A Tale o Frcton Basc Rollercoaster Physcs Fahrenhet Rollercoaster, Hershey, PA max heght = 11 t max speed = 58 mph
PLAY PLAY PLAY PLAY
Rotatonal Movement Knematcs Smlar to how lnear velocty s dened, angular velocty s the angle swept by unt o tme. Tangental velocty s the equvalent o lnear velocty or a partcle movng on a crcumerence. s r v T r d or v T r d a T dv T d d a T r a T r
Rotatonal Knetc Energy and Momentum o Inerta o a Rgd Body For a sngle partcle: Tangental knetc energy: Rotatonal knetc energy: K K 1 mv T 1 I Momentum o nerta: For a system o partcles: Momentum o nerta: For a rgd body: Momentum o nerta: I mr I I r mr dm
Angular Momentum and Torque o a Rgd Body Law o lever: F d Torque: r F Magntude: r F sn Newton s second law: F ma dv m Torque s a measure o how much a orce actng on an object causes that object to rotate. It s ormally dened as a vector comng rom the specal product o the poston vector o the pont o applcaton o the orce, and the orce vector. Its magntude depends on the angle between poston and orce vectors. I these vectors are parallel, the torque s zero.
Angular Momentum and Torque o a Rgd Body Lnear momentum: P mv d For m = constant: dv m ma Angular momentum: L r p m r v I r F : L mr v Force denton: F m v T Denng torque (orce producng rotaton) n a crcular movement (r constant) as the change n tme o the angular moment: dl mr dv T mr a T
Angular Momentum and Torque o a Rgd Body Lnear momentum: P mv d For m = constant: dv m ma Angular momentum: L r p m r v I r F : L mr v Force denton: F m v T Takng a T = r, and makng I = mr : mr a mr T or I
Frcton Force or a Rgd Sphere Rollng on an Inclne The sphere rolls because o the torque produced by the rcton orce s and the weght s component parallel to the nclne: F ma m g sn s and I the sphere s momentum o nerta s I = /5mr and = a/r: s r Wth ths value: 5 mr a r or s s r I ma 5 ma m g sn ma 5 Solvng or a n the above equaton, the acceleraton o the sphere rollng on the nclne s: 5 a g sn
Frcton Force or a Rgd Sphere Rollng on an Inclne Combnng: s 5 ma and 5 a g sn the statc rcton orce s now: s m g sn But by denton, the statc rcton orce s proportonal to the normal orce the body exerts on the surace : s F Takng F n rom the ree-body dagram: s n s m g cos s
Frcton Force or a Rgd Sphere Rollng on an Inclne Combnng the two expressons or s : m g sn s m g sn the coecent o statc rcton can be expressed as: s tan Ths expresson states that the coecent o statc rcton s a uncton o the nclne s angle only, speccally, a uncton o the slope o ths surace.
Frcton Force or a Rgd Sphere Rollng on a Varable Slope Path Let (x) a derentable uncton. I: m dy dx '( x) and then: tan (x) m tan At any pont o a curved path (x), a tangent lne can be vsualzed as a porton o an nclne. The slope m o ths nclne s the tangent o the angle between ths lne and the horzontal, tan. In calculus, ths slope s gven by the value o (x), the dervatve o the uncton (x) at that pont. The coecent o statc rcton s can be expressed as: s '( x) The statc rcton orce s s now: s m g '( x) cos
Frcton Force or a Rgd Sphere Rollng on a Varable Slope Path Because tan = (x), t s possble to dene a rght trangle wth sdes n terms o (x): tan '( x) 1 opposte adjacent s I: m g arctan( '( x)), then: '( x) cos(arctan( '( x))) Usng basc trgonometry: cos adjacent hypotenuse 1 1 ( '( x)) '( x) The statc rcton orce s now: s m g 1 ( '( x))
Frcton Force or a Rgd Sphere Rollng on a Varable Slope Path But, somethng needs to be xed n ths procedure. By denton, the statc rcton coecent s must always be postve, whle the slope o a path may be postve or negatve. So the requred correctons must be: s '( x) s m g '( x) 1 ( '( x)) Where: '( x) denotes the absolute value o the uncton (x)
Work-Energy or a Sphere Rollng on a Varable Slope Path wth Frcton The work-energy theorem states that the mechancal energy (knetc energy + potental energy) o an solated system under only conservatve orces remans constant: E K E U K or K U In a system under non-conservatve orces, lke rcton, the work-energy theorem states that work done by these orces s equvalent to the change n the mechancal energy: W U 0 E K U Addtonally, the work done by non-conservatve orces depends on the path or trajectory o the system, or n the tme these orces aect the system. E
Frcton Force or a Rgd Sphere Rollng on a Varable Slope Path By denton, mechancal work s the product o the dsplacement and the orce component along the dsplacement: For a varable slope path y = (x), the work done by the rcton s over a porton s o the path s: W s s m g '( x) 1 ( '( x)) s For a derental porton o the path: dw m g '( x) 1 ( '( x)) ds
Frcton Force or a Rgd Sphere Rollng on a Varable Slope Path Expressng ds n terms o the derentals dx and dy, the derental arc can be expressed n terms o the (x): ds dy dx dx dy 1 dx 1 '( x) dx The work along the derental porton o the path can be expressed as: dw m g m g '( x) 1 ( '( x)) '( x) 1 ( '( x)) ds 1 ( '( x)) dx dw m g '( x) dx
Frcton Force or a Rgd Sphere Rollng on a Varable Slope Path Because dx > 0, usng propertes o the absolute value and the denton o derental o a uncton: dw m g '( x) dx m g '( x) dx m g d ( x) Frcton orces always acts aganst the movement, so the work done by them must always be negatve: dw m g d ( x)
Takng small dsplacements nstead derentals: ) ( x g m W Usng ths expresson n the work-energy theorem: U K W h g m h g m v m v m x g m 1 1 ) ( Ths expresson relates the work done by rcton wth the mechancal energy o a sphere rollng on a lttle porton o a curved path. Vsualze ths porton as a lttle nclne. Heght h s gven by the uncton (x). Frcton Force or a Rgd Sphere Rollng on a Varable Slope Path
Then, dvdng by m: ) ( ) ( ) ( ) ( 1 1 x g x g v v x x g From ths expresson, we can determne nal velocty at the end o the nclne: ) ( ) ( 4 ) ( ) ( 1 x x g x x g v v Frcton Force or a Rgd Sphere Rollng on a Varable Slope Path
Frcton Force or a Rgd Sphere Rollng on a Varable Slope Path We can approxmate the rcton o a sphercal body on a curved path as the rollng o ths body on a sequence o nclnes. The nal velocty at the end o one nclne s the ntal velocty at the begnnng o the next nclne.
Are you ready to apply what you have learned?