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Solution Manual for Newtonian Tasks Inspired by Physics Education Research: nTIPERs, 1st Edition

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    SOLUTIONS MANUAL NEWTONIAN TASKS INSP IRED BY PHYSICS EDUCATION RE SEARCH n TIPER s Curtis J. Hieggelke Joliet Junior College

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    C ONTENTS Contents........................................................................................................................................................................1 Answering Ranking Tasks ..........................................................................................................................................2 nTex-RT1: Stacked Blocks—Mass of Stack ............................................................................................................2 nTpractice-RT2: Stacked Blocks—Number of Blocks.............................................................................................3 nTpractice-RT3: Stacked Blocks—Average Mass ...................................................................................................4 nT1 Preliminaries ........................................................................................................................................................5 nT1A-RT1: Cutting up a Block—Density ................................................................................................................5 nT1A-WWT2: Cutting up a Block—Density ...........................................................................................................6 nT1A-CCT3: Breaking up a Block—Density...........................................................................................................7 nT1A-QRT4: Slicing up a Block—Mass & Density ................................................................................................8 nT1A-QRT5: Cylindrical Rods with Same Mass—Volume, Area, and Density......................................................9 nT1A-BCT6: Four Blocks—Mass and Density......................................................................................................10 nT1B-CT7: Scale Model Planes—Surface Area and Weight .................................................................................11

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    2 A NSWERING R ANKING T ASKS N T EX -RT1: S TACKED B LOCKS —M ASS OF S TACK Shown below are stacks of various blocks. All masses are given in the diagram in terms of M, the mass of the smallest block. A B M 3 M 5 M C 3 M 3 M D 3 M M E M 3 M 5 M 3 M M Rank the total mass of each stack. Greatest 1 _______ 2 _______ 3 _______ 4 _______5 _______ Least OR, The magnitude of the total mass of each stack is the same but not zero. ___ OR, The magnitude of the total mass of each stack is zero. ___ OR, The ranking for the total mass of each stack cannot be determined. ___ Explain your reasoning. Example answer formats Stacks A and D have a total mass of 9M, C and E have a mass of 4M, and B has a mass of 6M or the ranking is A = D > B> C = E. Thus the ranking task answer should be expressed either as Greatest 1 ___AD__ 2 _______ 3 ___B___ 4 ___CE___5 _______ Least or Greatest 1 ___A___ 2 __D____ 3 ___B___ 4 ___C___5 ___E___ Least Note the order of equals is not important but it is easier if people are encouraged to use alphabetical order when possible. Other alternative formats are AD > B > CE or A = D > B > C = E An alternative but not preferred format is Greatest 1 ___AD__ 2 ___B___ 3 ___CE___4 ______ 5 ______ Least

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    3 N T PRACTICE -RT2: S TACKED B LOCKS —N UMBER OF B LOCKS Shown below are stacks of various blocks. All masses are given in the diagram in terms of M, the mass of the smallest block. A B M 3 M 5 M C 3 M 3 M D 3 M M E M 3 M 5 M 3 M M Rank the total number of blocks in each stack. Greatest 1 ___ A ___ 2 ___ D ___ 3 __ B ____ 4 ___ C ___5 ___ E ___ Least OR, The total number of is the same blocks for each stack. ___ OR, The ranking for the total number of blocks in each stack cannot be determined. ___ Explain your reasoning. There are 3 blocks in cases A and D, and two in all other cases. So a correct ranking is A = D > B = C = E.

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    4 N T PRACTICE -RT3: S TACKED B LOCKS —A VERAGE M ASS Shown below are stacks of various blocks. All masses are given in the diagram in terms of M, the mass of the smallest block. A B M 3 M 5 M C 3 M 3 M D 3 M M E M 3 M 5 M 3 M M Rank the average mass in each stack. Greatest 1 ___ A ___ 2 __ B ____ 3 __ D ____ 4 ___ C ___5 ___ E ___ Least OR, The average mass is the same for each stack. ___ OR, The ranking for the average mass in these stacks cannot be determined. ___ Explain your reasoning. Adding the mass of the individual boxes in each stack we find 9M for cases A and D each with three boxes, 6M for case B with two boxes, and 4M for cases C and E each with two boxes. The average mass in each stack is the total mass of each stack divided by the number of boxes in each stack giving an average of 3M for stacks A, B, and D. The average mass is 2M for stacks C and D. Thus the ranking is A = B = D > C = E.

