Sault Ste. Marie Area Public Schools / Overview



A.P. Physics Review Sheet 1

Chapters 1-13,15-18 in Walker, First Semester, 1/10/08

MECHANICS

• Review types of zeros and the rules for significant digits

• Vectors have both magnitude and direction whereas scalars have magnitude but no direction.

Velocity

• Two equations for average velocity: [pic] and [pic]

• Instantaneous velocity can be determined from the slope of a line tangent to the curve at a particular point on a position-versus-time graph.

Acceleration

• One equation for average acceleration: [pic]

• Instantaneous acceleration can be determined from the slope of a line tangent to the curve at a particular point on a velocity versus time graph.

• Acceleration due to gravity = 9.81 m/s2

• In a velocity-versus-time graph for constant acceleration, the slope of the line gives acceleration and the area under the line gives displacement. See Table 1.

| |

|Table 1: Graphing Changes in Position, Velocity, and Acceleration |

| |Constant |Constant |Constant Acceleration |Ball Thrown |

| |Position |Velocity | |Upward |

|Position Versus |[pic] |[pic] |[pic] |[pic] |

|Time: | | | | |

|Velocity Versus |[pic] |[pic] |[pic] |[pic] |

|Time: | | | | |

|Accelera-tion |[pic] |[pic] |[pic] |[pic] |

|Versus Time: | | | | |

| |

|Table 2: Relationship Between the Kinematic Equations and Projectile Motion Equations |

|Kinematic Equations |Missing Variable |Projectile Motion, |Projectile Motion, |

| | |Zero Launch Angle |General Launch Angle |

| | |Assumptions made: |Assumptions made: |

| | |[pic] and [pic] |[pic], [pic], |

| | | |and [pic] |

| | | | |

|[pic] |[pic] |[pic] |[pic] |

| | |where [pic] |where [pic] |

| | | | |

|[pic] |[pic] |[pic] |[pic] |

| | | | |

|[pic] | |[pic] |[pic] |

| |[pic] | | |

| | | | |

|[pic] |[pic] |[pic] |[pic] |

Projectile Motion

• The kinematic equations involve one-dimensional motion whereas the projectile motion equations involve two-dimensional motion. See Table 2 to better understand how the projectile motion equations can be derived from the kinematic equations.

• Recall that velocity is constant and acceleration is zero in the horizontal direction.

• Recall that acceleration is [pic]= 9.81 m/s2 in the vertical direction.

• For an object in free fall, the object stops accelerating when the force of air resistance, [pic], equals the weight, [pic]. The object has reached its maximum velocity, the terminal velocity.

• Projectiles follow a parabolic pathway governed by [pic]

• When a quarterback throws a football, the angle for a high, lob pass is related to the angle for a low, bullet pass when both footballs land in the same place. The sum of the angles is 90o.

• In distance contests for pumpkins launched by cannons, catapults, trebuchets, and similar devices, pumpkins achieve the farthest distance when launched at a 45o angle.

• The range of a projectile launched at initial velocity [pic] and angle [pic] is [pic]

• The maximum height of a projectile above its launch site is [pic]

Vectors

• Vectors have both magnitude and direction whereas scalars have magnitude but no direction.

• Examples of vectors are position, displacement, velocity, linear acceleration, tangential acceleration, centripetal acceleration, applied force, weight, normal force, frictional force, tension, spring force, momentum, gravitational force, and electrostatic force.

• Vectors can be moved parallel to themselves in a diagram.

• Vectors can be added in any order. See Table 3 for vector addition.

• For vector [pic] at angle [pic] to the x-axis, the x- and y-components for [pic] can be calculated from [pic] and [pic].

