A professor drives off with his car (mass 840 kg), but forgot to take his coffee mug (mass 0.37 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?
To find the maximum acceleration of the car, we need to compare the force of static friction to the force of gravity acting on the coffee mug.
First, let's calculate the force of gravity acting on the coffee mug. The force of gravity can be calculated using the formula:
Force (gravity) = mass × acceleration due to gravity
The acceleration due to gravity is approximately 9.8 m/s².
Force (gravity) = 0.37 kg × 9.8 m/s²
Next, let's calculate the maximum force of static friction between the mug and the roof. The formula for static friction is:
Force (static friction) = coefficient of static friction × Normal force
The normal force is equal to the weight of the mug, which is the same as the force of gravity acting on it.
Force (static friction) = 0.7 × (0.37 kg × 9.8 m/s²)
Now, we need to find the maximum acceleration that keeps the mug from sliding off the roof. This acceleration occurs when the force of static friction is equal to the force of gravity acting on the mug.
Set the equation for force of static friction equal to the force of gravity:
0.7 × (0.37 kg × 9.8 m/s²) = 0.37 kg × a
Simplify the equation:
a = 0.7 × 9.8 m/s²
a ≈ 6.86 m/s²
Therefore, the maximum acceleration of the car so the mug does not slide off is approximately 6.86 m/s².