: PHYS 55 (Pat, Topic ) Eample Solutions p. Review of Foce Eample ( ) ( ) What is the dot poduct fo F =,,3 and G = 4,5,6? F G = F G + F G + F G = 4 +... = 3 z z Phs55 -: Foce Fields Review of Foce Eample ( ) What is the magnitude of the vecto F =,3, 3? F = F F + F + F z = + 3 + ( 3) =... Clickes Phs55 -: Foce Fields
: PHYS 55 (Pat, Topic ) Eample Solutions p. Review of Foce Eample 3 Let F = 30 N, F = 0 N, F = 0 N. Find F z. with def. F = F F + F + F z F z = ± F F F = ± 0 N F = ( 0 N ) iˆ + ( 0 N ) ˆj + ( 0 N )kˆ o F = ( 0 N ) iˆ + ( 0 N ) ˆj ( 0 N )kˆ Phs55 -: Foce Fields 3 Review of Foce Eample 4 ( ) ( ) Find the angle between the two vectos F = and F =,,3 4,5,6. F F F F cos θ = = F F F F with F = F F + F F F = F F + F + F z F = F F + F + F = ( ) ( 4) + ( ) ( 5) + ( 3) ( 6) = 3 F + F z F z z 3 cos θ = = 0.9746 77 ( 0.9746) = 0.57 ad.93 deg θ = cos = = Phs55 -: Foce Fields 4
: PHYS 55 (Pat, Topic ) Eample Solutions p. 3 Foce Fields Eample 5 The chat shows the foce vecto acting on a kg mass at a few diffeent positions. The units of foce ae Newtons N. What is the foce on a kg mass located at = 5 metes, = 3 metes? foce vecto fo a kg mass at =5m, =3m is on the map. It has a slope of ½ and a magnitude of.. F( ) F( ) F iˆ F ˆ = 5, = 3 = 5,3 = + j ; slope: = ; magnitude F + F =. F signs fo F and F cannot be found fom these equations the ae obtained fom the vecto on the gaph: foce is acting down and left (neg. and diections) F and F will be negative F Phs55 -: Foce Fields 5 Foce Fields Eample 5 The chat shows the foce vecto acting on a kg mass at a few diffeent positions. The units of foce ae Newtons N. What is the foce on a kg mass located at = 5 metes, = 3 metes? F slope equation ields = F substituting F into the magnitude equation ields F will be negative F = -0.537 N fom slope equation: F = F = -.073 N F 5,3 =.073 N iˆ 0.537 N ˆ ( ) ( ) ( ) j Phs55 -: Foce Fields 6 F ( F ) F =. = ±.44 5 + = ± 0.537
: PHYS 55 (Pat, Topic ) Eample Solutions p. 4 Foce Fields Eample 6 Estimate the foce vecto at =4m, =m. we need to find F( 4,) = F iˆ + F ˆ j with F & F obtained fom 3 sepaated pieces of infomation: signs magnitude slope signs fo F and F obtained fom gaph: diection of foce is down & left (in neg. and diection) F and F ae negative point (4,) is located on a foce line about half wa between points maked with foce magnitudes of 6 and 7 6 N + 7 N magnitude of foce at (4,) is appoimatel the aveage F = = 6. 5 N Phs55 -: Foce Fields 7 Foce Fields Eample 6 Estimate the foce vecto at =4m, =m. we need to find F( 4,) = F iˆ + F ˆ j with F & F obtained fom 3 sepaated pieces of infomation: signs magnitude slope slope of the vecto is found (appoimatel) b dawing a line tangent to the foce line at point (4,) (dashed) and then finding the slope of the line ise Δ 6 0 6 slope of the foce vecto = = = un Δ 9.6 7.4 Phs55 -: Foce Fields 8
: PHYS 55 (Pat, Topic ) Eample Solutions p. 5 Foce Fields Eample 6 Estimate the foce vecto at =4m, =m. with slope = 6 / 7.4 6 F = F 7.4 and magnitude of 6.5 N F ( ) + F = 6. 5 N 6.5 N F = = 5.49 N 6 + 7.4 F = ± 5. 05 N, but sign must be negative (see above) F = 5. 05 N 6 with slope equation: F = F = 4. 09 N 7.4 F 4, 5.05 N iˆ + 4.09 N ˆ F 4, 5.05 iˆ 4.09 ˆj ( ) ( ) ( ) j ( ) ( ) N Phs55 -: Foce Fields 9 Eample 7 A foce with a magnitude of 3 N is applied to an object to slide it acoss a table. How much wok is done b the moving foce, if the object is moved 0.7 m? applied foce is diectl in line with (diection of) the path of the movement wok done on the object is W = F d = (3 N) (0.7 m) =. Nm =. J Phs55 -: Foce Fields 0
