X2 T06 06 banked curves (2011)
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Transcript of X2 T06 06 banked curves (2011)
![Page 1: X2 T06 06 banked curves (2011)](https://reader034.fdocuments.net/reader034/viewer/2022042700/5563bf2cd8b42ac70d8b578c/html5/thumbnails/1.jpg)
Motion Around A Banked Curve
![Page 2: X2 T06 06 banked curves (2011)](https://reader034.fdocuments.net/reader034/viewer/2022042700/5563bf2cd8b42ac70d8b578c/html5/thumbnails/2.jpg)
Motion Around A Banked Curve
![Page 3: X2 T06 06 banked curves (2011)](https://reader034.fdocuments.net/reader034/viewer/2022042700/5563bf2cd8b42ac70d8b578c/html5/thumbnails/3.jpg)
Motion Around A Banked Curve
![Page 4: X2 T06 06 banked curves (2011)](https://reader034.fdocuments.net/reader034/viewer/2022042700/5563bf2cd8b42ac70d8b578c/html5/thumbnails/4.jpg)
Motion Around A Banked Curve
N
![Page 5: X2 T06 06 banked curves (2011)](https://reader034.fdocuments.net/reader034/viewer/2022042700/5563bf2cd8b42ac70d8b578c/html5/thumbnails/5.jpg)
Motion Around A Banked Curve
mg
N
![Page 6: X2 T06 06 banked curves (2011)](https://reader034.fdocuments.net/reader034/viewer/2022042700/5563bf2cd8b42ac70d8b578c/html5/thumbnails/6.jpg)
Motion Around A Banked Curve
mg
N
rmv2
forces horizontal
![Page 7: X2 T06 06 banked curves (2011)](https://reader034.fdocuments.net/reader034/viewer/2022042700/5563bf2cd8b42ac70d8b578c/html5/thumbnails/7.jpg)
Motion Around A Banked Curve
mg
N
rmv2
forces horizontal
![Page 8: X2 T06 06 banked curves (2011)](https://reader034.fdocuments.net/reader034/viewer/2022042700/5563bf2cd8b42ac70d8b578c/html5/thumbnails/8.jpg)
Motion Around A Banked Curve
mg
N
rmv2
forces horizontal
![Page 9: X2 T06 06 banked curves (2011)](https://reader034.fdocuments.net/reader034/viewer/2022042700/5563bf2cd8b42ac70d8b578c/html5/thumbnails/9.jpg)
Motion Around A Banked Curve
mg
N
rmv2
forces horizontal
sinN
![Page 10: X2 T06 06 banked curves (2011)](https://reader034.fdocuments.net/reader034/viewer/2022042700/5563bf2cd8b42ac70d8b578c/html5/thumbnails/10.jpg)
Motion Around A Banked Curve
mg
N
rmv2
forces horizontal
sinN
rmvN
2
sin
![Page 11: X2 T06 06 banked curves (2011)](https://reader034.fdocuments.net/reader034/viewer/2022042700/5563bf2cd8b42ac70d8b578c/html5/thumbnails/11.jpg)
Motion Around A Banked Curve
mg
N
rmv2
forces horizontal
sinN
rmvN
2
sin
0forcesvertical
![Page 12: X2 T06 06 banked curves (2011)](https://reader034.fdocuments.net/reader034/viewer/2022042700/5563bf2cd8b42ac70d8b578c/html5/thumbnails/12.jpg)
Motion Around A Banked Curve
mg
N
rmv2
forces horizontal
sinN
rmvN
2
sin
0forcesvertical
![Page 13: X2 T06 06 banked curves (2011)](https://reader034.fdocuments.net/reader034/viewer/2022042700/5563bf2cd8b42ac70d8b578c/html5/thumbnails/13.jpg)
Motion Around A Banked Curve
mg
N
rmv2
forces horizontal
sinN
rmvN
2
sin
0forcesvertical
mgcosN
![Page 14: X2 T06 06 banked curves (2011)](https://reader034.fdocuments.net/reader034/viewer/2022042700/5563bf2cd8b42ac70d8b578c/html5/thumbnails/14.jpg)
Motion Around A Banked Curve
mg
N
rmv2
forces horizontal
sinN
rmvN
2
sin
0forcesvertical
mgcosN
mgNmgN
cos0cos
![Page 15: X2 T06 06 banked curves (2011)](https://reader034.fdocuments.net/reader034/viewer/2022042700/5563bf2cd8b42ac70d8b578c/html5/thumbnails/15.jpg)
rgv
mgrmv
2
2 1tan
![Page 16: X2 T06 06 banked curves (2011)](https://reader034.fdocuments.net/reader034/viewer/2022042700/5563bf2cd8b42ac70d8b578c/html5/thumbnails/16.jpg)
rgv
mgrmv
2
2 1tan
e.g. (i) A railway line has been constructed around a circular curve of radius 400m.The distance between the rails is 1.5m and the outside rail is 0.08m above the inside rail.Find the most favourable speed (the speed that eliminates a sideways force on the wheels) for a train on this curve.
