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Physics Ch. 4 and 5


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static stability
An object's stability at rest. To have static stability, an object needs a stable equilibrium.
stable equilibrium
an object experiences restoring influences—forces and/or torques—which push it back toward that equilibrium
Stable Equilibrium and Potential Energy
An object is in a stable equilibrium when any small shift increases its total potential energy.
base of support
The polygon defined by its contact points with the ground. If the object's center of gravity is positioned over its base of support, then tipping it raises its center of gravity and increases its gravitational potential energy. The tipped object will accelerate back toward that equilibrium and the equilibrium will be stable.
Unstable Equilibrium and Potential Energy
An object's equilibrium is unstable when a small shift can decrease its total potential energy. Instead of experiencing restoring forces or torques, the disturbed bicycle will tip farther and faster until it hits the ground.
gyroscopic precession
The pivoting of a gyroscope's rotational axis caused by a torque exerted perpendicular to its angular momentum.
The potential energy effect of bicycles depends on:
Fork shape and angle. To be stable, the front wheel must touch the ground behind the steering axis. The front fork of a typical adult's bicycle arcs forward so that the wheel touches the ground just behind the steering axis.
The bicyclist tries to keep himself and the bicycle in a:
rotational equilibrium, that is, experiencing zero net torque about their combined center of mass. A net torque would cause them to undergo angular acceleration and flip. (The pavement's upward support force on the wheels then produces a torque on them about their combined center of mass and that new torque opposes the frictional torque. When those two torques sum to zero, the bicyclist and bicycle are safely in rotational equilibrium.)
For example, when the bicycle is heading straight, the rider can avoid a net torque by:
keeping the bicycle upright. The pavement is then pushing the wheels straight upward, directly toward the combined center of mass. But when the bicycle is turning left, each wheel of the turning bicycle is experiencing not only an upward support force, but also the leftward frictional force it needs to accelerate through the turn.
Each bicycle crank has a pedal installed:
on its free end so that you can use your foot to push on it. With its axle suspended in bearings, the front wheel turns as you pedal it.
An early solution to the frantic pedaling problem was to use a gigantic front wheel...
But pedaling still interfered with steering and you couldn't stop pedaling while the bicycle was moving. Furthermore, this bicycle had a new problem: you couldn't push hard enough with your feet to keep its front wheel turning steadily on uphill stretches.
Solving early bicycle problems with the pennyfarthing:
These problems were solved by removing the cranks from the front wheel's axle and using an indirect drive scheme to convey power to the rear wheel. Employing toothed sprockets and a chain loop, that indirect drive allows the pedals and the wheels to turn at different rates. This change lets you to use mechanical advantage to choose how you supply power to the bicycle: whether you exert large forces on slowly moving pedals or small forces on rapidly moving pedals or, ideally, medium forces on medium-fast moving pedals. Whether you're zooming along a level road or grinding slowly up a steep hill, you can always find a drive setting or “gear” that lets you comfortably supply your maximum power. Lastly, the nonstop pedaling problem is solved by incorporating a one-way drive or freewheel in the hub of the rear wheel. It allows the rear wheel to turn freely in one direction so that you can stop pedaling as you coast forward.
A rocket's ability to propel itself forward:
It obtains a forward force, a thrust force, by pushing against its own limited store of fuel, and when that fuel runs out, it stops accelerating; Newton's third law.
The rocket uses a ___ reaction to create ____. Potential energy in ___ becomes:
It uses a chemical reaction to create very hot exhaust gas from fuels contained entirely within the rocket itself. What started as potential energy in the stored chemical fuels becomes thermal energy in the hot, burned exhaust gas. This thermal energy is mostly kinetic energy, hidden in the random motion of the tiny molecules themselves. (The rocket engine's nozzle steers most of this random motion in one direction, and the engine obtains thrust in the opposite direction.)
