Banking of Roads Formula & Importance

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Banking of roads refers to the process of elevating the edges of the outer surface of the road in comparison to the inner surface through a particular angle. Roads are designed in a way so that the outer edges of a curve are slightly raised than the inner edge and this provides the necessary centripetal force to the vehicles for them to make a safe turn. This decreases the chances of skidding and gives a smooth ride on the curvature and this method is known as the banking of roads.

Table of Content

Definition of banking of roads
Important formulae for banking of roads
Zero banking angle
Things to Remember
Sample Questions

Key Takeaways: Banking of road formula, Banking angle, Centripetal force, Inertia, Frictional force

Definition of Banking of Roads

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The phenomenon by which the outer edge of the curved road is raised at some angle with the horizontal ground is called banking of road.

The banking of roads makes the surface of the road to be in an inclined plane, and the angle at which the road is inclined is known as the banking angle.

The normal force acting on the vehicle while moving through such a curved road has a horizontal component that provides the centripetal force to avoid skidding.

The maximum speed is independent of the mass of the vehicle. It is restricted for a particular angle of inclination depending on the mass of the vehicle as per the banking angle, the coefficient of friction, and the radius of curvature.

Important Banking of Roads Formula

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Expression for the angle of banking: \(\Theta\) = tan^-1 v²/rg
Safety velocity of the unbanked road: V_max=\(\sqrt{\mu rg}\)
Safety velocity of banked road: V_max=\(\sqrt{rg tan\Theta}\)
Velocity of a vehicle on a curved banked road: v =\(\sqrt{[(rg(tan\Theta + \mu s))/1-\mu s tan\Theta }\)
The velocity of a vehicle on a curved banked road when tyre μs= tan\(\lambda\) and \(\lambda\) is the angle of the friction: v = \(\sqrt{rg tan(\Theta + \lambda)}\)
Centripetal force: F = mv²/r = m\(\omega\)²r

Zero Banking Angle

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The banking angle is zero if the surface of the road is flat. The normal force doesn’t contribute to the centripetal force anymore as it becomes vertical now and balances the weight of the vehicle.

Friction force acts only on a rough finish surface and provides a centripetal force. Thus the vertical component of the forces creates a balance.

So when the banking angle is 0,

V=\(\mu\)rg

Things to Remember

The inertia of vehicles driving on the road makes the necessity for banking of roads.
Banking assures the safety of vehicles while taking a turn.
Banking is done not only on roads but on railway tracks and air tracks so that the trains can take a safer turn while the aircraft wings get in a proper horizontal position.
The purpose of banking is to avoid skidding, prevent overturning of vehicles by providing appropriate centripetal force and prevent wear and tear of tyres.

Sample questions

Ques. A velocity of 72 km/ hour and a radius of 100 m. What is the banking angle? (3 Marks)

Ans.Maximum velocity , Vm=72 km/hr = 20 m/s

The radius of banking r=100 m

We know that Gravitational acceleration g = 9.8

From the expression of maximum velocity

tan\(\Theta\) = v²/rg

\(\Theta\) = tan^-1

= 20²/100*9.8

= tan^-1 40/98 = 0.4081

0.4081 = 22.2

Ques. Derive the formula to avoid skidding on the road? (3 Marks)

Ans. Let us consider,

The radius of turn - r

Frictional force on inside and outside wheels- F1 and F2

Normal reactions- N1 and N2

Weight - mg

So, (N1 + N2) = mg (eq 1)

Thus, F1 + F2 = μ(N1 + N2) = μgm (eq 2 )

This must be greater than or equal to the centripetal force required.

Therefore, μmg > mv2/r

or, v < μrg

or, V_max = μrg

Thus, if the velocity of the car is more than μrg , the car will skid.

Ques. An automobile of 1200 kg with a level curve of a radius of 200 m, on an unbanked road with a velocity of 72 km/hr. What is the minimum coefficient of friction between the tyres and road in order that the automobile may not skid? (g = 10 m/s2) (4 Marks)

Ans. On an unbanked road, the centripetal force is provided by the frictional force.

Friction = mv²/r

Or, μmg = friction or μmg = mv²/r

So, μmin = v²/g

= (20×20) / (10×200)

= 0.2.

Therefore, the minimum coefficient of friction between the tyres and road in order that the automobile may not skid is 0.2

Ques. A circular racetrack of a radius of 300 m is banked at a 15-degree angle. The coefficient of friction between the wheels of a racecar and the road is 0.2, what is the

(a) optimum speed of the racecar to avoid wear and tear on its tyres, and

(b) maximum permissible speed to avoid slipping? (4 Marks)

Solution-

(a) tan\(\theta\) = v²/rg

tan\(\theta\) = rg tan\(\theta\)

= 10 * 300 * tan 15°

v² = 780 m/s

v = 28 m/s

(b) v_m = \(\sqrt{Rg(\mu + tan\theta)/1-\mu s tan\theta }\)

= √ 300 * 10 * (0.2 + tan 15)/(1 - 0.2 tan15)

= 38.19 m/s

Ques. A curve has a radius of 50 meters and a banking angle of 15º. What is the ideal, or critical, speed (the speed for which no friction is required between the car's tires and the surface) for a car on this curve?(3 Marks)

Ans. Here, radius of curve, r = 50 m

banking angle, θ = 15º

free-fall acceleration, g = 9.8 m/s2

F_net = F_centripetal

mg tanθ = mv²/r

v² = rg tanθ

v = √rg tanθ

= √(50 m) (9.8 m/s2) (tan 15º) = 11 m/s

If the car has a speed of about 11 m/s, it can negotiate the curve without any friction.