![Page 17: X2 T06 06 banked curves (2011)](https://reader034.fdocuments.net/reader034/viewer/2022042700/5563bf2cd8b42ac70d8b578c/html5/thumbnails/17.jpg)
rgv
mgrmv
2
2 1tan
e.g. (i) A railway line has been constructed around a circular curve of radius 400m.The distance between the rails is 1.5m and the outside rail is 0.08m above the inside rail.Find the most favourable speed (the speed that eliminates a sideways force on the wheels) for a train on this curve.
![Page 18: X2 T06 06 banked curves (2011)](https://reader034.fdocuments.net/reader034/viewer/2022042700/5563bf2cd8b42ac70d8b578c/html5/thumbnails/18.jpg)
rgv
mgrmv
2
2 1tan
e.g. (i) A railway line has been constructed around a circular curve of radius 400m.The distance between the rails is 1.5m and the outside rail is 0.08m above the inside rail.Find the most favourable speed (the speed that eliminates a sideways force on the wheels) for a train on this curve.
1.5m
![Page 19: X2 T06 06 banked curves (2011)](https://reader034.fdocuments.net/reader034/viewer/2022042700/5563bf2cd8b42ac70d8b578c/html5/thumbnails/19.jpg)
rgv
mgrmv
2
2 1tan
e.g. (i) A railway line has been constructed around a circular curve of radius 400m.The distance between the rails is 1.5m and the outside rail is 0.08m above the inside rail.Find the most favourable speed (the speed that eliminates a sideways force on the wheels) for a train on this curve.
1.5m
0.08m
![Page 20: X2 T06 06 banked curves (2011)](https://reader034.fdocuments.net/reader034/viewer/2022042700/5563bf2cd8b42ac70d8b578c/html5/thumbnails/20.jpg)
rgv
mgrmv
2
2 1tan
e.g. (i) A railway line has been constructed around a circular curve of radius 400m.The distance between the rails is 1.5m and the outside rail is 0.08m above the inside rail.Find the most favourable speed (the speed that eliminates a sideways force on the wheels) for a train on this curve.
1.5m
0.08m
![Page 21: X2 T06 06 banked curves (2011)](https://reader034.fdocuments.net/reader034/viewer/2022042700/5563bf2cd8b42ac70d8b578c/html5/thumbnails/21.jpg)
rgv
mgrmv
2
2 1tan
e.g. (i) A railway line has been constructed around a circular curve of radius 400m.The distance between the rails is 1.5m and the outside rail is 0.08m above the inside rail.Find the most favourable speed (the speed that eliminates a sideways force on the wheels) for a train on this curve.
1.5m
0.08m
N
![Page 22: X2 T06 06 banked curves (2011)](https://reader034.fdocuments.net/reader034/viewer/2022042700/5563bf2cd8b42ac70d8b578c/html5/thumbnails/22.jpg)
rgv
mgrmv
2
2 1tan
e.g. (i) A railway line has been constructed around a circular curve of radius 400m.The distance between the rails is 1.5m and the outside rail is 0.08m above the inside rail.Find the most favourable speed (the speed that eliminates a sideways force on the wheels) for a train on this curve.
1.5m
0.08m
N
mg
![Page 23: X2 T06 06 banked curves (2011)](https://reader034.fdocuments.net/reader034/viewer/2022042700/5563bf2cd8b42ac70d8b578c/html5/thumbnails/23.jpg)
rgv
mgrmv
2
2 1tan
e.g. (i) A railway line has been constructed around a circular curve of radius 400m.The distance between the rails is 1.5m and the outside rail is 0.08m above the inside rail.Find the most favourable speed (the speed that eliminates a sideways force on the wheels) for a train on this curve.