Rocket nozzle
allows the rocket to obtain as much forward momentum as possible from its exhaust by directing that exhaust backward and accelerating it to the greatest possible speed. (Wider nozzles for thinner air/space)
With the help of the de Laval nozzle, exhaust gas leaves the rocket's engine at an:
exhaust velocity or backward-directed flow
Space shuttle's acceleration:
That means that the space shuttle can accelerate upward at about half the acceleration due to gravity! As the shuttle consumes its fuel, so that its weight and mass diminish, it can accelerate upward even more rapidly.
Common Misconceptions: Action and Reaction in Rockets
Resolution: While rocket propulsion does involve a pair of equal but opposite forces, action and reaction, the rocket is pushing its exhaust backward (action) and the exhaust is pushing the rocket forward (reaction). What this exhaust plume hits, if anything, makes no difference to the propulsion effect.
Space shuttle's speed:
(However, for the spacecraft to reach extremely high speed, the rocket must push the vast majority of its initial mass backward as exhaust.)
While the total momentum of the spacecraft and fuel still always sum to zero, this inopportune momentum transfer from the exhaust to the fuel reduces both the spacecraft's speed forward and the exhaust's average speed backward. Despite this problem, a spacecraft can still travel faster than the speed of its rocket exhaust; it just needs more fuel.
An orbit is:
the path an object takes as it falls freely around a celestial object. Although the spacecraft accelerates directly toward the earth's center at every moment, its huge horizontal speed prevents it from actually hitting the earth's surface. In effect, the spacecraft perpetually misses the earth as it falls.
The farther the spacecraft's orbit is from the earth's surface, the longer its orbital period:
the time it takes to complete one orbit. First, since the spacecraft must travel farther to complete the larger orbit, the trip takes longer. Second, the spacecraft must travel slower in order to follow a circular path around the earth because the earth's gravity becomes weaker with distance.
A more general formula relates the gravitational forces between two objects to their masses and the distance separating them.
These forces are equal to the gravitational constant times the product of the two masses, divided by the square of the distance separating them. This relationship, discovered by Newton and called the law of universal gravitation.
If the spacecraft has more than enough kinetic energy to reach this maximum gravitational potential energy:
It will be able to escape completely from the earth's gravity. The speed that a spacecraft needs in order to escape from the earth's gravity is called the escape velocity.
Misconceptions: Astronauts and “Weightlessness”
Resolution: The astronaut is still so near the earth's surface that she experiences almost her full earth weight. She only feels weightless because she is in free fall.
Kepler's first law
This law describes the shape of the spacecraft's looping orbit around the sun: it's an ellipse, with the sun at one focus of that ellipse.
Kepler's second law describes the area swept out by a line stretching from the sun to the spacecraft:
that line sweeps out equal areas in equal times. This observation demonstrates another physical law: conservation of angular momentum. Since the sun's gravity pulls the spacecraft directly toward the sun, it exerts no torque on the spacecraft about the sun and the spacecraft's angular momentum about the sun is constant. Remarkably, the rate at which this line sweeps out area is proportional to the spacecraft's angular momentum, so the steadiness of that sweep demonstrates the constancy of the spacecraft's angular momentum.
Kepler's third law describes the spacecraft's orbital period around the sun:
The square of its orbital period is proportional to the cube of its mean distance from the sun, that is, the average of its perihelion (closest distance to the sun) and aphelion (farthest distance to the sun).
Although extremely accurate at ordinary speeds, those laws falter near the speed of light.
low speeds, our isolated spacecraft's kinetic energy takes its usual Newtonian value: half its mass times the square of its speed (Eq. 2.2.1). But at high speeds, we must begin using relativistic energy. Relativistic energy is the product of the object's mass times the square of the speed of light times the relativistic factor.
The relativistic version of energy has two implications for our spacecraft.
First, the spacecraft's energy increases so quickly as it nears the speed of light that it can never reach that speed. Second, the spacecraft's initial store of energy before launch is associated with its initial mass.
General theory of relativity:
The physical rules governing all motion, even motion involving speeds comparable to the speed of light and occurring in the presence of massive objects.
When you're experiencing a weight, your: ______.
