6. Motion Around A
Banked Curve
N
mg
mv 2
horizontal forces
r
7. Motion Around A
Banked Curve
N
mg
mv 2
horizontal forces
r
8. Motion Around A
Banked Curve
N
mg
mv 2
horizontal forces
r
9. Motion Around A
Banked Curve
N
mg
mv 2
horizontal forces
r
N sin
10. Motion Around A
Banked Curve
N
mg
mv 2
horizontal forces
r
N sin 2
mv
N sin
r
11. Motion Around A
Banked Curve
N
mg
mv 2
horizontal forces vertical forces 0
r
N sin 2
mv
N sin
r
12. Motion Around A
Banked Curve
N
mg
mv 2
horizontal forces vertical forces 0
r
N sin 2
mv
N sin
r
13. Motion Around A
Banked Curve
N
mg
mv 2
horizontal forces vertical forces 0
r
N sin 2 N cos
mv
N sin mg
r
14. Motion Around A
Banked Curve
N
mg
mv 2
horizontal forces vertical forces 0
r
N sin 2 N cos N cos mg 0
mv N cos mg
N sin mg
r
16. mv 2 1
tan
r mg
v2
rg
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.
17. mv 2 1
tan
r mg
v2
rg
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.
18. mv 2 1
tan
r mg
v2
rg
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
19. mv 2 1
tan
r mg
v2
rg
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
20. mv 2 1
tan
r mg
v2
rg
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
21. mv 2 1
tan
r mg
v2
rg
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.
N
1 .5m 0.08m
22. mv 2 1
tan
r mg
v2
rg
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.
N
1 .5m mg 0.08m
23. mv 2 1
tan
r mg
v2
rg
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.
N
1 .5m mg 0.08m
31. mv 2
horizontal forces vertical forces 0
r
N sin 2 N cos N cos mg 0
mv N cos mg
N sin mg
r
32. mv 2
horizontal forces vertical forces 0
r
N sin 2 N cos N cos mg 0
mv N cos mg
N sin mg
r
mv 2 1
tan
r mg
v2
rg
33. mv 2
horizontal forces vertical forces 0
r
N sin 2 N cos N cos mg 0
mv N cos mg
N sin mg
r
mv 2 1
tan
r mg
v2
rg
0.08
But sin
1.5
4
75
34. mv 2
horizontal forces vertical forces 0
r
N sin 2 N cos N cos mg 0
mv N cos mg
N sin mg
r
mv 2 1
tan
r mg
v2
rg
0.08
But sin
1.5
4
75
4
tan
5609
35. mv 2
horizontal forces vertical forces 0
r
N sin 2 N cos N cos mg 0
mv N cos mg
N sin mg
r
mv 2 1 v2 4
tan
r mg 4009.8 5609
v2
rg
0.08
But sin
1.5
4
75
4
tan
5609
36. mv 2
horizontal forces vertical forces 0
r
N sin 2 N cos N cos mg 0
mv N cos mg
N sin mg
r
mv 2 1 v2 4
tan
r mg 4009.8 5609
v2 4400 9.8
rg v2
5609
0.08
But sin v 14.47 m/s
1.5 v 52km/h
4
75
4
tan
5609
37. (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 N to N ,
where is a positive constant and N is the
normal component; the sign of F is positive
when F is directed up the bank.
The acceleration due to gravity is2 g. The acceleration related to the
circular motion is of magnitude v and is directed towards the centre
of the track. R
38. a) By resolving forces horizontally and vertically, show that;
v 2 N sin F cos
Rg N cos F sin
39. a) By resolving forces horizontally and vertically, show that;
v 2 N sin F cos
Rg N cos F sin
40. a) By resolving forces horizontally and vertically, show that;
v 2 N sin F cos
Rg N cos F sin
mv 2
horizontal forces
R
41. a) By resolving forces horizontally and vertically, show that;
v 2 N sin F cos
Rg N cos F sin
mv 2
horizontal forces
R
42. a) By resolving forces horizontally and vertically, show that;
v 2 N sin F cos
Rg N cos F sin
mv 2
horizontal forces
R
N sin F cos
43. a) By resolving forces horizontally and vertically, show that;
v 2 N sin F cos
Rg N cos F sin
mv 2
horizontal forces
R
N sin F cos
mv 2
N sin F cos
R
44. a) By resolving forces horizontally and vertically, show that;
v 2 N sin F cos
Rg N cos F sin
mv 2
horizontal forces vertical forces 0
R
N sin F cos
mv 2
N sin F cos
R
45. a) By resolving forces horizontally and vertically, show that;
v 2 N sin F cos
Rg N cos F sin
mv 2
horizontal forces vertical forces 0
R
N sin F cos
mv 2
N sin F cos
R
46. a) By resolving forces horizontally and vertically, show that;
v 2 N sin F cos
Rg N cos F sin
mv 2
horizontal forces vertical forces 0
R
N sin F cos N cos F sin
mv 2 mg
N sin F cos
R
47. a) By resolving forces horizontally and vertically, show that;
v 2 N sin F cos
Rg N cos F sin
mv 2
horizontal forces vertical forces 0
R
N sin F cos N cos F sin N cos F sin mg 0
mv 2 mg N cos F sin mg
N sin F cos
R
48. a) By resolving forces horizontally and vertically, show that;
v 2 N sin F cos
Rg N cos F sin
mv 2
horizontal forces vertical forces 0
R
N sin F cos N cos F sin N cos F sin mg 0
mv 2 mg N cos F sin mg
N sin F cos
R
v 2 mv 2 1
Rg R mg
49. a) By resolving forces horizontally and vertically, show that;
v 2 N sin F cos
Rg N cos F sin
mv 2
horizontal forces vertical forces 0
R
N sin F cos N cos F sin N cos F sin mg 0
mv 2 mg N cos F sin mg
N sin F cos
R
v 2 mv 2 1
Rg R mg
v 2 N sin F cos
Rg N cos F sin
50. b) Show that the maximum speed vmaxat which the particle can travel
without slipping up the track is given by; vmax tan
2
Rg 1 tan
You may suppose that tan 1
51. b) Show that the maximum speed vmaxat which the particle can travel
without slipping up the track is given by; vmax tan
2
Rg 1 tan
You may suppose that tan 1
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. F N
52. b) Show that the maximum speed vmaxat which the particle can travel
without slipping up the track is given by; vmax tan
2
Rg 1 tan
You may suppose that tan 1
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. F N
vmax N sin N cos
2
Rg N cos N sin
53. b) Show that the maximum speed vmaxat which the particle can travel
without slipping up the track is given by; vmax tan
2
Rg 1 tan
You may suppose that tan 1
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. F N
vmax N sin N cos
2
Rg N cos N sin
sin cos
cos cos
cos sin
cos cos
tan
1 tan
54. c) Show that if tan , then the particle will not slide down the track,
regardless of its speed.
