On a winter day with the temperature hovering near freezing, the coefficent of static friction between the tires and an icy road is 0.27. What is the maximum incline that a four-wheel drive vehicle can climb with zero acceleration?
normal force = m g cos A
max friction force up slope = 0.27 m g cos A
weight component down slope = m g sin A
so
m g sin A = 0.27 m g cos A
or
tan A = 0.27
To find the maximum incline that a four-wheel-drive vehicle can climb with zero acceleration, we need to use the concept of friction and forces.
The maximum incline a vehicle can climb is limited by the force of gravity and the friction force between the tires and the road. When the vehicle is at the point of slipping or just about to slide down the incline, the force of static friction between the tires and the road is at its maximum.
The formula for static friction is:
F_friction = coefficent of static friction * normal force
In this case, since the vehicle is on an incline, the normal force (N) is the force of gravity acting on the vehicle, which can be calculated as:
N = m * g
where m is the mass of the vehicle and g is the acceleration due to gravity (approximately 9.8 m/s^2).
Therefore, the maximum force of static friction that can prevent the vehicle from slipping is:
F_friction_max = coefficent of static friction * (m * g)
To find the maximum incline, we equate the maximum force of static friction with the component of the force of gravity acting parallel to the incline:
m * g * sin(theta) = coefficent of static friction * (m * g)
Here, sin(theta) represents the sine of the angle of the incline.
Simplifying the equation:
sin(theta) = coefficent of static friction
Now, we can solve for the angle theta (in degrees) using the inverse sine function (sin^(-1)):
theta = sin^(-1)(coefficent of static friction)
Finally, the maximum incline that a four-wheel-drive vehicle can climb with zero acceleration is equal to theta.
Note: This calculation assumes ideal conditions and does not take into account other factors like tire condition, weight distribution, or other forces that may affect the vehicle's ability to climb an incline.