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    5 N T1 P RELIMINARIES N T1A-RT1: C UTTING UP A B LOCK —D ENSITY A block of material (labeled A in the diagram) with a width w, height h, and thickness t, has a mass of M o distributed uniformly throughout its volume. The block is then cut into three pieces, B, C, and D, as shown. h /3 2 h /3 w /3 2 w /3 A B C D Rank the density of the original block A, piece B, piece C, and piece D. Greatest 1 ______ 2 ______ 3 ______ 4 ______ Least OR, The density is the same for all these pieces. ___ OR, The ranking for the densities cannot be determined. ___ Please explain your reasoning. Answer: The density is the same for all four pieces. Since the mass is uniformly distributed, a piece with half the volume will also have half the mass but the density, the ratio of mass to volume, remains the same.

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    6 N T1A-WWT2: C UTTING UP A B LOCK —D ENSITY A block of material with a width w, height h, and thickness t, has a mass of M o distributed uniformly throughout its volume. The block is then cut into two pieces, A and B, as shown. A student makes the following statement: “The density is calculated by dividing the total mass by the volume. Since the volume is in the denominator, a large volume will give a small density. Therefore the block with the smallest volume, block B, will have the largest density.” What, if anything, is wrong with the above statement? If something is wrong, explain the error and how to correct it. If the statement is correct, explain why. Answer: The statement is incorrect. The smaller block has 1/3 the volume of the original block but also 1/3 the mass, so the ratio of mass to volume is the same as the original block. 2w h t 3 w 3

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    7 N T1A-CCT3: B REAKING UP A B LOCK —D ENSITY A block of material with a width w, height h, and thickness t, has a mass of M o distributed uniformly throughout its volume. The block is then broken into two pieces, A and B, as shown. Three students make the following statements: Andy: “The density is the mass divided by the volume, and the volume of B is smaller. Since the mass is uniform and the volume is in the denominator, the density is larger for B.” Badu: “The density of piece A is larger than the density of piece B since A is larger, thus it has more mass.” Coen: “They both have the same density. It’s still the same material.” Which, if any, of these three students do you agree with? Andy_____ Badu _____ Coen _____ None of them______ Please explain your reasoning. Answer: Coen is correct. Density is the ratio of mass to volume. The larger piece has twice the mass but also twice the volume of the smaller piece. Therefore the ratio of mass to volume is the same for both and the same as the density of the unbroken block. 2w h t 3 w 3

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    8 N T1A-QRT4: S LICING UP A B LOCK —M ASS & D ENSITY The block of material shown below has a length L o and a volume V o . An overall mass of M o is spread uniformly throughout the volume of the block to give a density ρ o and a linear density (in the direction of the measured length L o ) of λ o . V o M o ρ o L o Three possible ways to slice the block into unequal pieces are shown below. In each case, the larger piece has a volume 2 V o /3 and the smaller piece has a volume V o /3. E F V o 3 2 V o 3 C D V o 3 2 V o 3 A B V o 3 2 V o 3 Fill in the table below with the quantities indicated for the pieces of the block labeled A – F in terms of the variables M o , λ o , and ρ o . 1 3 λ o Mass per unit length Mass per unit volume λ o 2 3 λ o 1 3 λ o 2 3 λ o ρ o λ o λ o ρ o ρ o ρ o ρ o ρ o ρ o 1 3 M o M o 2 3 M o 1 3 M o 2 3 M o 2 3 M o 1 3 M o Original block Piece A Piece B Piece C Piece D Piece E Piece F Mass