1.5m
0.08m
N
mg
![Page 24: X2 T06 06 banked curves (2011)](https://reader034.fdocuments.net/reader034/viewer/2022042700/5563bf2cd8b42ac70d8b578c/html5/thumbnails/24.jpg)
rmv2
forces horizontal
![Page 25: X2 T06 06 banked curves (2011)](https://reader034.fdocuments.net/reader034/viewer/2022042700/5563bf2cd8b42ac70d8b578c/html5/thumbnails/25.jpg)
rmv2
forces horizontal
![Page 26: X2 T06 06 banked curves (2011)](https://reader034.fdocuments.net/reader034/viewer/2022042700/5563bf2cd8b42ac70d8b578c/html5/thumbnails/26.jpg)
rmv2
forces horizontal
sinN
![Page 27: X2 T06 06 banked curves (2011)](https://reader034.fdocuments.net/reader034/viewer/2022042700/5563bf2cd8b42ac70d8b578c/html5/thumbnails/27.jpg)
rmv2
forces horizontal
sinN
rmvN
2
sin
![Page 28: X2 T06 06 banked curves (2011)](https://reader034.fdocuments.net/reader034/viewer/2022042700/5563bf2cd8b42ac70d8b578c/html5/thumbnails/28.jpg)
rmv2
forces horizontal
sinN
rmvN
2
sin
0forcesvertical
![Page 29: X2 T06 06 banked curves (2011)](https://reader034.fdocuments.net/reader034/viewer/2022042700/5563bf2cd8b42ac70d8b578c/html5/thumbnails/29.jpg)
rmv2
forces horizontal
sinN
rmvN
2
sin
0forcesvertical
![Page 30: X2 T06 06 banked curves (2011)](https://reader034.fdocuments.net/reader034/viewer/2022042700/5563bf2cd8b42ac70d8b578c/html5/thumbnails/30.jpg)
rmv2
forces horizontal
sinN
rmvN
2
sin
0forcesvertical
mgcosN
![Page 31: X2 T06 06 banked curves (2011)](https://reader034.fdocuments.net/reader034/viewer/2022042700/5563bf2cd8b42ac70d8b578c/html5/thumbnails/31.jpg)
rmv2
forces horizontal
sinN
rmvN
2
sin
0forcesvertical
mgcosN
mgNmgN
cos0cos
![Page 32: X2 T06 06 banked curves (2011)](https://reader034.fdocuments.net/reader034/viewer/2022042700/5563bf2cd8b42ac70d8b578c/html5/thumbnails/32.jpg)
rmv2
forces horizontal
sinN
rmvN
2
sin
0forcesvertical
mgcosN
mgNmgN
cos0cos
rgv
mgrmv
2
2 1tan
![Page 33: X2 T06 06 banked curves (2011)](https://reader034.fdocuments.net/reader034/viewer/2022042700/5563bf2cd8b42ac70d8b578c/html5/thumbnails/33.jpg)
rmv2
forces horizontal
sinN
rmvN
2
sin
0forcesvertical
mgcosN
mgNmgN
cos0cos
rgv
mgrmv
2
2 1tan
754
5.108.0sinBut
![Page 34: X2 T06 06 banked curves (2011)](https://reader034.fdocuments.net/reader034/viewer/2022042700/5563bf2cd8b42ac70d8b578c/html5/thumbnails/34.jpg)
rmv2
forces horizontal
sinN
rmvN
2
sin
0forcesvertical
mgcosN
mgNmgN
cos0cos
rgv
mgrmv
2
2 1tan
754
5.108.0sinBut
56094tan
![Page 35: X2 T06 06 banked curves (2011)](https://reader034.fdocuments.net/reader034/viewer/2022042700/5563bf2cd8b42ac70d8b578c/html5/thumbnails/35.jpg)
rmv2
forces horizontal
sinN
rmvN
2
sin
0forcesvertical
mgcosN
mgNmgN
cos0cos
rgv
mgrmv
2
2 1tan
754
5.108.0sinBut
56094tan
56094
8.9400
2
v
![Page 36: X2 T06 06 banked curves (2011)](https://reader034.fdocuments.net/reader034/viewer/2022042700/5563bf2cd8b42ac70d8b578c/html5/thumbnails/36.jpg)
rmv2
forces horizontal
sinN
rmvN
2
sin
0forcesvertical
mgcosN
mgNmgN
cos0cos
rgv
mgrmv
2
2 1tan
754
5.108.0sinBut
56094tan
56094
8.9400
2
v
km/h52m/s47.145609
8.940042
vv
v
![Page 37: X2 T06 06 banked curves (2011)](https://reader034.fdocuments.net/reader034/viewer/2022042700/5563bf2cd8b42ac70d8b578c/html5/thumbnails/37.jpg)
(ii) (1995)A particle of mass m travels at a constant speed v round a circular track of radius R, centre C. The track is banked inwards at an angle , and the particle does not move up or down the bank.
