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Chapter 17 Onl ine HW

Chapter 17 Online HW Due: 12:45pm on Wednesday, September 7, 2016 To understand how points are awarded, read the Grading Policy for Policy for this assignment.

Heat versus Temperature The specific heat capacity of aluminum is about twice that of iron. Consider two blocks of equal mass, one made of aluminum and the other one made of iron, initially in thermal equilibrium.

Part A Heat is added added to each each block at the same constant constant rate until it reaches reaches a temperature temperature of 500 500

. Which of the following following statements is true?

Hint 1. How to approach the problem Heat is added to both blocks at the same constant rate. That is, the same amount of heat is added to each block per unit time. Therefore, the block that reaches the final temperature in the smallest amount of time is the block that requires the smallest amount of heat to undergo the given temperature change. Since both blocks have the same mass and undergo the same temperature change, you can relate the amount of heat absorbed by each block to the block's specific heat capacity.

Hint 2. Specific heat capacity Given a sample of mass

of a certain subst ance, the amount of heat

needed to change its temperature by an amount

is given by

, where is the specific heat capacity where capacity characteristic of that substance. substance. It follows that that the specific heat capacity capacity required to raise the temperature of one one gram of of the sample by by 1 .

of a sample is the amount amount of heat heat

Hint 3. Identify which material requires more heat Consider several one-gra Consider one-gram m samples of different materials. materials. Hea Heatt is adde added d to each sample to increase increase its t emper emperature ature by by 1 absorb the most heat?

. Which material will

Hint 1. Definition of specific heat capacity The specific specific hea heatt capacity

of a sample is the amount amount of heat heat required required to raise the tempera temperature ture of of one gram of that sample by 1

.

ANSWER: The material with the smallest specific heat capacity will absorb the most heat. The material with the largest specific heat capacity will absorb the most heat. All the materials will absorb the same amount of heat because they all have the same mass. All the materials will absorb the same amount of heat because they all undergo the same change in temperature.

ANSWER: The iron takes less time than the aluminum to reach the final temperature. The aluminum takes less time than the iron to reach the final temperature. The two blocks take the same amount of time to reach the final temperature.

Correct

Part B When the two materials have reached thermal equilibrium, the block of aluminum is cut in half and equal quantities of heat are added to the iron block and to each portion of the aluminum block. Which of the following statements is true?

Hint 1. How to approach the problem Since the same quantities of heat are added to samples that have different masses and different specific heat capacities, this may result in different final temperatures for each sample. However, you must keep in mind that each smaller block of aluminum now has half the mass of the

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Chapter 17 Online HW iron block, but about twice the specific heat capacity. To solv e this problem use proportional reasoning to find a relation between

, , and

.

Find the simplest equation that contains these variables and other known quantities from the problem. Write this equation twice: once to describe , , and and again to relate , , and , where the subscript i refers to iron and a to aluminum. Write each equation so that all the constants are on one side and the variables are on the other. In this problem the variable is so write your equations in the form . Finally, compare the two cases presented in the problem. For this question you should find the ratio .

Hint 2. Find the temperature change of the iron block When an amount of heat is absorbed by the block of iron, what is its change in temperature and the specific heat capacity of iron, respectively. Express your answer in terms of

,

, and

? Use

and

for the mass of the iron block

.

Hint 1. Specific heat capacity Given a mass

of a certain subst ance, the amount of heat

needed to change its temperature by an amount

is given by

, where

is a constant, called specific heat capacity, characteristic of that substance.

ANSWER:

=

Hint 3. Find the temperature change of the smaller aluminum block When an amount of heat is absorbed by the block of iron, what is its change in temperature ? Express the mass of the block in terms of the mass of the iron block and the specific heat capacity of aluminum in terms of the specific heat capacity of iron . Express your answer in terms of

,

, and

.

Hint 1. Specific heat capacity of aluminum Recall that the specific heat capacity of aluminum is about twice the specific heat capacity of iron.

ANSWER: =

ANSWER: The three blocks are no longer in thermal equilibrium; the iron block is warmer. The three blocks are no longer in thermal equilibrium; both the aluminum blocks are warmer. The blocks remain in thermal equilibrium.