When you're experiencing a feeling of acceleration, your : ________.
Gravitational mass is acting together with gravity to make you feel heavy. When you're experiencing a feeling of acceleration, your inertial mass is acting together with acceleration to make you feel heavy. But, in spite of their different roles, these two masses seem to be related.
principle of equivalence
that these two masses, gravitational and inertial, are truly identical and therefore that no experiment you perform inside your spacecraft can distinguish between free fall and the absence of gravity. The general theory of relativity is based on this principle of equivalence.
As long as your spacecraft stays in regions of weak gravity, Newtonian's law of universal gravitation will adequately describe its motion. But at the extremes of gravity, the general theory of relativity is necessary.
theory describes a universe in which massive objects distort the structure of nearby space and time, and in which extreme masses produce extreme distortions. One of the most startling predictions of this theory is the existence of objects so radical in their gravitational warping of nearby space and time that they are black holes—spherical or nearly spherical surfaces from which not even light can escape.
NOTES: Observations about Rockets
􀂄􀂄 Plumes of flame emerge from rockets
􀂄􀂄 Rockets can accelerate straight up
􀂄􀂄 Rockets can go very fast
􀂄􀂄 The flame only touches the ground initially
􀂄􀂄 Rockets can apparently operate in empty space
􀂄􀂄 Rockets usually fly nose-first
NOTES: Momentum Conservation
􀂄􀂄 A rocket’s momentum is initially zero
􀂄􀂄 That momentum is redistributed during thrust
􀂄􀂄 Ship pushes on fuel; fuel pushes on ship
􀂄􀂄 Fuel acquires backward momentum
􀂄􀂄 Ship acquires forward momentum
􀂄􀂄 Rocket’s total momentum remains zero
NOTES: Rocket Propulsion
􀂄􀂄 Neglecting gravity, then
􀂄􀂄 rocket’s total momentum is always zero
momentumfuel+ momentumship = 0
􀂄􀂄 The momenta of ship and fuel are opposite
􀂄􀂄 The ship’s momentum is equal but opposite to
􀂄􀂄 the velocity of the fuel
􀂄􀂄 times the mass of that fuel
NOTES: Rocket Engines
􀂄 Combustion produces hot, high-pressure gas
􀂄􀂄 The gas speeds up in a de Laval nozzle
􀂄􀂄 Gas reaches sonic speed
in the nozzle’s throat
􀂄􀂄 Beyond the throat, supersonic
gas expands to speed up further
NOTES: Stability and Orientation
􀂄 On the ground, a rocket needs static stability
􀂄􀂄 In the air, a rocket needs aerodynamic stability
􀂄􀂄 Center of aerodynamic forces behind center of mass
􀂄􀂄 In space, a spaceship is a freely rotating object
􀂄􀂄 Orientation governed by angular momentum
􀂄􀂄 Small rockets are used to exert torques on spaceship
􀂄􀂄 Spaceship’s orientation doesn’t affect its travel
NOTES: Ship’s Ultimate Speed
􀂄􀂄 Increases as
􀂄􀂄 the ratio of fuel mass to ship mass increases
􀂄􀂄 the fuel exhaust speed increases
􀂄􀂄 If fuel were released with the rocket at rest,
􀂄􀂄 Because rocket accelerates during thrust,
ultimate speed is less than given above
NOTES: Gravity (Part 1)
􀂄 The earth’s acceleration due to gravity is only
constant for small changes in height.
􀂄􀂄 The ship’s weight is only constant for small
changes in height
NOTES: Gravity (Part 3)
􀂄􀂄 Even far above earth, an object has weight
􀂄􀂄 Astronauts and spaceships have weights
􀂄􀂄 weights are somewhat less than normal
􀂄􀂄 weights depend on altitude
􀂄􀂄 Astronauts and spaceships are in free fall
􀂄􀂄 Astronauts feel weightless because they are falling
NOTES: Orbits (Part 1)
􀂄 An object that begins to fall from rest falls
directly toward the earth
􀂄􀂄 Acceleration and velocity
are in the same direction
NOTES: Orbits (Part 2)
􀂄 An object that has a sideways velocity follows a
trajectory called an orbit
􀂄􀂄 Orbits can be closed
or open, and are
ellipses, parabolas,
and hyperbolas.