• The magnitude of vector [pic] is [pic] and the direction angle for [pic] relative to the x-axis is [pic].

| |

|Table 3: Vector Addition |

|Vector Orientation |Calculational Strategy Used |

|Vectors are parallel: |Add or subtract the magnitudes (values) to get the resultant. |

| |Determine the direction by inspection. |

|Vectors are perpendicular: |Use the Pythagorean Theorem, [pic], to get the |

|[pic] |resultant, [pic], where [pic] is parallel to the x-axis and [pic] is |

| |parallel to the y-axis. |

| |Use [pic] to get the angle, [pic], made with the x-axis. |

|Vectors are neither parallel |Adding 2 Vectors |Adding 2 or More Vectors |

|nor perpendicular: | |(Vector Resolution Method) |

| |Limited usefulness |Used by most physicists |

|[pic] |(1) Use the law of cosines to |(1) Construct a vector table |

| |determine the resultant: |based on your diagram. |

| |[pic] |(Use vector, x-direction, |

| | |and y-direction for the |

| |(2) Use the law of sines to |column headings.) |

| |help determine direction: |(2) Determine the sum of the |

| |[pic] |vectors for each direction, |

| | |[pic] and [pic]. |

| | |(3) Use the Pythagorean Thm |

| | |to get the resultant, [pic]: |

| | |[pic] |

| | |(4) To get the angle, [pic], made |

| | |with the x-axis, use: |

| | |[pic] |

Vectors (continued)

• To subtract a vector, add its opposite.

• Multiplying or dividing vectors by scalars results in vectors.

• In addition to adding vectors mathematically as shown in the last table, vectors can be added graphically. Vectors can be drawn to scale and moved parallel to their original positions in a diagram so that they are all positioned head-to-tail. The length and direction angle for the resultant can be measured with a ruler and protractor, respectively.

Relative Motion

• Relative motion problems are solved by a special type of vector addition.

• For example, the velocity of object 1 relative to object 3 is given by [pic] where object 2 can be anything.

• Subscripts on a velocity can be reversed by changing the vector’s direction: [pic]

| |

|Table 4: Newton’s Laws of Motion |

| |Modern Statement for Law |Translation |

|Newton’s First Law: |If the net force on an object is zero, its |An object at rest will remain at rest. An |

|(Law of Inertia) |velocity is constant. |object in motion will remain in motion at |

|Recall that mass is a measure of inertia. | |constant velocity unless acted upon by an |

| | |external force. |

|Newton’s Second Law: |An object of mass [pic] has an acceleration |[pic] |

| |[pic] given by the net force [pic] divided by | |

| |[pic]. That is [pic] | |

|Newton’s Third Law: |For every force that acts on an object, there |For every action, there is an equal but |

|Recall action-reaction pairs |is a reaction force acting on a different |opposite reaction. |

| |object that is equal in magnitude and opposite | |

| |in direction. | |

Survey of Forces

• A force is a push or a pull. The unit of force is the Newton (N); 1 N = 1 kg-m/s2

• See Newton’s laws of motion in Table 4. Common forces on a moving object include an applied force, a frictional force, a weight, and a normal force.

• Contact forces are action-reaction pairs of forces produced by physical contact of two objects. Review calculations regarding contact forces between two or more boxes.

• Field forces like gravitational forces, electrostatic forces, and magnetic forces do not require direct contact. They are studied in later chapters.

• Forces on objects are represented in free-body diagrams. They are drawn with the tails of the vectors originating at an object’s center of mass.

• Weight, [pic], is the gravitational force exerted by Earth on an object whereas mass, [pic], is a measure of the quantity of matter in an object ([pic]). Mass does not depend on gravity.

• Apparent weight, [pic], is the force felt from contact with the floor or a scale in an accelerating system. For example, the sensation of feeling heavier or lighter in an accelerating elevator.

• The normal force, [pic], is perpendicular to the contact surface along which an object moves or is capable of moving. Thus, for an object on a level surface, [pic] and [pic] are equal in size but opposite in direction. However, for an object on a ramp, this statement is not true because [pic] is perpendicular to the surface of the ramp.

• Tension, [pic], is the force transmitted through a string. The tension is the same throughout the length of an ideal string.

• The force of an ideal spring stretched or compressed by an amount [pic] is given by Hooke’s Law, [pic]. Note that if we are only interested in magnitude, we use [pic] where k is the spring or force constant. Hooke’s Law is also used for rubber bands, bungee cords, etc.

Friction

• Coefficient of static friction =[pic] where[pic]is the max. force due to static friction.

• Coefficient of kinetic friction =[pic] where [pic] is the force due to kinetic friction.

• A common lab experiment involves finding the angle at which an object just begins to slide down a ramp. In this case, a simple expression can be derived to determine the coefficient of static friction: [pic]. Note that this expression is independent of the mass of the object.

