A motor vehicle has a mass of 1.8 tonnes and its wheelbase is 3 m. The centre of gravity of the vehicle is situated in the central plane 0.9 m above the ground and 1.7 m behind the front axle. When moving on the level at 90 km/h the brakes applied and it comes to a rest in a distance of 50 m. Calculate the normal reactions at the front and rear wheels during the braking period and the least coefficient of friction required between the tyres and the road. (Assume g = 10 m/s2)

normal reaction, front: 11,175 Nnormal reaction, rear: 6,825 Nminimum coefficient of friction: 0.625

Explanation:

The speed in meters per second is ...

  (90 km/h)(1000 m/km)(1 h/(3600 s)) = 25 m/s

The braking acceleration can be found from ...

  a = v²/(2d) = (25 m/s)²/(2×50 m) = 6.25 m/s²

Then the braking force is ...

  F = Ma = (1800 kg)(6.25 m/s²) = 11,250 N

The torque on the center of gravity is ...

  T = (11,250 N)(0.90 m) = 10,125 N·m

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If we let x and y represent the normal forces on the front and rear wheels, respectively, then we have ...

  x + y = (10 m/s²)(1800 kg) = 18000 . . . . . newtons

  1.7x -1.3y = 10,125 . . . . . . . . . . . . . . . . . . . newton-meters

The latter equation balances the torque due to the wheel normal forces with the torque due to braking forces.

Multiplying the first equation by 1.3 and adding that to the second, we have ...

  3.0x = (1.3)(18,000) + 10,125

  x = 33,525/3 = 11,175 . . . . . . . . . . . newtons normal force on front tyres

  y = 18000 -11175 = 6,825 . . . . . . . .newtons normal force on back tyres

The least coefficient of friction is the ratio of horizontal to vertical acceleration, 6.25/10 = 0.625.

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