The reaction exerted by the track on the particle has a normal component N, and a component F due to friction, directed up or down the bank. The force F lies in the range to , where is a positive constant and N is the normal component; the sign of F is positive when F is directed up the bank.
N N
The acceleration due to gravity is g. The acceleration related to the circular motion is of magnitude and is directed towards the centre of the track. R
v2
![Page 38: X2 T06 06 banked curves (2011)](https://reader034.fdocuments.net/reader034/viewer/2022042700/5563bf2cd8b42ac70d8b578c/html5/thumbnails/38.jpg)
a) By resolving forces horizontally and vertically, show that;
sincoscossin2
FNFN
Rgv
![Page 39: X2 T06 06 banked curves (2011)](https://reader034.fdocuments.net/reader034/viewer/2022042700/5563bf2cd8b42ac70d8b578c/html5/thumbnails/39.jpg)
a) By resolving forces horizontally and vertically, show that;
sincoscossin2
FNFN
Rgv
![Page 40: X2 T06 06 banked curves (2011)](https://reader034.fdocuments.net/reader034/viewer/2022042700/5563bf2cd8b42ac70d8b578c/html5/thumbnails/40.jpg)
a) By resolving forces horizontally and vertically, show that;
sincoscossin2
FNFN
Rgv
Rmv2
forces horizontal
![Page 41: X2 T06 06 banked curves (2011)](https://reader034.fdocuments.net/reader034/viewer/2022042700/5563bf2cd8b42ac70d8b578c/html5/thumbnails/41.jpg)
a) By resolving forces horizontally and vertically, show that;
sincoscossin2
FNFN
Rgv
Rmv2
forces horizontal
![Page 42: X2 T06 06 banked curves (2011)](https://reader034.fdocuments.net/reader034/viewer/2022042700/5563bf2cd8b42ac70d8b578c/html5/thumbnails/42.jpg)
a) By resolving forces horizontally and vertically, show that;
sincoscossin2
FNFN
Rgv
Rmv2
forces horizontal
sinN cosF
![Page 43: X2 T06 06 banked curves (2011)](https://reader034.fdocuments.net/reader034/viewer/2022042700/5563bf2cd8b42ac70d8b578c/html5/thumbnails/43.jpg)
a) By resolving forces horizontally and vertically, show that;
sincoscossin2
FNFN
Rgv
Rmv2
forces horizontal
sinN
RmvFN
2
cossin
cosF
![Page 44: X2 T06 06 banked curves (2011)](https://reader034.fdocuments.net/reader034/viewer/2022042700/5563bf2cd8b42ac70d8b578c/html5/thumbnails/44.jpg)
a) By resolving forces horizontally and vertically, show that;
sincoscossin2
FNFN
Rgv
Rmv2
forces horizontal
sinN
RmvFN
2
cossin
0forcesvertical
cosF
![Page 45: X2 T06 06 banked curves (2011)](https://reader034.fdocuments.net/reader034/viewer/2022042700/5563bf2cd8b42ac70d8b578c/html5/thumbnails/45.jpg)
a) By resolving forces horizontally and vertically, show that;
sincoscossin2
FNFN
Rgv
Rmv2
forces horizontal
sinN
RmvFN
2
cossin
0forcesvertical
cosF
![Page 46: X2 T06 06 banked curves (2011)](https://reader034.fdocuments.net/reader034/viewer/2022042700/5563bf2cd8b42ac70d8b578c/html5/thumbnails/46.jpg)
a) By resolving forces horizontally and vertically, show that;
sincoscossin2
FNFN
Rgv
Rmv2
forces horizontal
sinN
RmvFN
2
cossin
0forcesvertical
mgcosNcosF sinF
![Page 47: X2 T06 06 banked curves (2011)](https://reader034.fdocuments.net/reader034/viewer/2022042700/5563bf2cd8b42ac70d8b578c/html5/thumbnails/47.jpg)
a) By resolving forces horizontally and vertically, show that;
sincoscossin2
FNFN
Rgv
Rmv2
forces horizontal
sinN
RmvFN
2
cossin
0forcesvertical
mgcosN
mgFNmgFN
sincos0sincoscosF sinF
![Page 48: X2 T06 06 banked curves (2011)](https://reader034.fdocuments.net/reader034/viewer/2022042700/5563bf2cd8b42ac70d8b578c/html5/thumbnails/48.jpg)
a) By resolving forces horizontally and vertically, show that;
sincoscossin2
FNFN
Rgv
Rmv2
forces horizontal
sinN
RmvFN
2
cossin
0forcesvertical
mgcosN
mgFNmgFN
sincos0sincoscosF sinF
mgRmv
Rgv 122
![Page 49: X2 T06 06 banked curves (2011)](https://reader034.fdocuments.net/reader034/viewer/2022042700/5563bf2cd8b42ac70d8b578c/html5/thumbnails/49.jpg)