Newton’s Second Law Problems (Includes Ramp Problems)

1. Draw a free-body diagram to represent the problem.

2. If the problem involves a ramp, rotate the x- and y-axes so that the x-axis corresponds to the surface of the ramp.

3. Construct a vector table including all of the forces in the free-body diagram. For the vector table’s column headings, use vector, x-direction, and y-direction.

4. Determine the column total in each direction:

a. If the object moves in that direction, the total is [pic].

b. If the object does not move in that direction, the total is zero.

c. Since this is a Newton’s Second Law problem, no other choices besides zero and [pic] are possible.

5. Write the math equations for the sum of the forces in the x- and y-directions, and solve the problem. It is often helpful to begin with the y-direction since useful expressions are derived that are sometimes helpful later in the problem. Recall that the math equations regarding friction and weight are often substituted into the math equations to help solve the problem.

Equilibrium

• An object is in translational equilibrium if the net force acting on it is zero, [pic].

• Equivalently, an object is in equilibrium if it has zero acceleration.

• If a vector table is needed for an object in equilibrium, then [pic] and [pic].

• Typical problems involve force calculations for objects pressed against walls and tension calculations for pictures on walls, laundry on a clothesline, hanging baskets, pulley systems, traction systems, connected objects, etc.

Connected Objects

• Connected objects are linked physically, and thus, they are also linked mathematically. For example, objects connected by strings have the same magnitude of acceleration.

• When a pulley is involved, the x-y coordinate axes are often rotated around the pulley so that the objects are connected along the x-axis.

• A classic example of a connected object is an Atwood’s Machine, which consists of two masses connected by a string that passes over a single pulley. The acceleration for this system is given by [pic].

Work

• A force exerted through a distance performs mechanical work.

• When force and distance are parallel, [pic] with Joules (J) or Nm as the unit of work.

• When force and distance are at an angle, only the component of force in the direction of motion is used to compute the work: [pic]

• Work is negative if the force opposes the motion ([pic]>90o). Also, 1 J = 1 Nm = 1 kg-m2/s2.

• If more than one force does the work, then [pic]

• The work-kinetic energy theorem states that [pic]

• See Table 5 for more information about kinetic energy.

• In thermodynamics, [pic] for work done on or by a gas.

Determining Work from a Plot of Force Versus Position

• In a plot of force versus position, work is equal to the area between the force curve and the displacement on the x-axis. For example, work can be easily computed using [pic] when rectangles are present in the diagram.

• For the case of a spring force, the work to stretch or compress a distance [pic] from equilibrium is [pic]. On a plot of force versus position, work is the area of a triangle with base [pic] (displacement) and height [pic] (magnitude of force using Hooke’s Law, [pic]).

| |

|Table 5: Kinetic Energy |

|Kinetic Energy Type |Equation |Comments |

| | | |

|Kinetic Energy as a |[pic] |Used to represent kinetic energy in most |

|Function of Motion: | |conservation of mechanical energy problems. |

| | | |

|Kinetic Energy as a |[pic] since |Useful for a mass on a spring. |

|Function of Time in |[pic] and [pic] | |

|Oscillatory Motion: | | |

| | | |

|Kinetic Energy as a |[pic] |Kinetic theory relates the average kinetic |

|Function of Temperature: | |energy of the molecules in a gas to the Kelvin |

| | |temperature of the gas. |

Determining Work in a Block and Tackle Lab

• The experimental work done against gravity, [pic], is the same as the theoretical work done by the spring scale, [pic].

• [pic] where [pic] = distance the load is raised.

• [pic] where [pic] = force read from the spring scale and [pic] = distance the scaled moved from its original position.

• Note that the force read from the scale is ½ of the weight when two strings are used for the pulley system, and the force read is ¼ of the weight when four strings are used.

Power

• [pic] or [pic] with Watts (W) as the unit of Power.

• 1 W = 1 J/s and 746 W = 1 hp where hp is the abbreviation for horsepower.