Ques. A turn of radius 100 m is being designed for a speed of 25 m/s. At what angle should the turn be banked? (3 Marks)

Ans. Here, the radius of turn, r = 100 m

speed of the car, v = 25 m/s

free-fall acceleration, g = 9.8 m/s2

We have to find out the bank angle, \(\theta\).

F_net = F_centripetal

mg tanθ = mv²/r

tanθ = v²rg

\(\theta\) = tan^-1(v²rg)

= tan^-1[(25 m/s²) / (100 m) (9.8 m/s²)]

= 33º

The banked angle must be 33º.

Ques. Define banking angle. What is the best banking angle? Is banking possible on perfectly smooth roads? justify your answer. (3 Marks)

Ans. The angle at which the vehicle is inclined about its longitudinal axis with respect to the plane of its curved path is defined as the banking angle.

The optimum banking angle for a vehicle to perfectly glide along the turn is 45 degrees.

Banking is not possible on perfectly smooth roads as the banking angle of perfectly smooth roads is 0. Moreover, there is no turning involved in flat roads.

Ques. What are the criteria upon which the motion of a vehicle on a banked road depends? (3 Marks)

Ans. The criteria by which the motion of a vehicle depends are as follows

The radius of the road curvature
Coefficient of friction.
It must be noted that the maximum and minimum velocity of the car is independent of the mass.

Ques. Why should roads be banked? (2 Marks)

Ans. On a banked road, necessary centripetal force is supplied between the wheels and surface of the road thus ensuring that the vehicle does not skid. This allows safe movement along banked roads.

Also Read:

Acceleration	Elastic Collision	Pulley
Kinetic energy	Unit of Moment of Inertia	Torque
Angular Momentum	Angular Speed	Rotational kinetic energy
Radial acceleration	Unit of Power	Unit of work

Banking of Roads Formula & Importance

Definition of Banking of Roads

Important Banking of Roads Formula

Zero Banking Angle

Things to Remember

Sample questions

CBSE CLASS XII Related Questions

1.
Suppose a pure Si crystal has \( 5 \times 10^{28} \) atoms per \( \text{m}^3 \). It is doped with \( 5 \times 10^{22} \) atoms per \( \text{m}^3 \) of Arsenic. Calculate majority and minority carrier concentration in the doped silicon. (Given: \( n_i = 1.5 \times 10^{16} \, \text{m}^{-3} \))

3.
Two thin lenses of focal length \( f_1 \) and \( f_2 \) are placed in contact with each other coaxially. Prove that the focal length \( f \) of the combination is given by \[ f = \frac{f_1 f_2}{f_1 + f_2}. \]

4.
If both the number of protons and the neutrons are conserved in each nuclear reaction, in what way is mass converted into energy (or vice versa) in a nuclear reaction? Explain.

5.
Write any two features of nuclear forces.

6.
Two small identical metallic balls having charges \( q \) and \( -2q \) are kept far at a separation \( r \). They are brought in contact and then separated at distance \( \frac{r}{2} \). Compared to the initial force \( F \), they will now:

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Banking of Roads Formula & Importance

Definition of Banking of Roads

Important Banking of Roads Formula

Zero Banking Angle

Things to Remember

Sample questions

CBSE CLASS XII Related Questions

1.Suppose a pure Si crystal has \( 5 \times 10^{28} \) atoms per \( \text{m}^3 \). It is doped with \( 5 \times 10^{22} \) atoms per \( \text{m}^3 \) of Arsenic. Calculate majority and minority carrier concentration in the doped silicon. (Given: \( n_i = 1.5 \times 10^{16} \, \text{m}^{-3} \))

3.Two thin lenses of focal length \( f_1 \) and \( f_2 \) are placed in contact with each other coaxially. Prove that the focal length \( f \) of the combination is given by \[ f = \frac{f_1 f_2}{f_1 + f_2}. \]

4. If both the number of protons and the neutrons are conserved in each nuclear reaction, in what way is mass converted into energy (or vice versa) in a nuclear reaction? Explain.

5. Write any two features of nuclear forces.

6.Two small identical metallic balls having charges \( q \) and \( -2q \) are kept far at a separation \( r \). They are brought in contact and then separated at distance \( \frac{r}{2} \). Compared to the initial force \( F \), they will now:

Comments

SUBSCRIBE TO OUR NEWS LETTER

1.
Suppose a pure Si crystal has \( 5 \times 10^{28} \) atoms per \( \text{m}^3 \). It is doped with \( 5 \times 10^{22} \) atoms per \( \text{m}^3 \) of Arsenic. Calculate majority and minority carrier concentration in the doped silicon. (Given: \( n_i = 1.5 \times 10^{16} \, \text{m}^{-3} \))

3.
Two thin lenses of focal length \( f_1 \) and \( f_2 \) are placed in contact with each other coaxially. Prove that the focal length \( f \) of the combination is given by \[ f = \frac{f_1 f_2}{f_1 + f_2}. \]

4.
If both the number of protons and the neutrons are conserved in each nuclear reaction, in what way is mass converted into energy (or vice versa) in a nuclear reaction? Explain.

5.
Write any two features of nuclear forces.

6.
Two small identical metallic balls having charges \( q \) and \( -2q \) are kept far at a separation \( r \). They are brought in contact and then separated at distance \( \frac{r}{2} \). Compared to the initial force \( F \), they will now: