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प्रश्न
Derive an expression for maximum safety speed with which a vehicle should move along a curved horizontal road. State the significance of it.
Obtain an expression for maximum safety speed with which a vehicle should move along a curved horizontal road.
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उत्तर
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):
- Weight (mg) vertically downwards.
- The normal reaction (N) acts vertically upwards and balances the weight.
- 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 = mu_s = v_max^2/(rg)` ...[From equations (2) and (3)]
∴ `v_max = sqrt(mu_srg)`
This is an expression of the maximum safe speed at which a vehicle should move along a curved horizontal road.
Significance: The maximum safe speed of a vehicle on a curved road depends upon the friction between the tyres and the road, the radius of the curved road, and the acceleration due to gravity.
Notes
Students should refer to the answer according to their questions.
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