CK-12 People's Physics Book Version 2

by James H. Dann

  Bernoulli’s Principle is a restatement of the conservation of energy, but for fluids. The sum of pressure, kinetic energy density, and gravitational potential energy density is conserved. In other words, equals zero. One consequence of this is that a fluid moving at higher speed will exhibit a lower pressure, and vice versa. There are a number of common applications for this: when you turn on your shower, the moving water and air reduce the pressure in the shower stall, and the shower curtain is pulled inward; when a strong wind blows outside your house, the pressure decreases, and your shutters are blown open; due to the flaps on airplane wings, the speed of the air below the wing is lower than above the wing, which means the pressure below the wing is higher, and provides extra lift for the plane during landing. There are many more examples.

  Conservation of flux, , means that a smaller fluid-carrying pipe requires a faster moving fluid. Bernoulli’s Principle, which says that fast-moving fluids have low pressure, provides a useful result: pressure in a smaller pipe must be lower than pressure in a larger pipe.

  If the fluid is not in a steady state, energy can be lost in fluid flow. The loss of energy is related to viscosity, or deviation from smooth flow. Viscosity is related to turbulence, the tendency of fluids to become chaotic in their motion. In a high viscosity fluid, energy is lost from a fluid in a way that is quite analogous to energy loss due to current flow through a resistor. A pump can add energy to a fluid system also. The full Bernoulli Equation takes these two factors, viscosity and pumps, into account.

  Fluids Problem Set

  A block of wood with a density of is floating in a fluid of density . What fraction of the block is submerged, and what fraction is above the surface?

  A rectangular barge long, wide, and in height is floating in a river. When the barge is empty, only is submerged. With its current load, however, the barge sinks so that is submerged. Calculate the mass of the load.

  The density of ice is % that of water. Why does this fact make icebergs so dangerous?

  A form of the liquid naphthalene has a specific gravity of . What fraction of an ice cube would be submerged in a bath of naphthalene?

  A cube of aluminum with a specific gravity of and side length is put into a beaker of methanol, which has a specific gravity of . Draw a free body diagram for the cube.

  Calculate the buoyant force acting on the cube.

  Calculate the acceleration of the cube toward the bottom when it is released.

  A cube of aluminum (specific gravity of ) and side length is put in a beaker of liquid naphthalene (specific gravity of ). When the cube is released, what is its acceleration?

  Your class is building boats out of aluminum foil. One group fashions a boat with a square by bottom and sides high. They begin to put coins in the boat, adding them until it sinks. Assume they put the coins in evenly so the boat doesn’t tip. How many coins can they put in? (You may ignore the mass of the aluminum boat … assume it is zero.)

  You are riding a hot air balloon. The balloon is a sphere of radius and it is filled with hot air. The density of hot air depends on its temperature: assume that the density of the hot air is , compared to the usual for air at room temperature. The balloon and its payload (including you) have a combined mass of . Draw a free body diagram for the cube.

  Is the balloon accelerating upward or downward?

  What is the magnitude of the acceleration?

  Why do hot air ballooners prefer to lift off in the morning?

  What would limit the maximum height attainable by a hot air balloon?

  You are doing an experiment in which you are slowly lowering a tall, empty cup into a beaker of water. The cup is held by a string attached to a spring scale that measures tension. You collect data on tension as a function of depth. The mass of the cup is , and it is long enough that it never fills with water during the experiment. The following table of data is collected:

  String tension Depth Buoyant force

  Complete the chart by calculating the buoyant force acting on the cup at each depth.

  Make a graph of buoyant force vs. depth, find a best-fit line for the data points, and calculate its slope.

  What does this slope physically represent? (That is, what would a greater slope mean?)

  With this slope, and the value for the density of water, calculate the area of the circular cup’s bottom and its radius.

  Design an experiment using this apparatus to measure the density of an unknown fluid.

  A car is being lifted by a hydraulic jack attached to a flat plate. Underneath the plate is a pipe with radius . If there is no net force on the car, calculate the pressure in the pipe.

  The other end of the pipe has a radius of . How much force must be exerted at this end?

  To generate an upward acceleration for the car of , how much force must be applied to the small end of the pipe?

  A SCUBA diver descends deep into the ocean. Calculate the water pressure at each of the following depths.

  What happens to the gravitational potential energy density of water when it is siphoned out of a lower main ditch on your farm and put into a higher row ditch? How is this consistent with Bernoulli’s principle?

  Water flows through a horizontal water pipe in diameter into a smaller pipe. What is the ratio in water pressure between the larger and the smaller water pipes?

  A pump is required to pipe water from a well in depth to an open-topped water tank at ground level. The pipe at the top of the pump, where the water pours into the water tank, is in diameter. The water flow in the pipe is . What is the kinetic energy density of the water flow?

  What pressure is required at the bottom of the well? (Assume no energy is lost – i.e., that the fluid is traveling smoothly.)