a) By resolving forces horizontally and vertically, show that;
sincoscossin2
FNFN
Rgv
Rmv2
forces horizontal
sinN
RmvFN
2
cossin
0forcesvertical
mgcosN
mgFNmgFN
sincos0sincoscosF sinF
mgRmv
Rgv 122
sincoscossin2
FNFN
Rgv
![Page 50: X2 T06 06 banked curves (2011)](https://reader034.fdocuments.net/reader034/viewer/2022042700/5563bf2cd8b42ac70d8b578c/html5/thumbnails/50.jpg)
b) Show that the maximum speed at which the particle can travel without slipping up the track is given by;
maxv
tan1tan2
max
Rgv
1tanthat supposemay You
![Page 51: X2 T06 06 banked curves (2011)](https://reader034.fdocuments.net/reader034/viewer/2022042700/5563bf2cd8b42ac70d8b578c/html5/thumbnails/51.jpg)
b) Show that the maximum speed at which the particle can travel without slipping up the track is given by;
maxv
tan1tan2
max
Rgv
1tanthat supposemay You
As it is the friction that resists the particle moving up or down the slope, then if the particle is not slipping up, then friction must be at a maximum in the opposite direction, i.e. NF
![Page 52: X2 T06 06 banked curves (2011)](https://reader034.fdocuments.net/reader034/viewer/2022042700/5563bf2cd8b42ac70d8b578c/html5/thumbnails/52.jpg)
b) Show that the maximum speed at which the particle can travel without slipping up the track is given by;
maxv
tan1tan2
max
Rgv
1tanthat supposemay You
As it is the friction that resists the particle moving up or down the slope, then if the particle is not slipping up, then friction must be at a maximum in the opposite direction, i.e. NF
sincoscossin2
max
NNNN
Rgv
![Page 53: X2 T06 06 banked curves (2011)](https://reader034.fdocuments.net/reader034/viewer/2022042700/5563bf2cd8b42ac70d8b578c/html5/thumbnails/53.jpg)
b) Show that the maximum speed at which the particle can travel without slipping up the track is given by;
maxv
tan1tan2
max
Rgv
1tanthat supposemay You
As it is the friction that resists the particle moving up or down the slope, then if the particle is not slipping up, then friction must be at a maximum in the opposite direction, i.e. NF
sincoscossin2
max
NNNN
Rgv
tan1tan
cossin
coscos
coscos
cossin
![Page 54: X2 T06 06 banked curves (2011)](https://reader034.fdocuments.net/reader034/viewer/2022042700/5563bf2cd8b42ac70d8b578c/html5/thumbnails/54.jpg)
c) Show that if , then the particle will not slide down the track, regardless of its speed.
tan
![Page 55: X2 T06 06 banked curves (2011)](https://reader034.fdocuments.net/reader034/viewer/2022042700/5563bf2cd8b42ac70d8b578c/html5/thumbnails/55.jpg)
c) Show that if , then the particle will not slide down the track, regardless of its speed.
tan
minv is the minimum speed the particle can travel without sliding down the track. In this case friction must be a maximum up the slope i.e. NF
![Page 56: X2 T06 06 banked curves (2011)](https://reader034.fdocuments.net/reader034/viewer/2022042700/5563bf2cd8b42ac70d8b578c/html5/thumbnails/56.jpg)
c) Show that if , then the particle will not slide down the track, regardless of its speed.
tan
minv is the minimum speed the particle can travel without sliding down the track. In this case friction must be a maximum up the slope i.e. NF
tan1tan2
min
Rgv
![Page 57: X2 T06 06 banked curves (2011)](https://reader034.fdocuments.net/reader034/viewer/2022042700/5563bf2cd8b42ac70d8b578c/html5/thumbnails/57.jpg)
c) Show that if , then the particle will not slide down the track, regardless of its speed.
tan
minv is the minimum speed the particle can travel without sliding down the track. In this case friction must be a maximum up the slope i.e. NF
tan1tan2
min
Rgv
;tan If
![Page 58: X2 T06 06 banked curves (2011)](https://reader034.fdocuments.net/reader034/viewer/2022042700/5563bf2cd8b42ac70d8b578c/html5/thumbnails/58.jpg)
c) Show that if , then the particle will not slide down the track, regardless of its speed.
