a 2400-lb (=10.7-kn)car traveling at 30 mi/h (=13.4m/s)attempts to round an unbanked curve with a radius of 200ft (=61.0m).(a) what force of friction is required to keep the car on its path?(b) what minimum coefficient of static friction between the tires and road is required
To find the force of friction required to keep the car on its path, we need to consider the centripetal force acting on the car. The centripetal force is responsible for keeping the car in circular motion. In this case, the frictional force provides the necessary centripetal force.
(a) To calculate the force of friction, we can use the following formula:
F_friction = m * (v^2 / r),
where
m = mass of the car
v = velocity of the car
r = radius of the curve.
First, let's convert the given units:
2400 lb = 10.7 kN (1 lb = 0.00445 kN)
30 mi/h = 13.4 m/s
200 ft = 61.0 m
Using the formula, we can calculate the force of friction:
F_friction = (10.7 kN) * ((13.4 m/s)^2 / 61.0 m)
= (10.7 kN) * (179.56 m^2/s^2 / 61.0 m)
= (10.7 kN) * 2.944 m/s^2
= 31.5708 kN
Therefore, the force of friction required to keep the car on its path is approximately 31.5708 kN.
(b) To find the minimum coefficient of static friction required between the tires and the road, we can use the equation:
μ_static = F_friction / (m * g),
where
μ_static = coefficient of static friction
F_friction = force of friction calculated above
m = mass of the car
g = acceleration due to gravity (9.8 m/s^2).
Let's plug in the values:
μ_static = (31.5708 kN) / ((10.7 kN) * 9.8 m/s^2)
= (31.5708 kN) / (104.86 kN m/s^2)
≈ 0.3009
Therefore, the minimum coefficient of static friction required between the tires and the road is approximately 0.3009.
a)centripetal acceleration is v^2/r. In this case, v is 30 mi/hr and r = 200 ft. The inertial force would be:
F = ma = mv^2/r (1)
This would be the friction force required to keep the car on its path. b) The static friction is:
F = uN (2)
where u is the static friction coefficient and N is the normal force (in this case the weight of the car). Setting (1) equal to (2)
mv^2/r = uN (3a)
m = N (3b)
combine (3a) and (3b) and solve for u.