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