: PHYS 55 (Pat, Topic ) Eample Solutions p. 6 Eample 8 3) How much eneg is tansf. f. F to F? Solution (-3): diection of the foce in diection of the movement wok done b F will be positive W = F d = 0 N. m = Nm = J diection of the foce opposite to diection of the movement wok done b F will be negative W = - F d = - (6 N. m) = -7. Nm = -7. J of the J of wok done b F, 7. J ae done on F 7. J of eneg tansfeed fom F to F Phs55 -: Foce Fields Eample 8 4) If the suface of the table is fictionless, how much eneg is used to acceleate the block (i.e. how much eneg is tansfeed fom F to the block)? net foce acceleating the block is F - F eneg tansfeed to the block is W block = ( F - F ) d = W W = 4.8 J Note: the est of the J done b F (J - 4.8J = 7.J) is tansfeed to F, compae 3) Phs55 -: Foce Fields
: PHYS 55 (Pat, Topic ) Eample Solutions p. 7 Eample 9 A wooden object is dagged m The foce eeted on the ope is 0 N. wok done is W = θ ( F cos ) d = F d movement is m along the floo (positive -ais): d d = iˆ component of F in the diection of d, denoted F d, needs to be found: since d = iˆ : component of foce in i-diection to be found, i.e. Fd = F iˆ, F d = F with F = 0 N and F diection of foce (slope of ope is ½) esults in = F 0 N = F ( 0. 5 ) 0 N + F = F + F F = ± = ± 8. 94 N.5 Phs55 -: Foce Fields 3 Eample 9 A wooden object is dagged m The foce eeted on the ope is 0 N. mathematicall, foce can be positive o negative in this eample, ope foce is positive F = 8.94 N wok done b Obeli on the wooden object is W = Fd d = 8.94 N m = 7. 9 ( )( ) J Phs55 -: Foce Fields
: PHYS 55 (Pat, Topic ) Eample Solutions p. 8 Eample 0 A sail boat is moved fom position (,) = (0m, 0m) to (00m, 30m) b a wind that eets a foce of (0 N, -0 N) [i.e. a foce of (0i -0j) N ] on the sail boat. What is the wok done b the wind on the sail boat? W = F d (vectos incopoate angula infomation) with - foce F = ( 0 N ) iˆ ( 0 N ) ˆj - vecto d (distance between end and stat positions) d = ( 00 m 0 m) iˆ + ( 30 m 0 m) ˆj = ( 00 m) iˆ + ( 30 m) ˆj W = F d = 0 N 00 m + 0 N 30 m W = 000 Nm 300 Nm = 700 J ( )( ) ( ) ( ) Phs55 -: Foce Fields 5 Eample ) What is the gavitational foce on the mass? gavitational foce vecto F g and foce due to ope F peson as in fee bod diagam (a fee bod diagam isolates an object and shows all foces acting on it) fee bod diagam with mass m = 0 kg and gavitational constant on the eath g = 9.8 m / s magintude of gavitational foce vecto F g = m g = 98. N Phs55 -: Foce Fields 6
: PHYS 55 (Pat, Topic ) Eample Solutions p. 9 Eample ) How much foce must Obeli appl to the ope to lift the mass? Obeli must appl enough foce fee bod diagam to ovecome the gavitational foce. the foce must be an infinitesimal amount moe than the magnitude of the gav. foce we sa it must be equal in magnitude to the gavitational foce F peson = F g = 98. N Phs55 -: Foce Fields 7 Eample 3) How much wok is done b Obeli if the mass is hoisted m? foce eeted b Obeli (i.e. b the ope) is positive in the diection assumed in fee bod diagam fee bod diagam foce eeted b Obeli is in the diection of motion of the mass ( m upwads) wok done b Obeli is positive wok done b Obeli is W peson = F peson d = (98. N) ( m) = 96. J Note: Result would be the same if Obeli was pulling the mass veticall up, without pulle. Phs55 -: Foce Fields 8
: PHYS 55 (Pat, Topic ) Eample Solutions p. 0 Eample 4) How much wok is done b the foce due to the gavitational field (gavitational foce)? accoding to fee bod diagam: gavitational foce is in opposite diection to motion of the mass and same magnitude as foce eeted b Obeli fee bod diagam wok done b gavitational foce is negative W g = - F g d = -(98. N) ( m) = -96. J Phs55 -: Foce Fields 9 Eample 5) How much eneg is tansfeed fom Obeli to the mass? 96. J of eneg ae tansfeed fom Obeli to the mass as calculated in 3) fee bod diagam 6) Whee is the eneg that was added to the mass? eneg that was gained b the mass is stoed in the gavitational field Phs55 -: Foce Fields 0