Correct

Thermal Conductivity Ranking Task Six objects are placed in a 500 (260 ) oven and allowed to reach thermal equilibrium. Each object has a mass of 1.0 conductivity of each substance are denoted by and .

. The specific heat and thermal

Part A Rank these objects on the basis of their temperatures when removed from the oven.

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Rank from largest to smallest. To rank items as equivalent, overlap them.

Hint 1. Specific heat The specific heat of a substance is the amount of heat energy needed to raise the temperature of 1.0

of the substance by 1

.

Hint 2. Thermal conductivity The thermal conductivity of a substance is a measure of the rate at which heat energy can flow through a substance. A substance with large thermal conductivity allows the rapid flow of heat energy through it.

Hint 3. Equilibrium At equilibrium, an object and its surroundings are at the s ame temperature.

ANSWER:

Reset

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The correct ranking cannot be determined.

Correct

Part B When removing the objects from the oven, you accidentally touch each one with your hand. Rank these objects on the basis of how hot they feel. Rank from largest to smallest. To rank items as equivalent, overlap them.

Hint 1. Sensing temperature The rate at which heat energy flows into (or out of) your body will determine how hot (or cold) something feels when you touch it. For example, if heat energy flows rapidly from your hand when you touch an object, the object will feel very cold.

Hint 2. Rate of heat flow The rate of heat flow (via conduction) from one point to another depends on the temperature difference , between the points, the area and thickness of the contacting surface, and the thermal conductivity of the surface. Mathematically, this can be represented as .

ANSWER:

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The correct ranking cannot be determined.

Correct

Part C Each of the objects is immediately dunked in an identical tub of cold water. The tubs are quickly sealed and insulated. Rank the objects on the basis of their temperature on reaching equilibrium with the water. Rank from largest to smallest. To rank items as equivalent, overlap them.

Hint 1. Conservation of energy Once placed in contact with the cold water, heat energy will flow from the hot objects into the cold water, lowering the temperature of the objects and raising the temperature of the water. Objects that can transfer a large amount of heat energy with a small change in temperature will end up with the highest equilibrium temperatures.

Hint 2. Energy transfer and specific heat An object that can t ransfer a large amount of heat energy with a s mall change in t emperature has a large specific heat.

ANSWER:

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The correct ranking cannot be determined.

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The Size of Stars The hot glowing surfaces of stars emit energy in the form of electromagnetic radiation. It is a good approximation to assume that the emissivity for these surfaces.

is equal to 1

Part A Find the radius of the star Rigel, the bright blue star in the const ellation Orion that radiates energy at a rate of temperature of 11,000 . Assume that the star is spherical. Use

and has a surfac e

for the Stefan-Boltzmann constant and express your answer numerically in meters to two significant figures.

Hint 1. Equation for heat radiation The rate of heat radiation is given by , where is the surface area of the object, Boltzmann constant, and is the temperature of the object in kelvins.

is the emissivit y of the surface,

is the Stefan-

Hint 2. Surface area of the star Since the star is assumed to be spherical, the surface area is given by

, where

is the radius of the star.

ANSWER: = 5.1×1010

Correct This is over 50 times the size of our own sun and about a third of the orbital radius of the earth around the sun. Rigel is an example of a supergiant star.

Part B Find the radius of the star Procyon B, which radiates energy at a rate of that the star is spherical.

and has a surface temperature of 10,000

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. Assume

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Use

for the Stefan-Boltzmann constant and express your answer numerically in meters to two significant figures.

ANSWER: = 5.4×106

Correct This is slightly smaller than the size of the earth and much smaller than (less than 1%) the size of our sun. Procyon B is an example of a star. Like many small, dim stars it is visible only through a telescope.

white dwarf

Rankine Temperature Scale Like the Kelvin scale, the Rankine scale is an absolute temperature scale: Absolute zero is zero degrees Rankine (0 the same size as those on the Fahrenheit scale ( ) rather than the Celsius scale ( ).

