, such as a container wall or a body immersed in the fluid. This is the force that you feel pressing on your legs when you dangle them in a swimming pool. Even when a fluid as a whole is at rest, the molecules that make up the ...

## EXAMPLES ON GRAVITION

HERE ARE SOME SOLVED EXAMPLES TO CLEAR YOUR CONCEPTS

## Deriving Bernoullis equation

To derive Bernoulli’s equation, we apply the work–energy theorem to the fluid in a section of a flow tube. In Fig. 12.23 we consider the element of fluid that at some initial time lies between the two cross sections a and c. The speeds at the lower and upper ends are v1 and v2. In a small time in...

## A point Mass outside a spherical shell

We start by considering a ring on the surface of a shell , centered on the line from the center of the shell to m. We do this because all of the particles that make up the ring are the same distance s from the point mass m.

## PERIODIC MOTION

Many kinds of motion repeat themselves over and over: the vibration of a quartz crystal in a watch, the swinging pendulum of a grandfather clock, the sound vibrations produced by a clarinet or an organ pipe, and the back-and-forth motion of the pistons in a car engine. This kind of motion, calle...

## surface tension

We’ve seen that if an object is less dense than water, it will float partially submerged. But a paper clip can rest atop a water surface even though its density is several times that of water. This is an example of surface tension:

## Energy in simple Harmonic motion

We can learn even more about simple harmonic motion by using energy considerations. The only horizontal force on the body in SHM in Figs. 14.2 and 14.13 is the conservative force exerted by an ideal spring. The vertical forces do no work, so the total mechanical energy of the system is conserved....

## PASCAL LAW

Pressure applied to an enclosed fluid is transmitted undiminished to every portion of the fluid and the walls of the containing vessel.