  What power is being delivered to the water by the pump? (Hint: For the next part, refer to Chapter 12)

  If the pump has an efficiency of %, what is the pump’s electrical power consumption?

  If the pump is operating on a power supply (typical for large pieces of equipment like this), how much electrical current does the pump draw?

  f. At cents per kilowatt-hour, how much does it cost to operate this pump for a month if it is running % of the time?

  Ouch! You stepped on my foot! That is, you put a force of in an area of on the tops of my feet! What was the pressure on my feet?

  What is the ratio of this pressure to atmospheric pressure?

  A submarine is moving directly upwards in the water at constant speed. The weight of the submarine is . The submarine’s motors are off. Draw a sketch of the situation and a free body diagram for the submarine.

  What is the magnitude of the buoyant force acting on the submarine?

  You dive into a deep pool in the river from a high cliff. When you hit the water, your speed was . About seconds after hitting the water surface, you come to a stop before beginning to rise up towards the surface. Take your mass to be . What was your average acceleration during this time period?

  What was the average net force acting on you during this time period?

  What was the buoyant force acting on you during this time period?

  A glass of water with weight is sitting on a scale, which reads . An antique coin with weight is placed in the water. At first, the coin accelerates as it falls with an acceleration of . About half-way down the glass, the coin reaches terminal velocity and continues at constant speed. Eventually, the coin rests on the bottom of the glass. What was the scale reading when…

  … the coin had not yet been released into the water?

  … the coin was first accelerating?

  … the coin reached terminal velocity?

  … the coin came to rest on the bottom?

  You are planning a trip to the bottom of the Mariana Trench, located in the western Pacific Ocean. The trench has a maximum depth of , deeper than Mt. Everest is tall! You plan to use your bathysphere to descend to the bottom, and you want to make sure you design it to withstand the pressure. A bathysphere is a spherical
capsule used for ocean descent – a cable is attached to the top, and this cable is attached to a winch on your boat on the surface. Name and sketch your bathysphere.

  What is the radius of your bathysphere in meters? (You choose – estimate from your picture.)

  What is the volume of your bathysphere in ?

  What is the pressure acting on your bathysphere at a depth of ? The density of sea water is .

  If you had a circular porthole of radius on your bathysphere, what would the inward force on the porthole be?

  f. If the density of your bathysphere is , what is the magnitude of the buoyant force acting on it when it is at the deepest point in the trench? g. In order to stop at this depth, what must the tension in the cable be? (Draw an FBD!)

  Answers to Selected Problems

  a. % of the berg is underwater b. %

  b. c.

  6. coins

  b. upward c.

  d. Cooler air outside, so more initial buoyant force

  e. Thin air at high altitudes weighs almost nothing, so little weight displaced.

  a. At a depth of , the buoyant force is d. The bottom of the cup is in radius

  a. b.

  c.

  a. b.

  c.

  .

  a. b.

  c.

  d.

  e.

  f. $

  a. b.

  b.

  a. upward b.

  c.

  a. b.

  c.

  d.

  a. “The Thunder Road” b. (note: here and below, you may choose differently)

  c.

  e.

  f.

  Chapter 20: Thermodynamics and Heat Engines Version 2

  Introduction

  This chapter is an introduction to thermodynamics, the study of heat and its applications to work and energy. To discover this topic, we have to first 'formally' understand two important concepts: heat and temperature.

  First, let's look at the meaning of the word 'formally' above. In studying the mechanics of projectile motion much earlier, we realized that humans have a natural ability for vector manipulation. We know where and how hard to throw a ball or rock --- well, some of us at least --- so that it can be caught by a friend or hit a target. We do this without performing any calculations or breaking any vectors into components. The 'formal' approach to projectile motion presented in the book --- using the vectors and calculations mentioned above --- allowed us to describe the same situation mathematically.

  We also have a natural ability to perceive things like temperature, heat flow, and other thermodynamics-related phenomena, but the build up of thermodynamics from intuitive to formal is more difficult than the corresponding project for mechanics.

  In projectile motion, concepts like 'how hard' and 'in what direction' lead to the idea of vectors, which are just a translation of those concepts into math. In other words, the math corresponds directly to the intuition. At the same time, the formal vocabulary is largely independent of the informal way of thinking: before learning about them in physics most students have never heard of them; there is no need to 'unlearn' anything.

  In thermodynamics, we face two problems: the math is less intuitive than before, and at the same time the formal definitions of temperature and heat often confuse us (they were very confusing to me for a long time) because we already have preconceived notions of what these are.

  Two Roads to Thermodynamics

  Upon formulating his law of universal gravitation, Newton remarked that:

  I have not as yet been able to discover the reason for these properties of gravity from phenomena, and I do not feign hypotheses.

  In other words, Newton realized that his theory of gravity was descriptive: it could predict when and where an object would fall, but could not explain why this happened.