). However, the units of this s cale are

Part A Given that water at standard pressure freezes at 0 , which corresponds to 32 , and that it boils at 100 , which corresponds to 212 temperature difference in degrees Fahrenheit that corresponds to a temperature difference of 1 on the Kelvin scale.

, calculate the

Give your answer to two significant figures.

Hint 1. Relation of Celsius and Kelvin temperature scales A temperature increase of one kelvin corresponds to a t emperature increase of one degree als o on the Celsius scale. The Kelvin temperature s cale and the Celsius temperature scale differ only in their zero point.

ANSWER: = 1.8

Correct

Part B What is the numerical value of the triple-point temperature

of water on the Rankine scale?

Give your answer to three significant figures.

Hint 1. Triple-point temperature On the Kelvin temperature scale, water freezes and coexists in three phases (solid, liquid, and vapor) at 273.16 temperature is known as the triple point.

at standard pressure. This

ANSWER: = 492

Correct

PSS 17.2 Calorimetry Learning Goal: To practice Problem-Solving Strategy 17.2 Calorimetry. A 1.00 chunk of an unknown metal that has been i n boiling water for several minutes is quickly dropped into an insulating Styrofoam beaker containing 1.00 of water at 18.0 . After gently stirring for 5.00 , you observe that the water's temperature has reached a constant value of 22.0 . The specific heat capacity of water is = 4190 . Assuming that the Styrofoam absorbs a negligibly small amount of heat and that no heat was lost to the surroundings, what is the specific heat capacity of the metal, ?

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Problem-Solving Strategy: Calorimetry problems IDENTIFY the relevant concepts : When heat flow occurs between two bodies that are isolated from their surroundings, the amount of heat lost by one body must equal the amount gained by the other body. SET UP the problem using the following steps: 1. Identify which objects exchange heat. To avoid confusion with algebraic signs, take each quantity of heat added to a body as positive and each quantity leaving a body as negative. When two or more bodies interact, the algebraic sum of the quantities of heat transferred to all the bodies must be zero. 2. Each object will undergo a temperature change with no phase change, a phase change at constant temperature, or both. Use the equation to describe temperature changes and equations and to describe phase changes. 3. Consult tables for values of the specific heat, molar heat capacity, heat of fusion, or heat of vaporization. 4. Be certain to identify which quantities are known and which are the unknown target variables. EXECUTE the solution as follows: 1. Solve for the target variables. Of ten, you will need to find an unknown temperature . For example, if an object has an initial temperature of and an unknown final temperature , the temperature change for the object is . 2. In problems where a phase change takes place, as when ice melts, you may not know in advance whether all the material undergoes a phase change or only part of it. You can always assume one or the other, and if the resulting calculation gives an absurd result, you know the initial assumption was wrong. Back up and try again! EVALUATE your answer : A common error is to use t he wrong algebraic sign for eit her a physically sensible.

or

term. Double-check your c alculations, and make sure t hat t he final results are

IDENTIFY the relevant concepts Since it is assumed that the Styrofoam beaker absorbs a negligibly small amount of heat and that no heat is lost to the surroundings, you can think of the metal and the water as two bodies isolated from the surroundings that exchange a certain amount of heat. Calorimetry calculations will allow you to get a quantitative description of the heat exchange between the metal and the water.

SET UP the problem using the following steps

Part A In which direction does the heat flow?

Hint 1. Heat as energy transfer Heat is the amount of energy that is transferred from one body to another because of a difference in temperature. It always flows spontaneously from an object at higher temperature to one at lower temperature.

ANSWER: Heat enters the metal and enters the water. Heat leaves the water and enters the metal. Heat leaves the metal and leaves the water. Heat leaves the metal and enters the water.

Correct The principle of conservation of energy for an isolated system requires that the energy that leaves the warmer substance equal the energy that enters the colder substance. In our case, this means that , where is the heat gained by the water and is the heat lost by the metal. Note that according to the notation introduced in the strategy, will be a positi ve quantity, whereas will be negative.

EXECUTE the solution as follows

Part B Ass uming that the S tyrofoam absorbs a negligibly small amount of heat and that no heat was lost to the s urroundings, what is the s pecific heat capacity of the metal, ? Express your answer in joules per kilogram-kelvin.