55. c) Show that if tan , then the particle will not slide down the track,
regardless of its speed.
vmin 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. F N
56. c) Show that if tan , then the particle will not slide down the track,
regardless of its speed.
vmin 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. F N
vmin tan
2
Rg 1 tan
57. c) Show that if tan , then the particle will not slide down the track,
regardless of its speed.
vmin 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. F N
vmin tan
2
Rg 1 tan
If tan ;
58. c) Show that if tan , then the particle will not slide down the track,
regardless of its speed.
vmin 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. F N
vmin tan
2
Rg 1 tan
If tan ; 2
vmin
0
Rg
59. c) Show that if tan , then the particle will not slide down the track,
regardless of its speed.
vmin 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. F N
vmin tan
2
Rg 1 tan
If tan ; 2
vmin
0
Rg
vmin 0
2
vmin 0
60. c) Show that if tan , then the particle will not slide down the track,
regardless of its speed.
vmin 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. F N
vmin tan
2
Rg 1 tan
If tan ; 2
vmin
0
Rg
vmin 0
2
vmin 0
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.
61. (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/second 2, and
the mass of P is m kilograms.
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 F in
terms of the data. N
68. forces OP 0
F mg cos
mg cos F 0
F mg cos
69. forces OP 0
F mg cos
mg cos F 0
F mg cos
forces || OP mr 2
70. forces OP 0
F mg cos
mg cos F 0
F mg cos
forces || OP mr 2
71. forces OP 0
F mg cos
mg cos F 0
F mg cos
forces || OP mr 2
N mg sin
72. forces OP 0
F mg cos
mg cos F 0
F mg cos
forces || OP mr 2
N mg sin N mg sin mr 2
73. forces OP 0
F mg cos
mg cos F 0
F mg cos
forces || OP mr 2
N mg sin N mg sin mr 2
N mr 2 mg sin
74. forces OP 0
F mg cos
mg cos F 0
F mg cos
forces || OP mr 2
N mg sin N mg sin mr 2
N mr 2 mg sin
F mg cos
N mr 2 mg sin
F g cos
2
N r g sin
86. Resolving Forces Along The Bank mv 2
r N
mv 2 N
r F
F
mg
mg
mv 2
forces along the bank cos
r
87. Resolving Forces Along The Bank mv 2
r N
mv 2 N
r F
F
mg
mg
mv 2
forces along the bank cos
r
88. Resolving Forces Along The Bank mv 2
r N
mv 2 N
r F
F
mg
mg
mv 2
forces along the bank cos
r
mg sin
F
89. Resolving Forces Along The Bank mv 2
r N
mv 2 N
r F
F
mg
mg
mv 2
forces along the bank cos
r
mg sin
F
mv 2
F mg sin cos
r
90. Resolving Forces Along The Bank mv 2
r N
mv 2 N
r F
F
mg
mg
mv 2
forces along the bank cos
r
mg sin
F
mv 2
F mg sin cos
r
mv 2
F cos mg sin
r
91. Resolving Forces Along The Bank mv 2
r N
mv 2 N
r F
F
mg
mg
mv 2 mv 2
forces along the bank cos forces the bank sin
r r
mg sin
F
mv 2
F mg sin cos
r
mv 2
F cos mg sin
r
92. Resolving Forces Along The Bank mv 2
r N
mv 2 N
r F
F
mg
mg
mv 2 mv 2
forces along the bank cos forces the bank sin
r r
mg sin
F
mv 2
F mg sin cos
r
mv 2
F cos mg sin
r
93. Resolving Forces Along The Bank mv 2
r N
mv 2 N
r F
F
mg
mg
mv 2 mv 2
forces along the bank cos forces the bank sin
r r
mg sin N
F
mv 2 mg cos
F mg sin cos
r
mv 2
F cos mg sin
r
94. Resolving Forces Along The Bank mv 2
r N
mv 2 N
r F
F
mg
mg
mv 2 mv 2
forces along the bank cos forces the bank sin
r r
mg sin N
F
mv 2 mg cos
F mg sin cos mv 2
r N mg cos sin
mv 2 r
F cos mg sin
r
95. Resolving Forces Along The Bank mv 2
r N
mv 2 N
r F
F
mg
mg
mv 2 mv 2
forces along the bank cos forces the bank sin
r r
mg sin N
F
mv 2 mg cos
F mg sin cos mv 2
r N mg cos sin
mv 2 r
F cos mg sin
2
mv
r N sin mg cos
r