: PHYS 55 (Pat, Topic ) Eample Solutions p. Eample A peson with a mass of 70 kg snowboads down a mountain in a zig-zag fashion. How much wok is done b the gavitational field on the peson if the peson stats at a point on the mountain that is 500 m above the da lodge (vetical distance) & ends at the da lodge? gavitational field is a consevative field wok done does not depend on the path taken wok done onl depends on vetical diffeence fom beginning to end & on the mass the vetical distance tavelled is downwads (in the diection of the gavitational field) wok done b the gav. field is positive W = F g d = (mg) (500m) =... Clickes Phs55 -: Foce Fields Eample 3 A tuck mounted cane is used to lift a 500 kg wecking ball. The cane is mounted to the bed (deck) of the tuck. What is the potential eneg of the wecking ball with espect to the bed of the tuck if the cente of gavit of the ball is 0 m above the bed of the tuck? What is the potential eneg of the ball with espect to the bed of the tuck if the ball is loweed into a hole so that the cente of gavit is m below the bed of the tuck? the bed of the tuck is the efeence point when the ball is a height h above the efeence point, it has a pot. eneg of U = mgh case : m=500 kg, h=0 m case : m=500 kg, h= - m U = (500 kg) (9.8 m / s ) (0 m) = 98. kj U = (500 kg) (9.8 m / s ) (- m) = -9.8 kj Phs55 -: Foce Fields
: PHYS 55 (Pat, Topic ) Eample Solutions p. Eample An object is moved a distance of m b a foce. The object is moved along a cuve stating at a point 0 m fom a efeence point S 0 to a point m fom S 0. The tangential component of the foce is F s ( ) = N How much eneg is tansfeed fom the foce to the object, i.e. how much wok is done b the foce moving the object? ( is the distance fom S 0 in metes) foce is moving object in diection of motion wok will be positive D W = = m Fs ( )d = N d = = D = = 0 m 0 Nm = 60 Nm Phs55 -: Foce Fields 3 Eample 5 How much eneg is equied to move an object with the mass m fom the eath s suface to oute space? fo this eample let s assume oute space is an infinite distance fom the eath gavitational field is a consevative foce field wok does not depend on path, but onl on stat & end points choose a staight line path adiall outwad fom the cente of the eath (F s () known fo this path), with the cente of the eath as the efeence point = W = with - foce ( ) 4.0 0 m Fs ( )d F s = = eath - adius of eath eath = 6.4 0 6 m - distance [m] - mass m [kg] Phs55 -: Foce Fields 4
: PHYS 55 (Pat, Topic ) Eample Solutions p. 3 Eample 5 How much eneg is equied to move an object with the mass m fom the eath s suface to oute space? with all othe vaiables given in SI units (m, kg, s,...), F s () will be in units of Newtons N wok will be in units of Nm = Joules J = 4.0 0 m W = d 6 = 6.4 0 4.0 0 m 4.0 0 m = 6 6.4 0 W 7 = m 6.7 0 J (m is mass in kg) Phs55 -: Foce Fields 5 Eample 6 Geneate the eneg contous fo this foce field. The magnitude of the foce on the foce line labled S is constant at F/m = 4.0 N / kg. The efeence eneg contou should go though A. Othe eneg contous should be 0 J / kg apat. ) daw an eneg contou line though point A, pependicula to the foce lines ) find the distance to the net contou line, measued along a foce line: use foce line S as this is the line whee we know the foce that opposes (aids) the movement: F/m = 4.0 N/kg Let us fist find the eneg contou line that is +0 J / kg with espect to point A Phs55 -: Foce Fields 6
: PHYS 55 (Pat, Topic ) Eample Solutions p. Eample 6 Geneate the eneg contous fo this foce field. The magnitude of the foce on the foce line labled S is constant at F/m = 4.0 N / kg. The efeence eneg contou should go though A. Othe eneg contous should be 0 J / kg apat. ) find the point on foce line S (sa point B) such that it takes an eneg of 0 J to move an object of kg fom points A to B. W = F d with - wok W = 0 J fo a mass of kg - foce opposing the motion F = 4 N pe kg (F = 4N/kg kg = 4N) - distance d along the foce line W 0 J 0 Nm d = = = =.5 m F 4 N 4 N Phs55 -: Foce Fields 7 Eample 6 Geneate the eneg contous fo this foce field. The magnitude of the foce on the foce line labled S is constant at F/m = 4.0 N / kg. The efeence eneg contou should go though A. Othe eneg contous should be 0 J / kg apat. ) wok to be done b moving the kg mass though.5 m is positive if it is against the foce of the field against the foce aows (move mass up and ight) location of point B daw the +0 J / kg eneg contou line though point B (just as the line though point A) Phs55 -: Foce Fields 8
