Derive an expression for safety speed of vehicle along bank curved road
Answers
Answer:
Banking of roads : To avoid risk of skidding as well as to reduce the wear and tear of the car tyres, the road surface at a bend is tilted inwards, i.e., the outer side of road is raised above its inner side. This is called 'banking of roads'.
Consider a car taking a left turn along a road of radius r banked at an angle θ for a designed optimum speed V. Let m be the mass of the car. In general, the forces acting on the car are:
(a) Its weight
mg
, acting vertically down
(b) The normal reaction of the road
N
, perpendicular to the road surface
(c) The frictional force
f
s
along the inclined surface of the road.
Resolve
N
and
f
s
into two perpendicular components Ncosθ vertically up and
f
s
sinθ vertically down, Nsinθ and
f
s
cosθ horizontally towards the centre of the circular path.
If v
max
is the maximum safe speed without skidding.
r
mv
max
2
=Nsinθ+f
s
cosθ
=Nsinθ+μ
s
Ncosθ
r
mv
max
2
=N(sinθ+μ
s
cosθ)....(1)
and
Ncosθ=mg+f
s
sinθ
=mg+μ
s
Nsinθ
∴mg=N(cosθ−μ
s
sinθ)...(2)
Dividing eq. (1) by eq. (2),
r.mg
mv
max
2
=
N(cosθ−μ
s
sinθ)
N(sinθ+μ
s
cosθ)
∴
rg
v
max
2
=
cosθ−μ
s
sinθ
sinθ+μ
s
cosθ
=
1−μ
s
tanθ
tanθ+μ
s
∴v
max
=
1−μ
s
tanθ
rg(tanθ+μ
s
)
...,.(3)
This is the expression for the maximum safe speed on a banked road.
At the optimum speed, the friction between the car tyres and the road surface is not called into play. Hence, by setting μ
s
=0 in eq. (3), the optimum speed on a banked circular road is
v=
rgtanθ
...(4)
∴tanθ=
rg
v
2
or θ=tan
−1
(
rg
v
2
)
From this eq. we see that θ depends upon v,r and g. The angle of banking is independent of the mass of a vehicle negotiating the curve.
Explanation:
Banking of roads : To avoid risk of skidding as well as to reduce the wear and tear of the car tyres, the road surface at a bend is tilted inwards, i.e., the outer side of road is raised above its inner side. This is called 'banking of roads'.
Consider a car taking a left turn along a road of radius r banked at an angle θ for a designed optimum speed V. Let m be the mass of the car. In general, the forces acting on the car are:
(a) Its weight
mg
, acting vertically down
(b) The normal reaction of the road
N
, perpendicular to the road surface
(c) The frictional force
f
s
along the inclined surface of the road.
Resolve
N
and
f
s
into two perpendicular components Ncosθ vertically up and
f
s
sinθ vertically down, Nsinθ and
f
s
cosθ horizontally towards the centre of the circular path.
If v
max
is the maximum safe speed without skidding.
r
mv
max
2
=Nsinθ+f
s
cosθ
=Nsinθ+μ
s
Ncosθ
r
mv
max
2
=N(sinθ+μ
s
cosθ)....(1)
and
Ncosθ=mg+f
s
sinθ
=mg+μ
s
Nsinθ
∴mg=N(cosθ−μ
s
sinθ)...(2)
Dividing eq. (1) by eq. (2),
r.mg
mv
max
2
=
N(cosθ−μ
s
sinθ)
N(sinθ+μ
s
cosθ)
∴
rg
v
max
2
=
cosθ−μ
s
sinθ
sinθ+μ
s
cosθ
=
1−μ
s
tanθ
tanθ+μ
s
∴v
max
=
1−μ
s
tanθ
rg(tanθ+μ
s
)
...,.(3)
This is the expression for the maximum safe speed on a banked road.
At the optimum speed, the friction between the car tyres and the road surface is not called into play. Hence, by setting μ
s
=0 in eq. (3), the optimum speed on a banked circular road is
v=
rgtanθ
...(4)
∴tanθ=
rg
v
2
or θ=tan
−1
(
rg
v
2
)
From this eq. we see that θ depends upon v,r and g. The angle of banking is independent of the mass of a vehicle negotiating the curve.
solution
Answered By
toppr
19343 Views
How satisfied are you with the answer?
This will help us to improve better
answr
Get Instant Solutions, 24x7
No Signup required
girl
More Questions by difficulty
EASY
MEDIUM
HARD
Live Test
Practice from
20+ Live Tests
and see where you stand among the best in India
Physics
Chemistry
Mathematics
Biology
Practice important Questions
MH Board March 2012 Class 12 Physics I
25 Qs
Related Questions to study
A road is 8 m wide. Its radius of curvature is 40 m. The outer edge is above the lower edge by distance of 1.2 m. The most suited velocity on the road is nearly:
MEDIUM
Study later
View Answer
An aeroplane executes a horizontal loop at a speed of 720 kmph with its wings banked at 45
o
. What is the radius of the loop?
Take g = 10 ms
−2
.
MEDIUM
Study later
View Answer
VIEW MORE
Create custom Assignments
Customize assignments and download PDF’s
girl
BROWSE BY
ClassesBoardsExams
MODULES
Online ClassesMock TestsAdaptive PracticeLive Doubts
Stay upto date with our Newsletter!
Stay informed, stay ahead. Be a topper
Your email address
About Us
Brand Resources
Press
Customer Stories
Jobs
Educators
Fellowship
Learning Planet
Guides
Ask
Blog
Bytes
News
Terms of Service
Privacy Policy
Contact Us
FAQs
Google Play
App Store
IN
India