Conservative Forces Versus Nonconservative Forces

1. Conservative Forces

• A conservative force does zero total work on any closed path. In addition, the work done by a conservative force in going from point A to point B is independent of the path from A to B. In other words, we can use the conservation of mechanical energy principle to solve complex problems because the problems only depend on the initial and final states of the system.

• In a conservative system, the total mechanical energy remains constant: [pic]. Since [pic], it follows that [pic]. See Table 5 for kinetic energy, [pic], and Table 6 for potential energy, [pic], for additional information.

• For a ball thrown upwards, describe the shape of the kinetic energy, potential energy, and total energy curves on a plot of energy versus time.

• Examples of conservative forces are gravity and springs.

2. Nonconservative Forces

• The work done by a nonconservative force on a closed path is not zero. In addition, the work depends on the path going from point A to point B.

• In a nonconservative system, the total mechanical energy is not constant. The work done by a nonconservative force is equal to the change in the mechanical energy of a system: [pic] .

• Examples of nonconservative forces include friction, air resistance, tension in ropes and cables, and forces exerted by muscles and motors.

| |

|Table 6: Potential Energy |

|Potential Energy Type |Equation |Comments |

| | | |

|Gravitational Potential |[pic] |Good approximation for an object near sea level|

|Energy: | |on the Earth’s surface. |

| | | |

|Gravitational Potential |[pic] where |Works well at any altitude or distance between |

|Energy Between Two Point |[pic] 6.67 x 10-11 Nm2/kg2 |objects in the universe; recall that [pic] is |

|Masses: |= Universal Gravitation |the distance between the centers of the |

| |Constant |objects. |

| | | |

|Elastic Potential Energy |[pic] where [pic] is the force (spring) |Useful for springs, rubber bands, bungee cords,|

| |constant and [pic] is the distance the spring |and other stretchable materials. |

| |is stretched or compressed from equilibrium. | |

| | | |

|Potential Energy as a |[pic] since |Useful for a mass on a spring. |

|Function of Time in |[pic] | |

|Oscillatory Motion: | | |

Momentum

• Linear momentum is given by [pic] with kg-m/s as the unit of momentum.

• In a system having several objects, [pic].

• Newton’s second law can be expressed in terms of momentum. The net force acting on an object is equal to the rate of change in its momentum: [pic]

• When mass is constant, [pic] .

• The impulse-momentum theorem states that [pic] where the quantity [pic] is called the impulse, [pic]. A practical application of this equation involves auto air bags. In an automobile collision, the change in momentum remains constant. Thus, an increase in collision time, [pic], will result in a decreased force of impact, [pic], reducing personal injury.

• For a system of objects, the conservation of linear momentum principle states that the net momentum is conserved if the net external force acting on the system is zero. In other words, [pic]. Thus, for an elastic collision, [pic] and for a perfectly inelastic collision, [pic]. See the collision types in Table 7.

• In an elastic collision in one dimension where mass [pic] is moving with an initial velocity [pic], and mass [pic] is initially at rest, the velocities of the masses after the collision are: [pic] and [pic].

| |

|Table 7: Collision Types |

|Collision Type |Do the Objects |Is Momentum Conserved? |Is Kinetic Energy Conserved? |

|and Example |Stick Together? | | |

|Elastic: | | | |

|No permanent |No |Yes |Yes |

|deformation occurs; | | | |

|billiard balls collide. | | | |

|Inelastic: | | | |

|Permanent deforma- |No |Yes |No |

|tion occurs; most | | | |

|automobile collisions. | | | |

|Perfectly Inelastic: | | | |

|Permanent deforma- |Yes |Yes |No |

|tion occurs and | | | |

|objects lock together | | | |

|moving as a single | | | |

|unit; train cars collide and lock | | | |

|together. | | | |

Rotational Motion

• Angular position,[pic], in radians is given by [pic] where [pic] is arc length and [pic] is radius.

• Recall that [pic].

• Counterclockwise rotations are positive, and clockwise rotations are negative.

• In rotational motion, there are two types of velocities: angular velocity and tangential velocity, which are compared in Table 8.

• In rotational motion, there are three types of accelerations: angular acceleration, tangential acceleration, and centripetal acceleration, which are compared in Table 8.