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    9 N T1A-QRT5: C YLINDRICAL R ODS WITH S AME M ASS —V OLUME , A REA , AND D ENSITY Two cylindrical rods are shown. Rod A has a height H and a radius R and rod B has a height 2H and a radius 2R. Both rods have the same total mass. Rod A has a density ρ A and volume V A . (a) What is the volume of rod B in terms of the volume of rod A? (Your answer should look like V B = n V A , where n is some number.) Please explain. The volume of Rod B = 8 times the volume of Rod A since the volume is proportional to the product of the area (which is proportional to the square of the radius) and the height of the cylinder, and the radius and the height have doubled. (b) What is the surface area of rod B in terms of the surface area of rod A? (Your answer should look like SA B = n SA A , where n is some number.) Please explain. SA B = 4 SA A , The surface of each rod is made up of the circles at the top and bottom and the ‘side wall.’ The top and bottom of the rod are circles, and the area of a circle is proportional to the square of the radius. Since the radius is twice as big for Rod B as for Rod A, the area of the top and bottom is 4 times as big for Rod B. The ‘side wall’ of the cylinder has an area equal to the circumference times the height, and the circumference of a circle is proportional to the radius. Since both height and radius have doubled, the area of the side wall has also quadrupled. The overall surface area is 4 times as large for Rod B as for Rod A. (c) What is the density of Rod B in terms of the density of Rod A? (Your answer should look like ρ B = n ρ A , where n is some number.) Please explain. ρ A /8 Since B has 8 times the volume of A with the same total mass, its density will be one-eighth that of A. (d) What is the mass per unit length of Rod B to the mass per unit length of Rod A? (Your answer should look like λ B = n λ A , where λ is the mass per unit length and n is some number.) Please explain. Rod B has 8 times the mass of A and twice the length so the mass per unit length of Rod B is 4 times the density of A. A Radius R, Height H B Radius 2 R, Height 2 H

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    10 N T1A-BCT6: F OUR B LOCKS —M ASS AND D ENSITY The block of material shown to the right has a volume V o . An overall mass M o is spread evenly throughout the volume of the block so that the block has a uniform density ρ o . For each block shown below, the volume is given as well as either the mass or the density of the block. 2 V o 2 M o A 2 V o M o B 2 V o ρ o C 2 V o 2 ρ o D Construct two bar charts for the mass and density for the four blocks labeled A – D and for the pieces of the blocks if they were cut in half labeled A/2 – D/2. The mass and density for the original block is shown to set the scale of the chart. (Block B)/2 (Block C)/2 (Block A)/2 (Block D)/2 Mass Density Block A Block B Block C Block D Original block (Block B)/2 (Block C)/2 (Block A)/2 (Block D)/2 Block A Block B Block C Block D Original block Please explain. Answer The mass of blocks A and B are given. The mass of block C must be 2M o , since it has the same density as the original block and twice the volume. The mass of block D must be 4 times the mass of the original block, since it has twice the volume and twice the density and mass is volume times density. The mass of one-half of a block will be half the mass of the full block. The density of one-half of a block will be the same as the density of a full block, since the volume is halved as well as the mass, and the density is the ratio of these quantities. V o M o ρ o

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    11 N T1B-CT7: S CALE M ODEL P LANES —S URFACE A REA AND W EIGHT A woodworker has made four small airplanes and one large airplane. All airplanes are exactly the same shape, and all are made from the same kind of wood. The larger plane is twice as large in every dimension as one of the smaller planes. The planes are to be painted and then shipped as gifts. Case B Case A a) The amount of paint required to paint the planes is directly proportional to the surface area. Will the amount of paint required for the single plane in Case A be greater than, less than, or equal to the total amount of paint required for all four planes in Case B? Please explain your reasoning. Answer: Equal to or the same. Since the larger plane is twice as big in every dimension, any feature of the surface of the larger plane will have double the width and double the height of the equivalent piece of surface of the smaller plane. So the area of any piece of surface of the larger plane is four times the area of the equivalent surface of the smaller plane. Thus, the total surface area of four small planes is the same as the total surface area of one large plane. b) The shipping cost for the planes is proportional to the weight. Will the weight of the single plane in Case A be greater than, less than, or equal to the total weight of all four planes in Case B? Please explain your reasoning. Answer: Greater than. Since the larger plane is twice as big in every dimension, any piece of the larger plane will have double the width, double the height, and double the length of the equivalent piece of the smaller plane. So the volume of any piecelarger plane is eight times the volume of the equivalent piece of the smaller plane, and will weigh eight times as much. Thus, the weight of four small planes is only half the weight of one large plane.