􀂄􀂄 Improbable Dreams
􀂄􀂄 Rockets that rarely require refueling
􀂄􀂄 Rockets that can land and leave large planets
􀂄􀂄 Rockets that can turn on a dime in space
NOTES:Summary About Rockets
􀂄􀂄 Rockets are pushed forward by their fuel
􀂄􀂄 Total rocket impulse is basically the product of
exhaust speed times exhaust mass
􀂄􀂄 Rockets can be stabilized aerodynamically
􀂄􀂄 Rockets can be stabilized by thrust alone
􀂄􀂄 After engine burn-out, spaceships can orbit
Air is compressible, that is, you can squeeze a certain mass of it into almost any space. is a gas, a substance consisting of tiny, individual particles that travel around independently. These individual particles are atoms and molecules. A molecule's atoms are held together by chemical bonds, linkages formed by electromagnetic forces between the atoms.
Internal kinetic energy
The portion of an object's kinetic energy that involves only the relative motion of particles within the object and that excludes the object's overall translation or rotation.
- Measured as its temperature; the greater that energy per particle, the hotter the air. While air's thermal energy also includes a portion stored in the forces between particles, this internal potential energy is negligible because the average forces between air particles are so weak.
The random motions of individual particles in a material due to the internal or thermal energy of that material:
thermal motion
The total force depends on the wall's surface area:
larger its surface area, the more average force it experiences. In order to characterize the air, however, we don't really need to know the wall's surface area; instead, we can refer to the average force the air exerts on each unit of surface area, a quantity called pressure.
Pressure is measured in units: ______.
of force-per-area. Since the SI unit of surface area is the meter2 (abbreviated m2 and often referred to as the square meter), the SI unit of pressure is the newton-per-meter2. This unit is also called the pascal (abbreviated Pa).
Since air pressure is produced by bouncing air particles, it depends on:
how often, and how hard, those particles hit a particular region of surface. The more frequent or harder the impacts, the greater is the air pressure.
To increase the rate at which air particles hit a surface, we can pack them more tightly...
We add another 1 kg of air to our hypothetical box, we double the number of air particles in the same volume, which doubles the rate at which they hit each surface and therefore doubles the pressure. Air's pressure is thus proportional to its density, that is, its mass per unit of volume. Since the SI unit of volume is the meter3 (abbreviated m3 and often referred to as the cubic meter), the SI unit of density is the kilogram-per-meter3 (abbreviated kg/m3). The air around you has a density of about 1.25 kg/m3 (0.078 lbm/ft3). Water, in contrast, has a much greater density of about 1000 kg/m3
Air's pressure is thus proportional to:
Its density, that is, its mass per unit of volume. Since the SI unit of volume is the meter3 (abbreviated m3 and often referred to as the cubic meter), the SI unit of density is the kilogram-per-meter3 (abbreviated kg/m3). The air around you has a density of about 1.25 kg/m3 (0.078 lbm/ft3). Water, in contrast, has a much greater density of about 1000 kg/m3
We can also increase the rate at which air particles hit a surface by speeding them up...
If we double the internal kinetic energy of the air in our box, we double the average kinetic energy of each particle. Because a particle's kinetic energy depends on the square of its speed, doubling its kinetic energy increases its speed by a factor of square root of 2. (Air's pressure is thus proportional to the average kinetic energy of its particles—to their average internal kinetic energies.)
This average kinetic energy per particle is measured by the air's temperature;
the hotter the air, the larger the average kinetic energy per particle and the higher the air's pressure. But the most convenient scale for relating the temperature of air to its pressure isn't the common Celsius (°C) or Fahrenheit (°F) scale; instead, it's a special absolute temperature scale. The SI scale of absolute temperature is the Kelvin scale (K). When the air's temperature is 0 K (−273.15 °C or −459.67 °F), it contains no internal kinetic energy at all and has no pressure; this temperature is called absolute zero.