• The total acceleration of a rotating object is the vector sum of its tangential and centripetal accelerations.

| |

|Table 8: Comparing Angular and Tangential Velocity and |

|Angular, Tangential, and Centripetal Acceleration |

|Calculation |Equations |Units, Comments |

| | | |

|Angular Velocity: |[pic] |radians/s |

| | |Same value for horses A and B, side-by-side on |

| | |a merry-go-round. |

| | | |

|Tangential Velocity: |[pic] |m/s |

| | |Different values for horses A and B, |

| | |side-by-side on a merry-go-round. |

| | | |

|Angular Acceleration: |[pic] |radians/s2 |

| | |Same value for horses A and B, side-by-side on |

| | |a merry-go-round. |

| | | |

|Tangential Acceleration: |[pic] |m/s2 |

| | |Different values for horses A and B, |

| | |side-by-side on a merry-go-round. |

| | | |

|Centripetal Acceleration: |[pic] |m/s2 |

| | |[pic] is perpendicular to [pic] with [pic] |

| | |directed toward the center of the circle and |

| | |[pic] tangent to it. |

• Centripetal force, [pic], is a force that maintains circular motion: [pic]

• The period, [pic], is the time required to complete one full rotation. If the angular velocity is constant, then [pic].

• A comparison of linear and angular inertia, velocity, acceleration, Newton’s second law, work, kinetic energy, and momentum are presented in Table 9.

• Rotational kinematics is not covered on the AP Physics test.

• The moment of inertia, [pic], is the rotational analog to mass in linear systems. It depends on the shape or mass distribution of the object. In particular, an object with a large moment of inertia is hard to start rotating and hard to stop rotating. See Table 10-1 on p.298 for moments of inertia for uniform, rigid objects of various shapes and total mass.

• The greater the moment of inertia, the greater an object’s rotational kinetic energy.

• An object of radius [pic], rolling without slipping, translates with linear speed [pic] and rotates with angular speed [pic].

| |

|Table 9: Comparing Equations for Linear Motion and Rotational Motion |

|Measurement or Calculation |Linear Equations |Angular Equations |

| | | |

|Inertia: |Mass, m |[pic] where [pic]a constant |

| | | |

|Average Velocity: |[pic] |[pic] |

| | | |

|Average Acceleration: |[pic] |[pic] |

| | | |

|Newton’s Second Law: |[pic] |[pic] |

| | | |

|Work: |[pic] |[pic] |

| | | |

|Kinetic Energy: |[pic] |[pic] |

| | | |

|Momentum: |[pic] |[pic] |

Torque

• A tangential force, [pic], applied at a distance, [pic], from the axis of rotation produces a torque [pic] in Nm. (Since [pic] is perpendicular to [pic], this is sometimes written as [pic].)

• A force applied at an angle to the radial direction produces the torque [pic]

• Counterclockwise torques are positive, and clockwise torques are negative.

• The rotational analog of force, [pic], is torque, [pic], where [pic] moment of inertia and [pic] angular acceleration.

• The conditions for an object to be in static equilibrium are that the total force and the total torque acting on the object must be zero: [pic], [pic], [pic]. Related problems often involve bridges, scaffolds, signs, and rods held by wires.

Angular Momentum

• The rotational analog of momentum, [pic], is angular momentum, [pic] in kg-m2/s, where [pic]= moment of inertia and [pic] angular velocity.

• If the net external torque acting on a system is zero, its angular momentum is conserved and [pic].

Simple Machines

• All machines are combinations or modifications of six fundamental types of machines called simple machines.

• Simple machines include the lever, inclined plane, wheel and axle, wedge, pulley(s), and screw.

• Mechanical advantage, [pic], is defined as [pic]. It is a number describing how much force or distance is multiplied by a machine.

• Efficiency is a measure of how well a machine works, and [pic] is calculated using [pic] where [pic] is the work output and [pic] is the work input.

Gravity

• Newton’s Law of Universal Gravitation shows that the force of gravity between two point masses, [pic] and [pic], separated by a distance [pic] is [pic] where G is the universal gravitation constant, [pic] 6.67x10-11 Nm2/kg2 . Remember that [pic] is the distance between the centers of the point masses.

• In Newton’s Law of Universal Gravitation, notice that the force of gravity decreases with distance, [pic], as [pic]. This is referred to as an inverse square dependence.

