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 0.410 . 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? Use g = 9.81 m/s2.
the .410 coefficient means that the max acceleration is .410 g
To determine the maximum acceleration without the cup sliding backward on the table, we can use the equation for static friction:
Fs ≤ μs • N
Where:
Fs is the force of static friction
μs is the coefficient of static friction
N is the normal force
In this case, the force of gravity acting on the cup provides the normal force, given by:
N = m • g
Where:
m is the mass of the cup
g is the acceleration due to gravity
Let's assume the mass of the cup to be "m" kg. Therefore, the normal force is:
N = m • g
Now, we can determine the maximum value of the force of static friction as:
Fs ≤ μs • N
Fs ≤ μs • m • g
In this scenario, the maximum value of the force of static friction should be equal to the force required to accelerate the cup forward without it sliding backward.
So, the maximum acceleration that the plane can have without the cup sliding backward on the table is given by:
a = Fs / m
Now, let's substitute the values and calculate:
Given:
μs = 0.410
g = 9.81 m/s^2
Plugging in these values, the equation becomes:
a ≤ (μs • m • g) / m
Simplifying further:
a ≤ μs • g
Finally, substituting the value of μs = 0.410 and g = 9.81 m/s^2, we get:
a ≤ 0.410 • 9.81
a ≤ 4.01 m/s^2
Therefore, the maximum acceleration that the plane can have without the cup sliding backward on the table is 4.01 m/s^2.
To find the maximum acceleration that the plane can have without the cup sliding backward on the table, we need to consider the forces acting on the cup:
1. Weight (mg): This is the force due to gravity acting downward on the cup. The weight of the cup can be calculated using the formula weight = mass * acceleration due to gravity, where m is the mass of the cup and g is the acceleration due to gravity.
2. Normal force (N): This is the force exerted by the table on the cup perpendicular to the table's surface. Since the cup is at rest on the table, the normal force is equal in magnitude but opposite in direction to the weight of the cup.
3. Friction force (f): This is the force of static friction between the cup and the table. It acts parallel to the table's surface, opposing the motion of the cup.
The maximum static friction force can be calculated using the formula f = coefficient of static friction * N, where the coefficient of static friction is given as 0.410.
In order for the cup to remain in place and not slide backward, the static friction force should be equal to or greater than the force applied by the acceleration of the plane.
Let's assume that the maximum acceleration of the plane is a.
Since the altitude remains constant and there is no vertical motion, the normal force is equal to the weight (N = mg).
Now, we can set up the equation:
f = coefficient of static friction * N
ma = coefficient of static friction * mg
Rearranging this equation, we get:
a = (coefficient of static friction * g)
Plugging in the given values, we have:
a = (0.410 * 9.81)
Simplifying the calculation, we find:
a = 4.01 m/s^2
Therefore, the maximum acceleration that the plane can have without the cup sliding backward on the table is 4.01 m/s^2.