The normal force exerted on the car by the curved road is 13,139.7 N.
The radial acceleration of the car is 6.56 m/s².
The speed of the car is 21.43 m/s.
In the case of a banked curve with friction, the centripetal acceleration is increased by normal force and frictional force since it prevents the car from skidding.
The given parameters;
- radius of the curved path, r = 70 m
- banking angle, θ = 12⁰
- coefficient of friction, μ = 0.4
- mass of the race car, m = 1200 kg
The normal force exerted on the car by the curved road is calculated as follows;
[tex]\Sigma F_y = 0\\\\Ncos(\theta) - \mu_s Nsin(\theta) - mg = 0\\\\Ncos(\theta) - \mu_s Nsin(\theta) = mg\\\\N(cos\theta \ - \ \mu_s sin(\theta)) = mg\\\\N = \frac{mg }{cos\theta - \mu_s sin(\theta)} \\\\N = \frac{(1200 \times 9.8)}{cos (12) - \ 0.4\times sin(12)} \\\\N = 13,139.7 \ N[/tex]
The radial acceleration of the car is calculated as follows;
[tex]\Sigma F_x = ma_c\\\\ma_c = Nsin(\theta) + \mu_s N cos(\theta)\\\\a_c = \frac{Nsin(\theta) + \mu_s N cos(\theta)}{m} \\\\a_c = \frac{(13, 139.7)sin(12) \ + \ 0.4 \times 13,139.7cos(12)}{1200} \\\\a_c = 6.56 \ m/s^2[/tex]
The car's speed is calculated as follows;
[tex]a_c = \frac{v^2}{r} \\\\v^2 = a_c r\\\\v = \sqrt{a_c r} \\\\v = \sqrt{6.56 \times 70} \\\\v = 21.43 \ m/s[/tex]
In the case of a banked curve with friction, the centripetal acceleration is increased by the following;
[tex]ma_c = Nsin(\theta) + F_s\\\\ma_c = Nsin(\theta) + \mu_s Ncos (\theta)[/tex]
where;
- [tex]F_s[/tex] is the frictional force
- N is the normal force
Thus, In the case of a banked curve with friction, the centripetal acceleration is increased by normal force and frictional force since it prevents the car from skidding.
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