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Hint 1. Find the heat gained by the water Find a general expression for , the heat required to change the temperature of the water when it is heated by the metal. Use of the water, for the specific heat of water, and and for the initial and final temperatures of the water, respectively. Express your answer in terms of some or all of the variables

,

,

, and

for the mass

.

Hint 1. The heat required for a temperature change Recall that the amount of heat required to change the temperature of a given material by an amount the material and to the temperature change :

is proportional to the mass

of

, where

is the specific heat capacity of the material.

ANSWER: =

Hint 2. Find the heat lost by the metal Find an express ion for , the heat required to change the temperature of the metal when it is cooled by the water. Use for the mass of the piece of metal, for the specific heat of the metal, and and for the initial and final temperatures of the metal, respectively. Express your answer in terms of some or all of the variables

,

,

, and

.

Hint 1. The heat required for a temperature change Recall that the amount of heat required to change the temperature of a given material by an amount the material and to the temperature change :

is proportional to the mass

of

, where

is the specific heat capacity of the material.

ANSWER: =

Hint 3. The initial and final temperatures of the metal Before being dropped into the Styrofoam beaker, the piece of metal has been in boiling water for several minutes. This suggests that the piece of metal is at the boiling temperature of water when it is dropped into the insulating beaker. Since no phase changes occurred inside the Styrofoam beaker after the introduction of the piece of metal, when the temperature of the water in the insulating beaker has reached a constant value, the metal and the water are in thermal equilibrium.

Hint 4. Putting it all together Apply conservation of energy: Equate the total heat transferred into and out of the is olated sy stem to zero. This conservation is represented by the equation . Substituting the expressions you found for

and

into this expression yields .

Solve this equation for the unknown

.

ANSWER: = 215

Correct

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Part C You found that the unknown metal has a specific heat capacity close to, for example, that of silver [ ]. Let's compare silver to other known materials and see whether your results make sense. Imagine you have 1 of each of the subst ances listed below. If the same amount of heat is added to each, which subst ance will undergo the largest change in temperature experiences a phase change.

? Ass ume that no subst ance

Rank the samples from largest to smallest change in temperature. To rank items as equivalent, overlap them.

Hint 1. Determine the relationship between the specific heat capacity and the temperature change Recall that the amount of heat required to change the temperature of a given material by an amount material, the specific heat capacity of the material, and the temperature change , that is,

is proportional to the mass .

of the

Since and are the same for each subst ance in the table, you can relate the temperature change to the heat capacit y. Which of the following statements describes this relationship? ANSWER: is proportional to . is inversely proportional to . does not depend upon .

ANSWER:

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The correct ranking cannot be determined.

Correct Your results make sense. Silver, which is a metal, will experience a greater change in temperature than water when both substances absorb the same amount of heat because it has a lower specific heat capacity. This is exactly what occurred to the water and the piece of metal placed in the Sty rofoam c ontainer. Water has one of the highest specific heat capacities of all substances, so it stores more heat per degree of temperature change. This makes it an ideal substance for hot-water space-heating systems and other uses that require a minimum drop in temperature for a given amount of heat released into the environment.

PSS 17.3: Heat Conduction Learning Goal:

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To practice Problem-Solving Strategy 17.3 Heat conduction Animals in cold climates often depend on two layers of insulation: a layer of body fat (of thermal conductivity 0.200 in diameter having a layer of fat 4.00×10 −2

trapped inside fur or down. We can model a black bear as a sphere 1.50 studies of bear hibernation, it was found that the outer surface layer of the fur is at = 2.70 during hibernation, while the inner surface of the fat layer is at = 31.0 . How thick should the air layer (contained within the fur) be so that the bear loses heat at a rate of 50.0 ? Thermal conductivity of air is

= 2.40×10−2

) surrounded by a layer of air thick, as shown in the figure. In

.