: PHYS 55 (Pat, Topic ) Eample Solutions p. 5 Eample 6 Geneate the eneg contous fo this foce field. The magnitude of the foce on the foce line labled S is constant at F/m = 4.0 N / kg. The efeence eneg contou should go though A. Othe eneg contous should be 0 J / kg apat. 3) constuct the +0 J/kg eneg contou line, the -0 J/kg eneg contou line,... in the same wa Phs55 -: Foce Fields 9 Eample 7 Conside the gavitational foce field of the eath. Using the suface of a sphee with the adius i = 6.4 0 6 m as a efeence, find the adius of the sphee whose suface is at an eneg of 0 6 J/kg above the efeence. I.e., at what adius would be a mass of kg if it is lifted fom the suface of the eath (efeence) outwad until a wok of 0 6 J is done? final adius found b calculating the wok equied to move a kg mass move the mass on a path chosen so that the foce tangent to the path is known (i.e. foce opposing the movement must be known) choose adial path Phs55 -: Foce Fields 30
: PHYS 55 (Pat, Topic ) Eample Solutions p. 6 Eample 7 a kg mass is moved adiall fom an initial value i = 6.4 0 6 m to a final value of = f final value f calculated so that the wok done on the mass is W = 0 6 J gavitational foce is eactl opposite to the chosen adial path of movement gavitational foce [N] on a mass of kg moved metes fom the cente of the eath is with the wok done in moving the mass fom i to F f ( ) 4.0 0 g = f W = = Fg ( )d = i f 4.0 0 4.0 0 = i f 4.0 0 4.0 0 N m = = 4.0 0 4.0 0 N m W 0 6 i 6.4 0 m 6 = 6.504 0 m Phs55 -: Foce Fields 3 6 Nm Eample 8 ) A kg object is moved in a staight line fom A to B, which ae at a distance of m. find the aveage of the component of applied foce tangent to the path (in diection fom A to B) fom map: wok done is the potential eneg of a kg mass at point B with espect to point A W = 0 J 0 J kg J kg kg = 0 b definition, wok is W = F W ave d F ave = with Euclidian distance between A and B d= m d 0 J F ave = = 7. 07 N m Phs55 -: Foce Fields 3
: PHYS 55 (Pat, Topic ) Eample Solutions p. 7 Eample 8 ) Estimate the foce (i.e. F = F iˆ + F ˆj ) the field eets on a kg mass located at point D. daw a field foce line though D to get the diection of the foce estimate the magnitude of the foce using the appopiate aveage foce in the diection of the field line: with - wok W done b the field when kg is moved fom points P (0 J / kg ) to Q (0 J / kg ) - distance d = m = m W - equation deived ealie, F D d magnitude of foce F D appo. 0 N 0 J 0 J kg kg kg 0 J = = m m = 0 N Phs55 -: Foce Fields 33 Eample 8 ) Estimate the foce (i.e. F = F iˆ + F ˆj ) the field eets on a kg mass located at point D. the diection of the foce eeted b the field is the same as the diection of the field foce line diection of the field foce line is alwas fom a point of highe eneg to a point of lowe eneg to sepaate the magnitude of foce F D = 0 N into and components, slope of the field line (ise ove un) must be measued (e.g. with a ule) hee: ise = un = ( / ) m F slope = - = F = - F F fom diection of foce line, F is negative, F is positive Phs55 -: Foce Fields 34
: PHYS 55 (Pat, Topic ) Eample Solutions p. 8 Eample 8 ) Estimate the foce (i.e. F = F iˆ + F ˆj ) the field eets on a kg mass located at point D. with F D = 0 N, F = -F (F negative, F positive), and (definition of the magnitude of a vecto) F D = F + F = F FD F = 7. 07 N since F = -F, = 7.07 iˆ + 7.07 ˆj F D ( ) N Phs55 -: Foce Fields 35 Eample 8 ) Estimate the foce (i.e. F = F iˆ + F ˆj ) the field eets on a kg mass located at point D. Note: this esult is an appoimation because field foce at point D was appoimated with an aveage foce magnitude? potential eos dawing the field line though point D potential angula eos dawing the un and ise diection / components? potential measuement eos of the un and ise Phs55 -: Foce Fields 36