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?

maximum friction force = .7 *.37*9.8

=2.53 Newtons
2.53 = m a
a = 2.53/.37 = 6.86 or .7 g of course

As we all know, once it starts sliding, that is the ball game.
Moreover the mass of the vehicle is also a red herring.

Well, it seems like this professor is a real "mug-forgetter"! Let's calculate the maximum acceleration to keep that coffee mug from sliding off the roof.

To prevent the mug from sliding off, we need to find the maximum static frictional force between the mug and the roof. The formula to calculate this force is F_static = μ_static * N, where μ_static is the coefficient of static friction and N is the normal force.

The normal force acting on the mug is equal to its weight, which is given by F_weight = m * g, where m is the mass of the mug and g is the acceleration due to gravity.

So, F_weight = (0.37 kg) * (9.8 m/s^2) = 3.626 N.

Now, we can calculate the maximum static frictional force:

F_static = (0.7) * (3.626 N) = 2.5382 N.

Since the maximum static frictional force is acting in the opposite direction to the car's acceleration, we have:

F_static = m * a,

where m is the mass of the car and a is the car's acceleration.

Plugging in the given values, we can solve for the maximum acceleration:

2.5382 N = (840 kg) * a.

Simplifying the equation, we find:

a = 2.5382 N / 840 kg ≈ 0.003 N/kg.

Therefore, the maximum acceleration the car can have without the coffee mug sliding off is approximately 0.003 N/kg.

Now, let's hope this professor learns to "mug-check" before driving off next time!

To determine the maximum acceleration of the car without the mug sliding off the roof, we need to consider the forces acting on the mug.

1. Determine the force of gravity acting on the mug:
F_gravity = m * g
where m is the mass of the mug (0.37 kg) and g is the acceleration due to gravity (approximately 9.8 m/s^2).
Therefore, F_gravity = 0.37 kg * (9.8 m/s^2) = 3.626 N.

2. Determine the maximum force of static friction that can keep the mug from sliding:
F_static_friction = μ_static * F_normal
where μ_static is the coefficient of static friction (0.7) and F_normal is the normal force acting on the mug.
The normal force can be calculated as:
F_normal = m * g
Therefore, F_normal = 0.37 kg * (9.8 m/s^2) = 3.626 N.
Substituting these values, we get:
F_static_friction = 0.7 * 3.626 N = 2.5382 N.

3. The maximum acceleration occurs when the static friction force reaches its maximum value.
Therefore, the maximum force of static friction must equal the force of gravity acting on the mug:
F_static_friction = F_gravity
2.5382 N = 3.626 N.
Rearranging the equation, we find:
a = F_static_friction / m
a = 2.5382 N / 0.37 kg ≈ 6.86 m/s^2.

Therefore, the maximum acceleration of the car so that the mug does not slide off is approximately 6.86 m/s^2.

To find the maximum acceleration of the car without the mug sliding off, we need to consider the forces acting on the mug.

First, let's determine the force of static friction (F_static) between the mug and the roof. The formula for static friction is:

F_static = coefficient of static friction * normal force

The normal force acting on the mug is equal to its weight (mg), where m is the mass of the mug and g is the acceleration due to gravity (9.8 m/s²). Therefore, the normal force is:

Normal force = m * g

Substituting the given values:

Normal force = 0.37 kg * 9.8 m/s²

Next, we can calculate the maximum static friction force:

F_static = 0.7 * Normal force

Now, let's consider the force of kinetic friction (F_kinetic) once the mug begins to slide. The formula for kinetic friction is:

F_kinetic = coefficient of kinetic friction * normal force

Substituting the given values:

F_kinetic = 0.4 * Normal force

Since we want to find the maximum acceleration without the mug sliding off, the maximum static friction force should be equal to or greater than the force of kinetic friction.

F_static ≥ F_kinetic

0.7 * Normal force ≥ 0.4 * Normal force

Now, let's solve for the maximum acceleration (a) using Newton's second law:

F_applied - F_static = m * a

The force applied on the mug is the force exerted by the car's acceleration, which is equal to the product of the total mass (car's mass + mug's mass) and acceleration (a):

F_applied = (m_car + m_mug) * a

Substituting the given values:

F_applied = (840 kg + 0.37 kg) * a

Combining the equations:

(m_car + m_mug) * a - 0.7 * Normal force = m_mug * a

(840 kg + 0.37 kg) * a - 0.7 * (0.37 kg * 9.8 m/s²) = 0.37 kg * a

Now, let's solve the equation for the maximum acceleration (a).