tan
minv is the minimum speed the particle can travel without sliding down the track. In this case friction must be a maximum up the slope i.e. NF
tan1tan2
min
Rgv
;tan If 02min Rgv
![Page 59: X2 T06 06 banked curves (2011)](https://reader034.fdocuments.net/reader034/viewer/2022042700/5563bf2cd8b42ac70d8b578c/html5/thumbnails/59.jpg)
c) Show that if , then the particle will not slide down the track, regardless of its speed.
tan
minv is the minimum speed the particle can travel without sliding down the track. In this case friction must be a maximum up the slope i.e. NF
tan1tan2
min
Rgv
;tan If 02min Rgv
0
0
min
2min
v
v
![Page 60: X2 T06 06 banked curves (2011)](https://reader034.fdocuments.net/reader034/viewer/2022042700/5563bf2cd8b42ac70d8b578c/html5/thumbnails/60.jpg)
c) Show that if , then the particle will not slide down the track, regardless of its speed.
tan
minv is the minimum speed the particle can travel without sliding down the track. In this case friction must be a maximum up the slope i.e. NF
tan1tan2
min
Rgv
;tan If 02min Rgv
0
0
min
2min
v
v
Thus if the minimum velocity the particle can travel without sliding down the track is 0, the particle will not slide down the track, regardless of its speed.
![Page 61: X2 T06 06 banked curves (2011)](https://reader034.fdocuments.net/reader034/viewer/2022042700/5563bf2cd8b42ac70d8b578c/html5/thumbnails/61.jpg)
(iii) (1996)A circular drum is rotating with uniform angular velocity round a horizontal axis. A particle P is rotating in a vertical circle, without slipping, on the inside of the drum.The radius of the drum is r metres and its angular velocity is radians/second. Acceleration due to gravity is g metres/ , and the mass of P is m kilograms.
2second
The centre of the drum is O, and OP makes an angle of to the horizontal.The drum exerts a normal force N on P, as well as frictional force F, acting tangentially to the drum, as shown in the diagram.By resolving forces perpendicular to and parallel to OP, find an expression for in terms of the data.
NF
![Page 62: X2 T06 06 banked curves (2011)](https://reader034.fdocuments.net/reader034/viewer/2022042700/5563bf2cd8b42ac70d8b578c/html5/thumbnails/62.jpg)
![Page 63: X2 T06 06 banked curves (2011)](https://reader034.fdocuments.net/reader034/viewer/2022042700/5563bf2cd8b42ac70d8b578c/html5/thumbnails/63.jpg)
![Page 64: X2 T06 06 banked curves (2011)](https://reader034.fdocuments.net/reader034/viewer/2022042700/5563bf2cd8b42ac70d8b578c/html5/thumbnails/64.jpg)
0forces OP
![Page 65: X2 T06 06 banked curves (2011)](https://reader034.fdocuments.net/reader034/viewer/2022042700/5563bf2cd8b42ac70d8b578c/html5/thumbnails/65.jpg)
0forces OP
![Page 66: X2 T06 06 banked curves (2011)](https://reader034.fdocuments.net/reader034/viewer/2022042700/5563bf2cd8b42ac70d8b578c/html5/thumbnails/66.jpg)
0forces OP
F cosmg
![Page 67: X2 T06 06 banked curves (2011)](https://reader034.fdocuments.net/reader034/viewer/2022042700/5563bf2cd8b42ac70d8b578c/html5/thumbnails/67.jpg)
0forces OP
F0cos Fmg
cosmg
![Page 68: X2 T06 06 banked curves (2011)](https://reader034.fdocuments.net/reader034/viewer/2022042700/5563bf2cd8b42ac70d8b578c/html5/thumbnails/68.jpg)
0forces OP
F0cos Fmg
cosmg
cosmgF
