Calculate the maximum acc. of a car that is heading up a 4 degree slope. Assume that only half the weight of the car is supported by the two drive wheels and the static coefficient friction is involved- the tires can not slip during the acceleration
a) on dry concrete, us=10 uk=.7
b) on wet concrete, us=.7 uk=.5
c) on ice us=.1 uk=.05
To calculate the maximum acceleration of a car heading up a 4-degree slope, we need to consider the forces acting on the car and the maximum friction force that can be exerted by the tires.
The force acting on the car in the vertical direction is its weight, which can be calculated by multiplying the car's mass by the acceleration due to gravity (9.8 m/s^2). Since only half the weight is supported by the drive wheels, we divide the weight by 2.
Force vertical = (m * g) / 2
The force acting on the car in the horizontal direction is the friction force exerted by the tires. We need to determine the maximum static friction force (Fs) and the maximum kinetic friction force (Fk) and use the lower value between the two.
The maximum static friction force can be calculated by multiplying the coefficient of static friction (us) by the normal force (force vertical).
Fs = us * force vertical
The maximum kinetic friction force can be calculated by multiplying the coefficient of kinetic friction (uk) by the normal force.
Fk = uk * force vertical
Now, let's calculate the maximum acceleration for each scenario:
a) On dry concrete (us = 1.0, uk = 0.7):
Fs = 1.0 * (m * g) / 2
Fk = 0.7 * (m * g) / 2
b) On wet concrete (us = 0.7, uk = 0.5):
Fs = 0.7 * (m * g) / 2
Fk = 0.5 * (m * g) / 2
c) On ice (us = 0.1, uk = 0.05):
Fs = 0.1 * (m * g) / 2
Fk = 0.05 * (m * g) / 2
Once we have the maximum static friction force or maximum kinetic friction force, we can use Newton's second law (F = ma) to calculate the maximum acceleration. Since the friction force acts in the opposite direction of the applied force, we subtract it from the force vertical.
Max acceleration = (force vertical - friction force) / m
Now, using the values of Fs or Fk calculated above, you can substitute them back into the equation to obtain the maximum acceleration for each scenario.
Note: The units used in these calculations are arbitrary. You can use either the metric system (kilograms, meters, and seconds) or the imperial system (pounds, feet, and seconds), but make sure to be consistent throughout the calculations.