A professor drives off with his car (mass 800 kg), but forgot to take his coffee mug (mass 0.35 kg) off the roof. The coefficient of static friction between the mug and the roof is 1.1, and the coefficient of kinetic friction is 0.5. What is the maximum acceleration of the car, so the mug does not slide off?
nevermind, i figured it out.
To determine the maximum acceleration of the car so that the mug does not slide off, we need to consider the forces acting on the mug. These forces include the gravitational force acting downward on the mug and the static friction force acting upward due to the contact between the mug and the roof of the car.
First, let's calculate the gravitational force acting on the mug:
Gravitational force = mass x gravitational acceleration
Gravitational force = 0.35 kg x 9.8 m/s^2 ≈ 3.43 N
Next, let's determine the maximum static friction force that can be exerted on the mug:
Maximum static friction force = coefficient of static friction x normal force
To find the normal force acting on the mug, we need to consider the vertical forces on the mug. Since the car is not accelerating vertically, the normal force is equal in magnitude and opposite in direction to the gravitational force acting on the mug.
Normal force = gravitational force = 3.43 N
Maximum static friction force = 1.1 x 3.43 N ≈ 3.78 N
The maximum static friction force is the maximum amount of force that the roof can exert on the mug to prevent it from sliding.
Now, we can determine the maximum acceleration of the car using Newton's second law, which states that the sum of the forces acting on an object is equal to the mass of the object multiplied by its acceleration:
Sum of forces = mass x acceleration
For the mug:
Sum of forces = maximum static friction force - gravitational force
3.78 N - 3.43 N = 0.35 kg x acceleration
0.35 kg x acceleration = 0.35 N
acceleration ≈ 1 m/s^2
Therefore, the maximum acceleration of the car so that the mug does not slide off is approximately 1 m/s^2.