![Page 69: X2 T06 06 banked curves (2011)](https://reader034.fdocuments.net/reader034/viewer/2022042700/5563bf2cd8b42ac70d8b578c/html5/thumbnails/69.jpg)
0forces OP
F0cos Fmg
cosmg
cosmgF 2||forces mrOP
![Page 70: X2 T06 06 banked curves (2011)](https://reader034.fdocuments.net/reader034/viewer/2022042700/5563bf2cd8b42ac70d8b578c/html5/thumbnails/70.jpg)
0forces OP
F0cos Fmg
cosmg
cosmgF 2||forces mrOP
![Page 71: X2 T06 06 banked curves (2011)](https://reader034.fdocuments.net/reader034/viewer/2022042700/5563bf2cd8b42ac70d8b578c/html5/thumbnails/71.jpg)
0forces OP
F0cos Fmg
cosmg
cosmgF 2||forces mrOP
sinmgN
![Page 72: X2 T06 06 banked curves (2011)](https://reader034.fdocuments.net/reader034/viewer/2022042700/5563bf2cd8b42ac70d8b578c/html5/thumbnails/72.jpg)
0forces OP
F0cos Fmg
cosmg
cosmgF 2||forces mrOP
sinmgN2sin mrmgN
![Page 73: X2 T06 06 banked curves (2011)](https://reader034.fdocuments.net/reader034/viewer/2022042700/5563bf2cd8b42ac70d8b578c/html5/thumbnails/73.jpg)
0forces OP
F0cos Fmg
cosmg
cosmgF 2||forces mrOP
sinmgN2sin mrmgN
sin2 mgmrN
![Page 74: X2 T06 06 banked curves (2011)](https://reader034.fdocuments.net/reader034/viewer/2022042700/5563bf2cd8b42ac70d8b578c/html5/thumbnails/74.jpg)
0forces OP
F0cos Fmg
cosmg
cosmgF 2||forces mrOP
sinmgN2sin mrmgN
sin2 mgmrN
sincos
sincos
2
2
grg
NF
mgmrmg
NF
![Page 75: X2 T06 06 banked curves (2011)](https://reader034.fdocuments.net/reader034/viewer/2022042700/5563bf2cd8b42ac70d8b578c/html5/thumbnails/75.jpg)
Resolving Forces Along The Bank
![Page 76: X2 T06 06 banked curves (2011)](https://reader034.fdocuments.net/reader034/viewer/2022042700/5563bf2cd8b42ac70d8b578c/html5/thumbnails/76.jpg)
Resolving Forces Along The Bank
![Page 77: X2 T06 06 banked curves (2011)](https://reader034.fdocuments.net/reader034/viewer/2022042700/5563bf2cd8b42ac70d8b578c/html5/thumbnails/77.jpg)
Resolving Forces Along The Bank
![Page 78: X2 T06 06 banked curves (2011)](https://reader034.fdocuments.net/reader034/viewer/2022042700/5563bf2cd8b42ac70d8b578c/html5/thumbnails/78.jpg)
Resolving Forces Along The Bank
N
![Page 79: X2 T06 06 banked curves (2011)](https://reader034.fdocuments.net/reader034/viewer/2022042700/5563bf2cd8b42ac70d8b578c/html5/thumbnails/79.jpg)
Resolving Forces Along The Bank
mg
N
![Page 80: X2 T06 06 banked curves (2011)](https://reader034.fdocuments.net/reader034/viewer/2022042700/5563bf2cd8b42ac70d8b578c/html5/thumbnails/80.jpg)
Resolving Forces Along The Bank
mg
N
F
![Page 81: X2 T06 06 banked curves (2011)](https://reader034.fdocuments.net/reader034/viewer/2022042700/5563bf2cd8b42ac70d8b578c/html5/thumbnails/81.jpg)
Resolving Forces Along The Bank
mg
N
Fmg
N
F
mg
N
F
![Page 82: X2 T06 06 banked curves (2011)](https://reader034.fdocuments.net/reader034/viewer/2022042700/5563bf2cd8b42ac70d8b578c/html5/thumbnails/82.jpg)
Resolving Forces Along The Bank
mg
N
Fmg
N
F
mg
N
F
![Page 83: X2 T06 06 banked curves (2011)](https://reader034.fdocuments.net/reader034/viewer/2022042700/5563bf2cd8b42ac70d8b578c/html5/thumbnails/83.jpg)
Resolving Forces Along The Bank
mg
N
Fmg
N
F
mg
N
F
r
mv2
![Page 84: X2 T06 06 banked curves (2011)](https://reader034.fdocuments.net/reader034/viewer/2022042700/5563bf2cd8b42ac70d8b578c/html5/thumbnails/84.jpg)
Resolving Forces Along The Bank
mg
N
Fmg
N
F
mg
N
F
r
mv2r
mv2
