CHAPTER 5 Forces in Two Dimensions

1 Physics: Principles and ProblemsSolutions Manual87 Copyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, in Two DimensionsCHAPTERP ractice 119 125page car is driven km due west, then km due south. What is the magnitude of its displacement? Solve this problem both graphically and mathematically, and check your answers against each !A2"B2R!!A2"B"2"!!( "km)2""(12" km")2"!141 shoppers walk from the door of the mall to their car, which is m down a lane of cars, and then turn 90 to the right and walk an additional m.

2 What is the magnitude of the displacement of the shoppers car from the mall door? Solve this problem both graphically and mathematically, and check your answers against each !A2"B2R!!A2"B"2"!!( "m)2""( "m)2"!257 hiker walks km in one direction, then makes a 45 turn to the right and walks another km. What is the magnitude of her displacement?R2!A2"B2#2 ABcos !R!!A2"B"2#2A"Bcos"!"!!( k"m)2""( k"m)2#"2( "km)(6".4 km")(cos"135 )"! $ ant is crawling on the sidewalk. At one moment, it is moving south a distance of mm. It then turns southwest and crawls mm.

3 What is the magnitude of the ant s displacement?R2!A2"B2#2 ABcos !R!!A2""B2#"2 ABc"os !"!!( m"m)2""( m"m)2#"2( "mm)(" m"m)(co"s 135" )"! m257 km141 kmpage 125 Solve problems 5 10 algebraically. You may also choose to solve some of them graphically to check your walks km in a direction west of north, then goes km due west. What is his displacement?Identify north and west as the positive !d1sin !!( km)(sin ) ! km d1N!d1cos !!( km)(cos ) ! km d2W! km d2N! kmRW!d1W"d2W! km " km ! kmRN!d1N"d2N! km " km !

4 KmR!!RW2""RN2"!!( "km)2""( "km)2"! km!!tan#1!"!tan#1!! !"!77 R! km at 77 west of and Chrissy are going to sleep overnight in their tree house and are using some ropes to pull up a box containing their pillows and blankets, which have a total mass of kg. The girls stand on different branches, as shown in Figure 5-6,and pull at the angles and with the Forces indicated. Find the x- and y-components of the net force on the box. Hint: Draw a free-body diagram so that you do not leave out a up and right as on box,x!FA on boxcos !

5 A!( N)(cos 120 )!# NFA on box,y!FA on boxsin !A!( N)(sin 120 )! N FC on box,x!FC on boxcos !A!( N)(cos 55 )RW!RN88 Solutions ManualPhysics: Principles and ProblemsCopyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, 5 continued Figure 5-6 Physics: Principles and ProblemsSolutions Manual89 Copyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.! NFC on box,y!FC on boxsin !A!( N)(sin 55 )! NFg,x! NFg,y!#mg!#( kg)( m/s2) !# NFnet on box,x!FA on box,x"FC on box,x"Fg,x!# N " N " N !

6 NFnet on box,y!FA on box,y"FC on box,y"Fg,y! N " N # N ! NThe net force is N in the first walk km north from home, thenwalk east until your displacement from homeis km. How far east did you walk?The resultant is km. Using thePythagorean Theorem, the distance eastis R2!A2"B2,soB!!R2#"A2"!!( "km)2"#( "km)2"! child s swing is held up by two ropes tiedto a tree branch that hangs from thevertical. If the tension in each rope is N,what is the combined force (magnitude anddirection) of the two ropes on the swing?

7 The force will be straight up. Becausethe angles are equal, the horizontalforces will be equal and opposite andcancel out. The magnitude of thisvertical force is Fcombined!Frope1 on swingcos !"Frope2 on swingcos !!2 Frope2 on swingcos !!(2)( N)(cos )! N a vector ever be shorter than one ofits components? Equal in length to one ofits components? could never be shorter than one of itscomponents, but if it lies along eitherthe x- or y-axis, then one of its compo-nents equals its a coordinate system in which the x-axis iseast, for what range of angles is the x-compo-nent positive?

8 For what range is it negative?The x-component is positive for anglesless than 90 and for angles greaterthan 270 . It s negative for anglesgreater than 90 but less than 270 .Section 119 125page v. DisplacementIs the distancethat you walk equal to the magnitude ofyour displacement? Give an example thatsupports your necessarily. For example, you couldwalk around the block (one km perside). Your displacement would be zero,but the distance that you walk would be4 DifferenceSubtract vector Kfromvector L,shown in Figure 5 continued Figure #(# )!

9 To the the components of vector M,shown in Figure 5-7. Mx!m cos !!( )(cos )! to the rightMy!msin !!( )(sin )! SumFind the sum of the three vectors shown in Figure 5-7. Rx!Kx"Lx"Mx!# " " ! !Ky"Ly"My! " " ! !!Rx2""Ry2"!! "" "! !!tan#1!!RRyx!"!tan#1!!36!"!27 R! at 27 OperationsThe order inwhich vectors are added does not say that vector addition iscommutative. Which ordinary arithmeticoperations are commutative? Which are not?Addition and multiplication are commu-tative. Subtraction and division are ThinkingA box is moved throughone displacement and then through a sec-ond displacement.

10 The magnitudes of thetwo displacements are unequal. Could thedisplacements have directions such that theresultant displacement is zero? Suppose thebox was moved through three displace-ments of unequal magnitude. Could theresultant displacement be zero? Supportyour conclusion with a , but if there are three displacements, the sum canbe zero if the three vectorsform a triangle when theyare placed tip-to-tail. Also,the sum of three displace-ments can be zero withoutforming a triangle if the sum of two displacements in one direction equalsthe third in the opposite 126 130page girl exerts a 36-N horizontal force as shepulls a 52-N sled across a cement sidewalkat constant speed.