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    1 C ONTENTS ( BOOK PAGES 7-25) Contents ....................................................................................................................................................................... 1 nT2 Vectors.................................................................................................................................................................. 2 nT2A-QRT1: Vectors on a Grid I—Magnitudes...................................................................................................... 2 nT2A-RT2: Vectors on a Grid I—Magnitudes ........................................................................................................ 3 nT2A-QRT3: Vectors on a Grid II—Directions ...................................................................................................... 4 nT2B-CCT4: Adding Two Vectors—Magnitude of the Resultant........................................................................... 5 nT2B-QRT5: Vectors on a Grid III—Graphical Representation of Sum................................................................. 6 nT2B-RT6: Vectors I—Resultant Magnitudes of Adding Two Vectors.................................................................. 7 nT2B-CCT7: Combining Two Vectors—Resultant................................................................................................. 8 nT2B-CT8: Combining Vectors—Magnitude of Resultant ..................................................................................... 9 nT2B-QRT9: Vector Combination II—Direction of Resultant.............................................................................. 10 nT2C-CCT10: Two Vectors—Vector Difference .................................................................................................. 11 nT2C-CT11: Two Vectors—Vector Sum and Difference...................................................................................... 12 nT2C-RT12: Addition and Subtraction of Three Vectors I—Magnitude............................................................... 13 nT2C-RT13: Addition and Subtraction of Three Vectors II—Direction of Resultant ........................................... 14 nT2C-RT14: Addition and Subtraction of Three Vectors II—Magnitude of Resultant ......................................... 15 nT2D-QRT15: Vector Combinations III—Components of the Resultant Vector .................................................. 16 nT2D-QRT16: Force Vectors—Properties of Components ................................................................................... 17 nT2D-QRT17: Velocity Vectors—Properties of Components............................................................................... 18 nT2D-CCT18: Vector—Resolution into Components ........................................................................................... 19 nT2D-CT19: Vector on Rotated Axes—Components............................................................................................ 20 nT2D-CCT20: Vector Components—Resultant Vectors ....................................................................................... 21 nT2E-WBT21: Calculations with Four Vectors—Vector Operation ..................................................................... 22 nT2E-WBT22: Vectors on a Grid—Scalar (Dot) Product Expression................................................................... 23 nT2E-WBT23: Vectors on a Grid—Vector (Cross) Product ................................................................................. 24 nT2E-QRT24: Vectors on a Grid—Product Expressions That Are Zero............................................................... 25

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    nTIPERs 2 N T2 V ECTORS N T2A-QRT1: V ECTORS ON A G RID I—M AGNITUDES Eight vectors are shown below superimposed on a grid. A E H B C F G D G G G G G G G G (a) List all of the vectors that have the same magnitude as vector G A . Answer: D and G since their magnitudes are 3 units like A. (b) List all of the vectors that have the same magnitude as vector G B . Answer: None of them, since none of the other vectors have length 5 units. (c) List all of the vectors that have the same magnitude as vector G C . Answer: F since its magnitude is 4 units like C. (d) List all of the vectors that have the same magnitude as vector G D . Answer: A and G since their magnitudes are 3 units like D. (e) List all of the vectors that have the same magnitude as vector − G A . Answer: D and G since their magnitudes are 3 units like A. Vector –A has the same length as vector A and so has the same magnitude. (f) List all of the vectors that have the same magnitude as vector − G B . Answer: None of them, since none of the other vectors have length 5 units. (g) List all of the vectors that have the same magnitude as vector − G C . Answer: F since its magnitude is 4 units like C and like –C. (h) List all of the vectors that have the same magnitude as vector − G D . Answer: A and G since their magnitudes are 3 units like D and like –D.

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    3 N T2A-RT2: V ECTORS ON A G RID I—M AGNITUDES Eight vectors are shown below superimposed on a grid. A E H B C F G D G G G G G G G G Rank the magnitudes of the vectors. Greatest 1_______ 2_______ 3_______ 4_______ 5_______ 6_______ 7_______ 8______ Least OR, All of these vectors have the same magnitude. ____ OR, We cannot determine the ranking for the magnitudes of the vectors. ____ Please explain your reasoning. Answer: B > F = C > A = D = G > E = H based on the relative lengths of the vectors, which represent their magnitudes.

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