• The superposition principle can be applied to gravitational force. If more than one mass exerts a gravitational force on a given object, the net force is simply the vector sum of each individual force. (The superposition principle is also used for electrostatic forces, electric fields, electric potentials, electric potential energies, and wave interference.)

• Replacing the Earth with a point mass at its center, the acceleration due to gravity at the surface of the Earth is given by [pic]. A similar equation is used to calculate the acceleration due to gravity at the surface of other planets and moons in the Solar System.

• At some altitude, [pic], above the Earth, [pic] can be used to calculate the acceleration due to gravity.

• In 1798, Henry Cavendish first determined the value of G, which allowed him to calculate the mass of the Earth using [pic].

• Using Tycho Brahe’s observations concerning the planets, Johannes Kepler formulated three laws for orbital motion as shown in Table 10. Newton later showed that each of Kepler’s laws follows as a direct consequence of the universal law of gravitation.

• As previously mentioned in the potential energy table, the gravitational potential energy, [pic], between two point masses [pic] and [pic] separated by a distance [pic] is [pic]. This equation is used in mechanical energy conservation problems for astronomical situations.

• Energy conservation considerations allow the escape speed to be calculated for an object launched from the surface of the Earth: [pic] 11,200 m/s [pic] 25,000 mi/h.

| |

|Table 10: Kepler’s Laws of Orbital Motion |

|Law |Modern Statement for Law |Alternate Description |

|1st Law: |Planets follow elliptical orbits, with the Sun at one |The paths of the planets are ellipses, with the center of |

| |focus of the ellipse. |the Sun at one focus. |

|2nd Law: |As a planet moves in its orbit, it sweeps out an equal |An imaginary line from the Sun to a planet sweeps out equal |

| |amount of area in an equal amount of time. |areas in equal time intervals. |

|3rd Law: |The period, [pic], of a planet increases as its mean |The square of a planet’s period, [pic], is proportional to |

| |distance from the Sun, [pic], raised to the 3/2 power. |the cube of its radius, [pic]. That is, |

| |That is, [pic](constant)[pic] |[pic](constant)[pic] |

| | |where [pic] mass of the Sun. |

Simple Harmonic Motion (SHM)

• Periodic motion repeats after a definite length of time. The period is the time required for a motion to repeat (one “cycle”), and the frequency is the number of oscillations (cycles) per unit time. Period, [pic], frequency, [pic], and angular frequency, [pic], are related to each other by: [pic] and [pic]. Rapid motion has a short period and a large frequency.

• Classic examples of simple harmonic motion (SHM) include the oscillation of a mass attached to a spring and the periodic motion of a pendulum. See Tables 11 and 12 for additional information like the period for each type of system.

• The position, [pic], of an object undergoing simple harmonic motion varies with time, [pic], as [pic] where the amplitude, [pic], is the maximum displacement from equilibrium.

• Simple harmonic motion is the projection of uniform circular motion onto the [pic]axis.

• The maximum speed of an object in simple harmonic motion is [pic] and the maximum acceleration is [pic].

• From [pic] and [pic], it can be shown that the amplitude, [pic], is [pic] and the period, [pic], is [pic].

• In an ideal oscillatory system, the total energy remains constant. Thus, for a spring-mass system in simple harmonic motion (SHM), [pic] which comes from [pic] since [pic] for all [pic].

| |

|Table 11: Relative Velocity, Acceleration, and Restoring Force |

|at Various Positions for a Spring-Mass System or Pendulum in SHM |

|Quantity |Maximum Displacement Left |Equilibrium Position |Maximum Displacement Right |

| | | | |

|Velocity, [pic] |[pic] |[pic] |[pic] |

| | | | |

|Acceleration, [pic] |[pic] |[pic] |[pic] |

| | | | |

|Restoring Force, [pic] |[pic] |[pic] |[pic] |

| | |since [pic] change in | |

| | |distance from the | |

| | |equilibrium position. | |

| |

|Table 12: Periods for Oscillating Objects |

|Period Type |Equation |Comments |

| | | |

|Period of a Mass on a Spring: |[pic] |When a spring obeys Hooke’s Law, the period is |

| | |independent of the amplitude. For a vertical |

| | |spring, Hooke’s Law gives [pic] so [pic]. |

| | | |

|Period of a Simple Pendulum |[pic] |For a pendulum with an amplitude less than 10o,|

|with a Small Amplitude: | |the period is independent of the amplitude and |

|( stress.