Problem-Solving Strategy 17.3: Heat conduction IDENTIFY the relevant concepts : The concept of heat conduction comes into play whenever two objects at different temperature are placed in contact. SET UP the problem using the following steps 1. Identify the direction of heat flow in the problem (from hot to cold). In the equation

is always measured along that directi on and is always an area perpendicular to that directi on. Often, when a box or other container has an irregular shape, but uniform wall thickness, you can approximate it as a flat slab with the same thickness and total wall area. 2. Identify the target variable. EXECUTE the solution as follows: 1. If heat flows through a single object, use the above equation to solve for the target variable. 2. In some problems, the heat flows through two different materials in succession. The temperature at the interface between the two materials is then intermediate between and ; In steady-state heat flow, the same heat has to pass through both materials in success ion, so the heat current must be the same in both materials. 3. If there are two parallel heat flow paths, so that some heat flows through each, then the total is the sum of the quantities and for the separate paths. 4. As always, it is essential to use a consistent set of units. EVALUATE your answer : As always, ask yourself whether the results are physic ally reasonable.

IDENTIFY the relevant concepts In this problem there is heat flow through two different materials in succession. The temperature at the interface for the fur and fat must be some number between the external and internal temperature of the bear. Assuming the data in the problem are for a steady state, then the heat current in both materials must be the same.

SET UP the problem using the following steps

Part A As mentioned i n t he introduction, the bear c an be modeled as a s phere with different layers. In this model, in which direction does the heat flow?

Hint 1. Direction of heat flow Heat transfer occurs only between regions at different temperatures, and the direction of heat flow is always from higher to lower temperature.

ANSWER: Inward along the radius of the sphere. Outward along the radius of the sphere. Tangent perpendicular to the radius of the sphere.

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Correct By modeling the bear as a sphere, the problem translates into the study of heat flow through two spherical shells in succession: an internal shell representing the layer of fat, and an external shell representing the layer of air trapped in the fur. To apply the heat conduction model discussed in the strategy above, assume that both layers are made of an homogeneous material and that their cross sections perpendicular to the direction of flow are constant. This is reasonable because the thickness of the layers of insulation is small compared to the radius of the sphere. Your target variable is the thickness of the external layer.

EXECUTE the solution as follows

Part B What is the thickness of the air layer (contained within the fur),

?

Express your answer in meters.

Hint 1. How to approach the problem Recall that in a steady-state heat flow through two materials in succession, the heat current must be the same in both materials. Thus, in the bear model, the heat current through the fat layer must be the same as the heat current through the layer of air trapped within the fur. You also know the temperature of the outer surface layer of air. You must find the temperature at the interface between the fat and the fur. Then you can use these values in the heat-current equation to calculate the thickness of the air layer.

Hint 2. Equation for heat conduction through the layer of air The heat current

where

through the air layer is

is the thermal conductivity of the air and

and

are, respectively, the thickness and total surface area of the air layer.

Hint 3. Find the surface area of the fat layer Note that you are modeling the bear as a sphere. As explained in Part A, the thickness of the layers of insulation is small compared to the radius of the sphere so you can consider the surface area perpendicular to the direction of heat flow as constant.What is the surface area of the layer of fat? Express your answer in meters squared.

Hint 1. Surface area of a sphere

As shown in the f igure, the heat flows along the radius of the s phere. The area perpendicular to that direction is the s urface area of the sphere, which is

where

is the radius of the sphere.

ANSWER: = 7.07

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Correct For the rest of the problem, we will assume that also the second layer of insulation has the same surface area

as the layer of fat.

Hint 4. Find the temperature of the outer surface layer of the fur What is the temperature of the outer surface layer of the fur,

.

Express your answer in degrees Celsius. ANSWER: = 2.70

Correct Even though the thermal conductivity of body fat, , is given in , its numerical value remains unchanged when you express it in because the Celsius and Kelvin scales have the same size unit. Therefore, you can continue to work using temperatures in degrees Celsius.

Hint 5. Find the temperature at the interface between the fat and fur Find the temperature

at the interface between the fat and fur layers.

Express your answers in degrees Celsius.

