A professor drives off with his car (mass 850 kg), but forgot to take his coffee mug (mass 0.39 kg) off the roof. The coefficient of static friction between the mug and the roof is 0.7, and the coefficient of kinetic friction is 0.4. What is the maximum acceleration of the car, so the mug does not slide off?
.7 = a/g = a/9.81
a = 6.87 m/s^2
To find the maximum acceleration of the car, we need to determine the maximum value of the static friction force that can act on the coffee mug.
The static friction force is given by the equation Fs = μs * N, where Fs is the static friction force, μs is the coefficient of static friction, and N is the normal force.
In this case, the normal force is equal to the weight of the coffee mug, which can be calculated as N = m * g, where m is the mass of the coffee mug and g is the acceleration due to gravity (approximately 9.8 m/s^2).
N = (0.39 kg) * (9.8 m/s^2) = 3.822 N
Therefore, the maximum static friction force that can act on the coffee mug is:
Fs_max = μs * N = (0.7) * (3.822 N) = 2.6754 N
Now, let's consider the forces acting on the mug when the car accelerates. There is the gravitational force pointing downward, and the static friction force pointing upward (preventing the mug from sliding off). The net force acting on the mug in the vertical direction is:
F_net = F_gravity - Fs_max
Since the car is accelerating, there is an additional force acting on the mug in the horizontal direction, which is the force of static friction.
F_net_horizontal = F_friction
Since the mug is not sliding, the static friction force must equal the maximum static friction force:
F_friction = Fs_max = 2.6754 N
Now, we can consider Newton's second law in the horizontal direction:
F_net_horizontal = m * a
Where m is the mass of the car and a is the acceleration of the car. Rearranging the equation, we can solve for the maximum acceleration:
a = F_net_horizontal / m = F_friction / m
Substituting the values, we get:
a = 2.6754 N / 850 kg = 0.0031 m/s^2 (approximated to 4 significant figures)
Therefore, the maximum acceleration of the car so that the coffee mug does not slide off is approximately 0.0031 m/s^2.