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Question
Derive an expression for maximum safety speed with which a vehicle should move along a curved horizontal road. State the significance of it.
Solution
- Consider a vertical section of a car moving on a horizontal circular track having a radius ‘r’ with ‘C’ as the centre of the track.
- Forces acting on the car (considered to be a particle):
a. Weight (mg), vertically downwards,
b. Normal reaction (N), vertically upwards that balances the weight
c. Force of static friction (fs) between the road and the tyres.
Since normal reaction balances the weight
∴ N = mg ...(1)
While working in the frame of reference attached to the vehicle, the frictional force balances the centrifugal force.
`f_s = (mv^2)/r` ...(2)
Dividing equation (2) by equation (1),
∴ `f_s/"N" = v^2/rg` ...(3) - However, fs has an upper limit (fs)max = µsN, where μs is the coefficient of static friction between the road and the tyres of the vehicle. This imposes an upper limit to the speed v.
At the maximum possible speed,
`(f_s)_max/N = µ_s = v_max^2/(rg)` ...[From equations (2) and (3)]
∴ `v_max = sqrt(µ_srg)`
This is an expression of maximum safety speed with which a vehicle should move along a curved horizontal road. - Significance: The maximum safe speed of a vehicle on a curve road depends upon friction between tyres and road, the radius of the curved road and acceleration due to gravity.
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