Hint 1. How to approach the problem Because a steady state has been reached, the heat currents through the layer of fat and the layer of air trapped in the fur must be equal. Note that in the steady state, the intermediate temperature at the interface between the two layers doesn't change. Thus, set up the heat conduction equation for the fat layer only and solve for the fat-fur interface temperature . Make sure to use consistent units for the temperature. Note that the heat current is proportional to the temperature gradient (that is, the change in temperature per unit length). Since the Celsius and Kelvin temperature scales have the same size unit, you can equally express the temperature gradient in either or .

Hint 2. Equation for heat conduction through the layer of fat The heat current

where

through the layer of fat is

is the thermal conductivity of body fat and

and

are, respectively, the surface area and thickness of the fat layer.

Hint 3. Find the temperature at the inner surface of the fat layer What is the temperature at the inner surface of the fat layer,

?

Express your answer in degrees Celsius. ANSWER: = 31.0

Correct Even though the thermal conductiv ity of body fat, , is given in , its numerical value remains unchanged when you express it in because the Celsius and Kelvin scales have the same size unit. Therefore, you can continue to work using temperatures in degrees Celsius.

ANSWER: =

29.6

Correct

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ANSWER: = 9.12×10 −2

Correct Note that animals' fur serves only as a layer of insulation. The air trapped within the fur has a low thermal conductivity and reduces the transfer of energy away from body due to conduction. In a similar way, blankets and coats keep us warm in cold weather by trapping the warmer air in regions close to our bodies and hence reducing energy loss by convection and conduction. In other words, what keeps us warm is not the clothing itself but the air trapped in the clothing.

EVALUATE your answer

Part C How many inches or centimeters thick should the bear's fur layer (

) be to hold a layer of air

so that adequate insulation is provided?

Express your answer using inches (in) or centimeters (cm).

Hint 1. Inches and centimeters Recall that

.

ANSWER: = 9.12

Correct Your calculations make sense! In fact, it is found that typically in winter black bears have a fur that can grow up to four inches thick or about ten centimeters.

± A Sliding Crate of Fruit A crate of fruit with a mass of 30.5 the horizontal.

and a specific heat c apacity of 3550

slides 8.80

down a ramp inclined at an angle of 36.4

below

Part A If the crate was at rest at the top of the incline and has a speed of 2.40 Use 9.81

at the bottom, how much work

was done on the crate by fricti on?

for the acceleration due to gravity and express your answer in joul es.

Hint 1. How to approach the problem If no friction were acting, then the kinetic energy of the crate at the bottom of the incline would equal the difference in gravitational potential energy of the crate between its initial and final positions. The nonconservative (nc) frictional force is responsible for the difference. To find the work done by friction, apply energy conservation: .

Hint 2. Find the initial and final kinetic energies What is the kinetic energy of the crate before it starts to slide (

) and after it reaches the bottom of the ramp (

)?

Express your answer in joules as two terms separated by commas. ANSWER: ,

= 0, 87.8

Hint 3. Find the difference between initial and final potential energy What is

, the change in the potential energy of the crate from when it starts to slide to after it reaches the bottom of the ramp?

Express your answer in joules.

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Hint 1. A helpful formula and a helpful diagram The difference in the potential energy of the crate is given by the value of ?

, where

is the height shown in the figure below. What is

Express your answer in meters.

ANSWER: = 5.22 m

ANSWER: = -1560

ANSWER: = -1470

Correct The frictional force opposes the motion of the crate, so the work done on the crate by friction must be a negative quantity.

Part B If an amount of heat equal to the magnitude of the work done by friction is absorbed by the crate of fruit and the fruit reaches a uniform final temperature, what is its temperature change ?

Hint 1. Equation for temperature change The quantity of heat needed to increase the temperature of an object by a certain amount is given by , where is the object's mass , is its specific heat, and is the temperature change (in kelvins) of the object. In this case is a positive quantity since the temperature of the crate is increasing.

ANSWER: = 1.36×10 −2

Correct Of course, the assumptions of "total heat absorption" and "uniform temperature change" are not very realistic; still, this simplified model provides a useful reminder about the transformation of mechanical energy into thermal energy when nonconservative forces are present.

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