The atmosphere stays on the earth's surface because of:
Gravity. Every air particle, as we've seen, has a weight. Only the lightest and fastest moving particles in the atmosphere, hydrogen molecules and helium atoms, occasionally manage to escape from earth's gravity and drift off into interplanetary space.
While gravity pulls the atmosphere downward, air pressure...
pushes the atmosphere upward. As the air particles try to fall to the earth's surface, their density increases and so does their pressure.
Atmospheric pressure
The pressure of air in the earth's atmosphere. Atmospheric pressure reaches a maximum of about 100,000 Pa near sea level and diminishes with increasing altitude.
As we've seen, the air in the earth's atmosphere is a fluid:
a shapeless substance with mass and weight. This air has a pressure and exerts forces on the surfaces it touches; that pressure is greatest near the ground and decreases with increasing altitude. Air pressure and its variation with altitude allow air to lift a hot-air or helium balloon through an effect known as buoyancy.
Archimedes' Principle
An object partially or wholly immersed in a fluid is acted on by an upward buoyant force equal to the weight of the fluid it displaces.
Without gravity the forces would cancel each other perfectly because the pressure of a stationary fluid would be uniform throughout. But gravity causes:
a stationary fluid's pressure to decrease with altitude. For example, when nothing is moving, the air pressure beneath an object is always greater than the air pressure above it. Thus air pushes upward on the object's bottom more strongly than it pushes downward on the object's top, and the object consequently experiences an upward overall force from the air—a buoyant force.
How large is the buoyant force on this object?
equal in magnitude to the weight of the fluid that the object displaces.
But a portion of fluid suspended in more of the same fluid doesn't accelerate anywhere; it just sits there, so the net force on it is...
clearly zero. It has a downward weight, but that weight must be canceled by some upward force that can only come from the surrounding fluid. This upward force is the buoyant force, and it's always equal in magnitude to the weight of the object-shaped portion of fluid, the fluid displaced by the object.
An object placed in a fluid experiences two forces:
its downward weight and an upward buoyant force. If its weight is more than the buoyant force, it will accelerate downward; if its weight is less than the buoyant force, it will accelerate upward. And if the two forces are equal, it won't accelerate at all and will maintain a constant velocity.
The total volume of an object is less important than:
How its density compares to that of the surrounding fluid.
Since air is very light, with a density of only:
1.25 kg/m3 (0.078 lbm/ft3) few objects float in it. An empty balloon won't last long floating upward. Because it's surrounded by atmospheric pressure air, each square meter of its envelope will experience an inward force of 100,000 N.
Filling our balloon with hot air...
to the overall pressure than does a cold-air particle. A hot-air balloon contains fewer particles, has less mass, and weighs less than it would if it contained cold air. Now we have a practical balloon with an average density less than that of the surrounding air.
Because the air pressure inside a hot-air balloon is the same as the air pressure outside the balloon...
the air has no tendency to move in or out (an issue we will cover in the next section), and the balloon doesn't need to be sealed. . Although the balloon's weight decreases as the air thins out, the buoyant force on it decreases even more rapidly, and it becomes less effective at lifting its cargo.
Particle density
The number of particles in an object divided by its volume. The particle density of water is about 3.35 × 1028 molecules-per-meter3. The particle density of air at sea level is about 2.687 × 1025 molecules-per-meter3.
Since each helium atom weighs:
14% as much as the average air particle, 1 m3 of helium weighs only 14% as much as 1 m3 of air. Thus a helium-filled balloon has only a fraction of the weight of the air it displaces, and the buoyant force carries it upward easily. Lighter but faster-moving helium atoms are just as effective at creating pressure as heavier but slower-moving air particles.
The pressure of a gas is proportional to:
the product of its particle density and its absolute temperature. This proportionality holds regardless of the gas's chemical composition.