• At the elastic limit, greater stress results in permanent deformation. This represents a spring or rubber band that has been stretched too far or a car fender that has been dented.

Phase Changes and Phase Diagrams

• Be able to sketch a phase diagram for water, label the axes (pressure vs. temperature), label the regions of the diagram (solid, liquid, and gas), label the triple point and critical point, and show how the melting and boiling points can be determined at atmospheric pressure (1 atm).

• At the triple point in a phase diagram, all three phases are in equilibrium.

• Beyond the critical point in a phase diagram, the liquid and gas phases become indistinguishable and are referred to as a fluid.

• Be able to sketch a heating curve for water, label the axes (temperature vs. heat), label the phases in the diagram (solid, liquid, and gas), label the phase changes (melting, vaporizing), and clearly show where the melting and boiling points are in relation to the curve.

• The latent heat, [pic], is the amount of heat per unit mass that must be added to or removed from a substance to convert it from one phase to another. Latent heats of fusion, [pic], and latent heats of vaporization, [pic], for various substances are given in Table 17-4 on p.571 of the course text.

• A common heat transfer problem is calculating the heat needed to raise the temperature of ice from -15oC to steam at 115oC. Using the heating curve for water, the heat needed to increase the temperature of ice, liquid water, and steam can be calculated using [pic] and the heat needed for the phase transfers can be calculated using [pic] and [pic]. Add the heats together, [pic], to get the final answer.

Conversion of Gravitational Potential Energy to Thermal Energy

• A common problem is to calculate the specific heat for an object that has fallen. Ideally, the gravitational potential energy lost, [pic], is equal to the thermal energy gained, [pic]. This allows us set these expressions equal to each other. Thus, [pic].

Laws of Thermodynamics

• The laws of thermodynamics are described in Table 19.

• [pic] and [pic]are state functions and depend only on the initial and final states of the system. [pic] and [pic] are not state functions and depend on the pathway between the initial and final states..

| |

|Table 19: Laws of Thermodynamics |

|Law |Physics Description (Walker) |Chemistry Description (Chang) |

|0th Law: |When two objects have the same temperature, they are in |Not described. |

| |thermal equilibrium. | |

|1st Law: |[pic] |Energy can be converted from one form to another, but it |

| |[pic] pos. [pic] system gains heat. |cannot be created or destroyed. |

| |[pic] neg. [pic] system loses heat. |[pic] |

| |[pic] pos. [pic] work done by system. |[pic] pos. [pic] system gains heat. |

| |[pic] neg. [pic] work done on system. |(endothermic process). |

| |[pic] and [pic] are not state functions. |[pic] neg. [pic] system loses heat. |

| | |(exothermic process). |

| | |[pic] pos. [pic] work done on system. |

| | |[pic] neg. [pic] work done by system. |

| | |[pic] and [pic] are not state functions. |

|2nd Law: |When objects of different temperatures are brought into |The entropy of the universe increases in a spontaneous |

| |thermal contact, the spontaneous flow of heat that |process and remains unchanged in an equilibrium process. |

| |results is always from the high temperature object to the | |

| |low temperature object. | |

|3rd Law: |It is impossible to lower the temperature of an object to |The entropy of each element in a perfect crystalline state |

| |absolute zero in a finite number of steps. |is zero at absolute zero, 0 K. |

Thermal Processes

• Isobaric, isochoric, isothermal, and adiabatic processes and their characteristics are described in Table 20.

• In general, the work done during a process is equal to the area under the process curve in a PV plot. Recall that [pic].

• In a complete cycle on a PV plot, the system returns to its original state, which means that the internal energy, [pic], must return to its original value. Therefore, the net change in internal energy must be zero, [pic].

• Using [pic] and the thermodynamic processes in Table 20, the results in Table 21 are obtained for [pic] moles of a monatomic ideal gas.

• Don’t forget that the work done in an isothermal expansion is [pic].