  Thermodynamics started in the same manner: scientists studied various heat-related phenomena and found certain empirical relationships that seemed to always hold. However, there is a difference between thermodynamics and Newtonian Mechanics: the systems consist of trillions of interacting objects, rather than just a few.

  In fact, thermodynamics can be considered a statistical science: all the macroscopic (large-scale) results are a product of interactions of the microscopic constituents of systems. It is the goal of this chapter to outline the relationship between the properties of the molecules or atoms of a substance and its macroscopic qualities like temperature.

  Heat, Temperature, and Entropy

  Let's start with a simple scenario: when you touch something at a higher temperature than your hand, the temperature of your hand rises; your brain registers this as comfort, discomfort, or pain. We might guess that some kind of 'thermal energy' is transferred from the object to your hand to cause this temperature increase. This kind of transfer is called a heat flow. Informally, heat (for this case, at least) is often thought of as something that raises your hand's temperature, and temperature as something that measures the readiness of a substance to give up heat.

  To understand the above scenario from the perspective of physics, however, we will need to define the concepts mentioned --- temperature, thermal energy, and heat --- in the 'formal' manner described above.

  Thermal Energy

  We will first define the thermal energy of a substance in familiar terms. All substances (gases, solids, fluids, etc) have some internal thermal energy, which consists of:

  The kinetic energy from the random motion of the molecules or atoms of the substance (called Sensible Energy), and

  The energy associated with changes in the phase of a system --- for example the energy it takes to break the bonds between water molecules in melting ice (called Latent Energy).

  The concept of thermal energy underlies phenomena that we colloquially associate with 'heat', 'hotness', coldness', etc. Accordingly, these are fundamentally based on the energies present in the molecules or atoms of the substances in question.

  Solids and liquids have both types of thermal energy, while most gases only have sensible energy. In the ideal gas approximation, a monatomic (consisting of single atoms, not molecules --- helium gas, for instance) gas is said to consist of a large number of identical particles that fly in random directions and collide elastically with each other and all other objects (the walls of a container, etc).

  For monatomic gases, it's clear where this thermal energy comes from: particles are literally flying around with some velocities; the thermal energy depends on their kinetic energies from translational motion. But what about solids and liquids? In these substances atoms and molecules can still have kinetic energy due to vibrational and/or rotational motion. For instance, many solids can be modeled as atoms in a lattice connected by springs. Then, each pair of atoms is like a harmonic oscillator, which can have different energy levels. We learned earlier that the kinetic energy of a single harmonic oscillator depends on its position, but there are trillions of them in any macroscopic amount of matter and we can use the average value of kinetic energy over an oscillation.

  Heat

  The term heat is formally defined as a transfer of thermal energy between substances. Note that heat is not the same as thermal energy. Before the concept of thermal energy, physicists sometimes referred to the 'heat energy' of a substance, that is, the energy it received from actual 'heating' (heating here can be understood as it is defined above, though for these early physicists and chemists it was a more 'common sense' idea of heating: think beaker over Bunsen burner). The idea was then to try to explain thermodynamic phenomena through this concept.

  The reason this approach fails is that --- as stated in the paragraphs above --- it is in fact thermal energy that is most fundamental to the science, and 'heating' is not the only way to change the thermal energy of a substance. For example, if you rub your palms together, you increase the thermal energy of both palms.

  Once heat (a transfer of thermal energy) is absorbed by a substance, it becomes indis
tinguishable from the thermal energy already present: what methods achieved that level of thermal energy is no longer relevant. In other words, 'to heat' is a well defined concept as a verb: its use automatically implies some kind of transfer. When heat using as a noun, one needs to be realize that it must refer to this transfer also, not something that can exist independently.

  Temperature

  Now that we understand heat as a transfer of thermal energy, we would like to have some way of understanding when and why such transfers occur. This is achieved through the concept of temperature, which is a measure of the Sensible Energy of the substance in question.

  The definition above immediately raises a question: what do we mean by 'measure of the Sensible Energy,' can't we find the actual value? Remember, Sensible Energy is the kinetic energy associated with the motion of atoms or molecules in a substance. For questions of heat flows, it isn't just the total amount of energy that matters, but also how this kinetic energy can be distributed.

  For instance: the only way to increase the kinetic energies of the atoms in a monatomic gas is to increase their velocities. On the other hand, other gases --- called diatomic --- consist of two atoms held by a bond. This bond can be modeled as a spring, and the two atoms and bond together as a harmonic oscillator. Now, a single molecule's kinetic energy can be increased either by increasing its speed, by making it vibrate in simple harmonic motion, or by making it rotate around its center of mass. This difference is understood in physics through the concept of degrees of freedom: each degree of freedom for a molecule or particle corresponds to a possibility of increasing its kinetic energy independently of the kinetic energy in other degrees.