Ideal gas law
The pressure of a gas is equal to the product of the Boltzmann constant times the particle density times the absolute temperature
A balloon's lifting capacity is the difference between the upward buoyant force it experiences and its downward weight. Although the gas in a hydrogen balloon weighs half that in a similar helium balloon, the balloons experience the same buoyant force...
Thus the hydrogen balloon's lifting capacity is only slightly more than that of the helium balloon.
Water distribution systems require two things:
plumbing and water pressure. Plumbing is what delivers the water, and water pressure is what starts that water flowing. Water pressure is important because, like everything else, water has mass and accelerates only when pushed. If nothing pushed on the water when you opened a faucet, the water simply wouldn't budge.
As we've seen with the atmosphere, gravity creates pressure gradients in fluids:
distributions of pressure that vary continuously with position. Pressure decreases with altitude and increases with depth, and these vertical pressure gradients complicate plumbing in hilly cities and skyscrapers. With no significant changes in height, we can safely ignore gravity, since no water is supporting the weight of water above it and gravity's effects are minimal.
In this simplified situation, water accelerates only in response to:
unbalanced pressures. Just as unbalanced forces make a solid object accelerate, so unbalanced pressures make a fluid accelerate. If the water inside a pipe is exposed to a uniform pressure throughout, then each portion of the water feels no net force and doesn't accelerate; it either remains stationary or coasts in a straight line at a steady pace.
This acceleration doesn't mean that the water will instantly begin moving toward the lowest pressure. Because of its inertia...
water changes velocity gradually: it speeds up, slows down, or turns to the side, depending on where the lowest pressure is located.
Static variation and
dynamic variations
Changes in a physical quantity such as pressure that are not caused by motion. Changes in a physical quantity such as pressure that are caused by motion.
To start water flowing through the plumbing in a level house or city, you need a water pump:
a device that uses mechanical work to deliver pressurized water through a pipe. At its most basic level, a water pump squeezes a portion of water to raise its local pressure and keeps squeezing as that water accelerates and flows out toward regions of lower pressure elsewhere in the plumbing.
Like all liquids, water is incompressible:
its volume doesn't change as its pressure increases—so the bottle won't get smaller. But the rise in water pressure inside it can be substantial. It doesn't take much force, exerted with your thumb on a small area of the bottle, to increase the pressure in the bottle from atmospheric to twice that value or more.
The pressure increases uniformly throughout the water bottle, an observation known as Pascal's principle:
a change in the pressure of an enclosed incompressible fluid is conveyed undiminished to every part of the fluid and to the surfaces of its container. This uniform pressure rise leads to a large upward force on the bottle's cap. If the cap were wider and had more surface area, the upward force on it might be large enough to blow it off the bottle. That effect is the basis for hydraulic systems and lifts, where pressure produced in an incompressible fluid by a small force exerted on a small area of the fluid's container results in a large force exerted on a large area of the fluid's container.
Pumps do work when:
they deliver pressurized water, and pressurized water carries with it the energy associated with that work.
Water is pumped from a region of low pressure to a region of high pressure by a reciprocating piston pump. (a) As the piston is drawn outward...
water flows into the cylinder from the low-pressure region. (b) As the piston is pushed inward, the inlet one-way valve closes and water is driven out of the cylinder and into the high-pressure region.
The amount of work you do is equal to:
the product of the water pressure times the volume of water you pump. This simple relationship between work, pressure, and volume comes about because the inward force you exert on the water with the piston is equal to the water pressure times the surface area of the piston, and because the distance the water moves in the direction of that force is equal to the volume of water pumped divided by the surface area of the piston. Force times distance equals work.
While the water never really stores any energy, the pump gives each liter of water a certain amount of energy as it leaves the hose so we can imagine...
that this energy is associated with the water and not with the pump. We create a useful fiction: pressure potential energy. Water that's under pressure has a pressure potential energy equal to the product of the water's volume times its pressure.
Pressure potential energy is most meaningful in:
steady-state flow—a situation in which fluid flows continuously and steadily through a stationary environment, without starting or stopping or otherwise changing its characteristics anywhere.