• Specific heats have different values depending on whether they apply to a process at constant pressure or a process at constant volume.

• The molar specific heat, [pic], is defined by [pic] where [pic] is the number of moles.

• Table 22 compares the specific heats for a monatomic ideal gas at constant pressure or constant volume.

• Table 23 compares isotherms and adiabats for monatomic ideal gases on a PV diagram.

| |

|Table 20: Thermodynamic Processes and Their Characteristics |

|Process |Assumption(s) |Work, [pic], = ? |Heat, [pic], = ? |

| | | | |

|Constant Pressure: |[pic] and |[pic] |[pic] |

|(Isobaric) |[pic] | | |

| | | | |

|Constant Volume: |[pic] so |[pic] |[pic] |

|(Isochoric) |[pic] | | |

| | | | |

|Constant Temperature: |[pic] so |[pic] |[pic] since [pic] |

|(Isothermal) |[pic] | | |

| | | | |

|Adiabatic: |[pic] |[pic] |[pic] |

|(No heat flow) | | | |

| |

|Table 21: Results for Various Thermodynamic Processes Involving |

|n Moles of a Monatomic Ideal Gas |

|Process |Heat, [pic], = ? |Work, [pic], = ? |[pic] = ? |

| | | | |

|Constant Pressure: |[pic] |[pic] |[pic] |

|(Isobaric) | | | |

| | | | |

|Constant Volume: |[pic] |[pic] since [pic] |[pic] |

|(Isochoric, [pic]) | | | |

| | | | |

|Constant Temperature: |[pic] |[pic] from a P-V diagram where |[pic] since [pic] |

|(Isothermal, [pic]) | |[pic] | |

| | | | |

|Adiabatic: |[pic] |[pic] |[pic] |

|(No heat flow,[pic]) | | | |

| |

|Table 22: Specific Heats for an Ideal Gas: Constant Pressure, Constant Volume |

|(Recall that we are still considering monatomic ideal gases) |

|Process |Heat, [pic], = ? |Heat, [pic], = ? |Specific Heat, [pic], = ? |

| | | | |

|Constant Pressure: |[pic] |[pic] |[pic] |

|(Isobaric) | | | |

| | | | |

|Constant Volume: |[pic] |[pic] |[pic] |

|(Isochoric, [pic]) | | | |

| |

|Table 23: Comparing Isotherms and Adiabats on a P-V Diagram |

|Process |Comparison |

| | |

|Isotherms: |[pic]constant |

| | |

| |[pic]constant where [pic] for monatomic ideal gases |

|Adiabats: |(The value of [pic] is different for gases that are diatomic, triatomic, etc.) |

Heat Engines and Heat Pumps

• A heat engine is a device that converts heat into work; for example, a steam engine.

• Be able to draw a schematic diagram for a heat engine and label the reservoirs, engine, [pic], [pic], [pic], [pic], and [pic].

• The efficiency [pic] of a heat engine that takes in the heat [pic] from a hot reservoir, exhausts a heat [pic] to a cold reservoir, and does the work [pic] is [pic].

• Carnot’s theorem states that if an engine operating between two constant-temperature reservoirs is to have maximum efficiency, it must be an engine in which all processes are reversible. In addition, all reversible engines operating between the same two temperatures have the same efficiency.

• The maximum efficiency of a heat engine operating between the Kelvin temperatures [pic] and [pic] is [pic].

• Refrigerators, air conditioners, and heat pumps are devices that use work to make heat flow from a cold region to a hot region.

• Be able to draw a schematic diagram for a heat pump and label the reservoirs, engine, [pic], [pic], [pic], [pic], and [pic].

Entropy

• Entropy is a measure of the disorder or randomness of a system. As entropy increases, a system becomes more disordered.

• The total entropy of the universe increases whenever an irreversible process occurs.

• The change in entropy during a reversible exchange of heat [pic] at the Kelvin temperature [pic]is [pic] where the unit of entropy is J/K.

-----------------------

[pic]

[pic]

[pic]

[pic]

a = -9.81m/s2

a

t

a

t

a

t

a

t

Slope = -9.81 m/s2

v

t

Slope = aave

v

t

v

t

v

t

x

t

x

t

Slope = vave

x

t

x

t

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

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