X2 t06 06 banked curves (2012)
96
Motion Around A Banked Curve
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Transcript of X2 t06 06 banked curves (2012)
Motion Around A Banked Curve
Motion Around A Banked Curve
Motion Around A Banked Curve
Motion Around A Banked Curve
N
Motion Around A Banked Curve
mg
N
Motion Around A Banked Curve
mg
N
rmv2
forces horizontal
Motion Around A Banked Curve
mg
N
rmv2
forces horizontal
Motion Around A Banked Curve
mg
N
rmv2
forces horizontal
Motion Around A Banked Curve
mg
N
rmv2
forces horizontal
sinN
Motion Around A Banked Curve
mg
N
rmv2
forces horizontal
sinN
rmvN
2
sin
Motion Around A Banked Curve
mg
N
rmv2
forces horizontal
sinN
rmvN
2
sin
0forcesvertical
Motion Around A Banked Curve
mg
N
rmv2
forces horizontal
sinN
rmvN
2
sin
0forcesvertical
Motion Around A Banked Curve
mg
N
rmv2
forces horizontal
sinN
rmvN
2
sin
0forcesvertical
mgcosN
Motion Around A Banked Curve
mg
N
rmv2
forces horizontal
sinN
rmvN
2
sin
0forcesvertical
mgcosN
mgNmgN
cos0cos
rgv
mgrmv
2
2 1tan
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.
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.
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
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
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
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
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
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
rmv2
forces horizontal
rmv2
forces horizontal
rmv2
forces horizontal
sinN
rmv2
forces horizontal
sinN
rmvN
2
sin
rmv2
forces horizontal
sinN
rmvN
2
sin
0forcesvertical
rmv2
forces horizontal
sinN
rmvN
2
sin
0forcesvertical
rmv2
forces horizontal
sinN
rmvN
2
sin
0forcesvertical
mgcosN
rmv2
forces horizontal
sinN
rmvN
2
sin
0forcesvertical
mgcosN
mgNmgN
cos0cos
rmv2
forces horizontal
sinN
rmvN
2
sin
0forcesvertical
mgcosN
mgNmgN
cos0cos
rgv
mgrmv
2
2 1tan
rmv2
forces horizontal
sinN
rmvN
2
sin
0forcesvertical
mgcosN
mgNmgN
cos0cos
rgv
mgrmv
2
2 1tan
754
5.108.0sinBut
rmv2
forces horizontal
sinN
rmvN
2
sin
0forcesvertical
mgcosN
mgNmgN
cos0cos
rgv
mgrmv
2
2 1tan
754
5.108.0sinBut
56094tan
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
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
(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
a) By resolving forces horizontally and vertically, show that;
sincoscossin2
FNFN
Rgv
a) By resolving forces horizontally and vertically, show that;
sincoscossin2
FNFN
Rgv
a) By resolving forces horizontally and vertically, show that;
sincoscossin2
FNFN
Rgv
Rmv2
forces horizontal
a) By resolving forces horizontally and vertically, show that;
sincoscossin2
FNFN
Rgv
Rmv2
forces horizontal
a) By resolving forces horizontally and vertically, show that;
sincoscossin2
FNFN
Rgv
Rmv2
forces horizontal
sinN cosF
a) By resolving forces horizontally and vertically, show that;
sincoscossin2
FNFN
Rgv
Rmv2
forces horizontal
sinN
RmvFN
2
cossin
cosF
a) By resolving forces horizontally and vertically, show that;
sincoscossin2
FNFN
Rgv
Rmv2
forces horizontal
sinN
RmvFN
2
cossin
0forcesvertical
cosF
a) By resolving forces horizontally and vertically, show that;
sincoscossin2
FNFN
Rgv
Rmv2
forces horizontal
sinN
RmvFN
2
cossin
0forcesvertical
cosF
a) By resolving forces horizontally and vertically, show that;
sincoscossin2
FNFN
Rgv
Rmv2
forces horizontal
sinN
RmvFN
2
cossin
0forcesvertical
mgcosNcosF sinF
a) By resolving forces horizontally and vertically, show that;
sincoscossin2
FNFN
Rgv
Rmv2
forces horizontal
sinN
RmvFN
2
cossin
0forcesvertical
mgcosN
mgFNmgFN
sincos0sincoscosF sinF
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
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
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
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
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
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
c) Show that if , then the particle will not slide down the track, regardless of its speed.
tan
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
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
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
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
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
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.
(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
0forces OP
0forces OP
0forces OP
F cosmg
0forces OP
F0cos Fmg
cosmg
0forces OP
F0cos Fmg
cosmg
cosmgF
0forces OP
F0cos Fmg
cosmg
cosmgF 2||forces mrOP
0forces OP
F0cos Fmg
cosmg
cosmgF 2||forces mrOP
0forces OP
F0cos Fmg
cosmg
cosmgF 2||forces mrOP
sinmgN
0forces OP
F0cos Fmg
cosmg
cosmgF 2||forces mrOP
sinmgN2sin mrmgN
0forces OP
F0cos Fmg
cosmg
cosmgF 2||forces mrOP
sinmgN2sin mrmgN
sin2 mgmrN
0forces OP
F0cos Fmg
cosmg
cosmgF 2||forces mrOP
sinmgN2sin mrmgN
sin2 mgmrN
sincos
sincos
2
2
grg
NF
mgmrmg
NF
Resolving Forces Along The Bank
Resolving Forces Along The Bank
Resolving Forces Along The Bank
Resolving Forces Along The Bank
N
Resolving Forces Along The Bank
mg
N
Resolving Forces Along The Bank
mg
N
F
Resolving Forces Along The Bank
mg
N
Fmg
N
F
mg
N
F
Resolving Forces Along The Bank
mg
N
Fmg
N
F
mg
N
F
Resolving Forces Along The Bank
mg
N
Fmg
N
F
mg
N
F
r
mv2
Resolving Forces Along The Bank
mg
N
Fmg
N
F
mg
N
F
r
mv2r
mv2
Resolving Forces Along The Bank
mg
N
Fmg
N
F
mg
N
F
r
mv2r
mv2
Resolving Forces Along The Bank
mg
N
Fmg
N
F
mg
N
F
r
mv2r
mv2
cosbank thealong forces2
rmv
Resolving Forces Along The Bank
mg
N
Fmg
N
F
mg
N
F
r
mv2r
mv2
cosbank thealong forces2
rmv
Resolving Forces Along The Bank
mg
N
Fmg
N
F
mg
N
F
r
mv2r
mv2
cosbank thealong forces2
rmv
Fsinmg
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
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
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
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
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
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
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
Exercise 9D; all