Without gravity, the energy in a certain volume of water in steady-state flow is equal to:
the sum of its pressure potential energy and its kinetic energy. We've already seen that the pressure potential energy is the product of the water's volume times its pressure. The water's kinetic energy is one-half the product of its mass times the square of its speed.
The particular path that a volume of water takes is called:
a streamline, and the energy-per-volume of fluid along a streamline is constant.
Bernoulli's equation
For an incompressible fluid in steady-state flow, the sum of its pressure potential energy, its kinetic energy, and its gravitational potential energy is constant along a streamline. Because energy is conserved, an incompressible fluid such as water that's in steady-state flow can exchange pressure for speed or speed for pressure as it flows along a streamline. The water's total energy is constant.
Gravity creates a pressure gradient in water:
the deeper the water, the more weight there is overhead and the greater the pressure. Since water is much denser than air, water pressure increases rapidly with depth. The shape of the pipe doesn't affect the relationship between pressure and depth. uniform pressure gradient creates an upward buoyant force on anything immersed in the water. In fact, that buoyant force is what supports the water itself.
The dependence of water pressure on depth has a number of important implications for water distribution.
First, water pressure at the bottom of a tall pipe is substantially higher than at the top of that same pipe. Second, pressure in a city water main does more than just accelerate water out of a showerhead; it also supports water in the pipes of multistory buildings. Third, its pressure varies with height.
But if you seal off part of the isolated plumbing and reduce the pressure above one of the water's free surfaces, that surface will:
Rise higher than all of the others. It will rise until the added pressure produced by the taller column of water replaces the missing pressure above the water's free surface. The less pressure there is above that surface, the higher the water must rise to make up for the missing pressure. This effect lifts water in a drinking straw and allows it to travel between two open containers in order to “seek its level” through an elevated pipe known as a siphon. However, removing all of the air pressure above water's free surface inside a long straw or siphon will raise its height only about 10 m (33 feet) above the level of the water elsewhere in an open container. Instead, a pump must be attached to the bottom of the pipe to pressurize the water and push it all the way to the top of the pipe.
Water's gravitational potential energy is equal to:
its weight times its height (the force required to lift it times the distance it has been lifted), and its gravitational potential energy-per-volume is its weight-per-volume times its height. Since its weight-per-volume is its density times the acceleration due to gravity, water's gravitational potential energy-per-volume is its density times the acceleration due to gravity times its height.
This is a revised version of Bernoulli's equation, one that includes gravity. It correctly describes steady-state flow in streamlines that change height.
For an incompressible fluid in steady-state flow, the sum of its pressure potential energy, its kinetic energy, and its gravitational potential energy is constant along a streamline.
Because energy is conserved, an incompressible fluid such as water that's in steady-state flow can:
exchange its speed, pressure, and height for one another. Water is at atmospheric pressure at the top of the water tower, but the pressure is much higher at the bottom; at the base of a 50-m-high water tower, for example, the pressure is about 600,000 Pa or six times atmospheric pressure. a water tower stores energy efficiently and can deliver that energy quickly. When water is drawn out of the water tower, its gravitational potential energy at the top becomes pressure potential energy at the bottom. The water tower replaces a pump, supplying a steady flow of water at an almost constant high pressure.
(A) Compare the dentability of this helium-filled volleyball to one filled with twice-atmospheric-pressure air. (B) What two individual forces does this volleyball experience when you let go of it and why does it accelerate upward as a result?
A) The two volleyballs are equally dentable. (B) The volleyball experiences (1) an upward buoyant force that is stronger than (2) its downward weight, so the net force on the helium-filled volleyball is upward and it accelerates upward.

Why: Dentability depends only on internal pressure and the helium-filled volleyball has the same internal pressure as the air-filled volleyball. But helium at twice-atmospheric pressure has only one-seventh the mass (and weight) of twice-atmospheric pressure air. So the volleyball weighs much less than the air it displaces and floats upward when released.

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