![Page 85: X2 T06 06 banked curves (2011)](https://reader034.fdocuments.net/reader034/viewer/2022042700/5563bf2cd8b42ac70d8b578c/html5/thumbnails/85.jpg)
Resolving Forces Along The Bank
mg
N
Fmg
N
F
mg
N
F
r
mv2r
mv2
![Page 86: X2 T06 06 banked curves (2011)](https://reader034.fdocuments.net/reader034/viewer/2022042700/5563bf2cd8b42ac70d8b578c/html5/thumbnails/86.jpg)
Resolving Forces Along The Bank
mg
N
Fmg
N
F
mg
N
F
r
mv2r
mv2
cosbank thealong forces2
rmv
![Page 87: X2 T06 06 banked curves (2011)](https://reader034.fdocuments.net/reader034/viewer/2022042700/5563bf2cd8b42ac70d8b578c/html5/thumbnails/87.jpg)
Resolving Forces Along The Bank
mg
N
Fmg
N
F
mg
N
F
r
mv2r
mv2
cosbank thealong forces2
rmv
![Page 88: X2 T06 06 banked curves (2011)](https://reader034.fdocuments.net/reader034/viewer/2022042700/5563bf2cd8b42ac70d8b578c/html5/thumbnails/88.jpg)
Resolving Forces Along The Bank
mg
N
Fmg
N
F
mg
N
F
r
mv2r
mv2
cosbank thealong forces2
rmv
Fsinmg
![Page 89: X2 T06 06 banked curves (2011)](https://reader034.fdocuments.net/reader034/viewer/2022042700/5563bf2cd8b42ac70d8b578c/html5/thumbnails/89.jpg)
Resolving Forces Along The Bank
mg
N
Fmg
N
F
mg
N
F
r
mv2r
mv2
cosbank thealong forces2
rmv
F cossin
2
rmvmgF
sinmg
![Page 90: X2 T06 06 banked curves (2011)](https://reader034.fdocuments.net/reader034/viewer/2022042700/5563bf2cd8b42ac70d8b578c/html5/thumbnails/90.jpg)
Resolving Forces Along The Bank
mg
N
Fmg
N
F
mg
N
F
r
mv2r
mv2
cosbank thealong forces2
rmv
F cossin
2
rmvmgF
sinmg
sincos2
mgr
mvF
![Page 91: X2 T06 06 banked curves (2011)](https://reader034.fdocuments.net/reader034/viewer/2022042700/5563bf2cd8b42ac70d8b578c/html5/thumbnails/91.jpg)
Resolving Forces Along The Bank
mg
N
Fmg
N
F
mg
N
F
r
mv2r
mv2
cosbank thealong forces2
rmv
F cossin
2
rmvmgF
sinmg
sincos2
mgr
mvF
sinbank the forces2
rmv
![Page 92: X2 T06 06 banked curves (2011)](https://reader034.fdocuments.net/reader034/viewer/2022042700/5563bf2cd8b42ac70d8b578c/html5/thumbnails/92.jpg)
Resolving Forces Along The Bank
mg
N
Fmg
N
F
mg
N
F
r
mv2r
mv2
cosbank thealong forces2
rmv
F cossin
2
rmvmgF
sinmg
sincos2
mgr
mvF
sinbank the forces2
rmv
![Page 93: X2 T06 06 banked curves (2011)](https://reader034.fdocuments.net/reader034/viewer/2022042700/5563bf2cd8b42ac70d8b578c/html5/thumbnails/93.jpg)
Resolving Forces Along The Bank
mg
N
Fmg
N
F
mg
N
F
r
mv2r
mv2
cosbank thealong forces2
rmv
F cossin
2
rmvmgF
sinmg
sincos2
mgr
mvF
sinbank the forces2
rmv
cosmg
N
![Page 94: X2 T06 06 banked curves (2011)](https://reader034.fdocuments.net/reader034/viewer/2022042700/5563bf2cd8b42ac70d8b578c/html5/thumbnails/94.jpg)
Resolving Forces Along The Bank
mg
N
Fmg
N
F
mg
N
F
r
mv2r
mv2
cosbank thealong forces2
rmv
F cossin
2
rmvmgF
sinmg
sincos2
mgr
mvF
sinbank the forces2
rmv
cosmg
N
sincos2
rmvmgN
![Page 95: X2 T06 06 banked curves (2011)](https://reader034.fdocuments.net/reader034/viewer/2022042700/5563bf2cd8b42ac70d8b578c/html5/thumbnails/95.jpg)
Resolving Forces Along The Bank
mg
N
Fmg
N
F
mg
N
F
r
mv2r
mv2
cosbank thealong forces2
rmv
F cossin
2
rmvmgF
sinmg
sincos2
mgr
mvF
sinbank the forces2
rmv
cosmg
N
sincos2
rmvmgN
cossin2
mgr
mvN
![Page 96: X2 T06 06 banked curves (2011)](https://reader034.fdocuments.net/reader034/viewer/2022042700/5563bf2cd8b42ac70d8b578c/html5/thumbnails/96.jpg)
Exercise 9D; all