A cup of coffee is on a table in an airplane flying at a constant altitude and a constant velocity. The coefficient of static friction between the cup and the table is 0.18. Suddenly, the plane accelerates forward, its altitude remaining constant. What is the maximum acceleration that the plane can have without the cup sliding backward on the table?

A 301-kg motorcycle is accelerating up along a ramp that is inclined 30.0º above the horizontal. The propulsion force pushing the motorcycle up the ramp is 3420 N, and air resistance produces a force of 230 N that opposes the motion. Find the magnitude of the motorcycle's acceleration.

ma=F(fr) =μ•N = μ•m•g.

a= μ •g = 0.18•9.8=1.764 m/s².

To determine the maximum acceleration that the plane can have without the cup sliding backward on the table, you can use the concept of static friction.

1. Obtain the expression for the maximum static friction force:
The maximum static friction force can be calculated using the equation: Fs(max) = μs * N
where Fs(max) is the maximum static friction force, μs is the coefficient of static friction, and N is the normal force.

2. Determine the normal force acting on the cup:
The normal force is equal to the weight of the cup. Since the altitude is constant, the weight of the cup remains the same. Thus, the normal force is equal to the weight, which can be calculated using the formula N = m * g
where N is the normal force, m is the mass of the cup, and g is the acceleration due to gravity (approximately 9.8 m/s^2).

3. Calculate the maximum acceleration:
The maximum acceleration can be calculated by dividing the maximum static friction force by the mass of the cup (using Newton's second law: F = m * a). Thus, a = Fs(max) / m.

4. Substitute the given values into the equation:
- Coefficient of static friction (μs) = 0.18
- Acceleration due to gravity (g) = 9.8 m/s^2

Using these values, you can substitute them into the equations to calculate the maximum acceleration.

To determine the maximum acceleration that the plane can have without the cup sliding backward on the table, we need to consider the force of friction between the cup and the table.

The force of friction can be calculated using the equation:

frictional force = coefficient of static friction × normal force

The normal force is equal to the weight of the cup, which is the force of gravity acting on it. On an airplane flying at a constant altitude, the weight of the cup remains constant.

The maximum static frictional force is given by:

maximum static frictional force = coefficient of static friction × normal force

If the cup is on the verge of sliding, the maximum static frictional force should equal or be slightly less than the horizontal force exerted by the plane's acceleration. Therefore, we can equate these two forces:

force of acceleration = maximum static frictional force

Now we can substitute the relevant equations:

force of acceleration = coefficient of static friction × normal force

Plugging in values:

force of acceleration = 0.18 × (weight of the cup)

It is important to note that the weight of the cup is given by the formula:

weight = mass × acceleration due to gravity

Since the altitude remains constant, the acceleration due to gravity remains constant as well.

Now, we can rearrange the equation to solve for the acceleration:

acceleration = force of acceleration / weight

Substituting the respective values:

acceleration = (0.18 × weight of the cup) / weight of the cup = 0.18

Hence, the maximum acceleration that the plane can have without the cup sliding backward on the table is 0.18.