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 0.100 . 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.
.1/9.81
sorry
a/9.81 = .1
a = .981 m/s^2
To find the maximum acceleration of the plane without the cup sliding backward on the table, we need to consider the forces acting on the cup.
1. Determine the weight force acting on the cup:
The weight force is given by the equation: F_weight = mass * gravitational acceleration
Since the weight force only acts downward and the altitude remains constant, it does not contribute to the cup sliding backward.
2. Find the maximum static friction force:
The maximum static friction force is given by the equation: F_friction_max = coefficient of static friction * normal force
The normal force, in this case, is equal to the weight force acting on the cup.
3. Calculate the maximum acceleration using Newton's second law:
The net force acting on the cup is the difference between the maximum static friction force and any external forces causing the cup to move backward. In this case, the net force is equal to zero because the cup does not slide backward.
Therefore, the maximum static friction force is equal to the external force, and we can equate this to mass * acceleration:
F_friction_max = mass * acceleration
Substituting the equations from steps 2 and 3, we get:
coefficient of static friction * normal force = mass * acceleration
Since the normal force is equal to the weight force, we can rewrite the equation as:
coefficient of static friction * mass * gravitational acceleration = mass * acceleration
Using the equation for gravitational acceleration (g = 9.81 m/s^2), we can simplify further:
coefficient of static friction * mass * 9.81 m/s^2 = mass * acceleration
Simplifying the equation, we find:
acceleration = coefficient of static friction * 9.81 m/s^2
Given that the coefficient of static friction is 0.100, we can substitute it into the equation:
acceleration = 0.100 * 9.81 m/s^2
Calculating the value, we find:
acceleration = 0.981 m/s^2
Therefore, the maximum acceleration that the plane can have without the cup sliding backward on the table is 0.981 m/s^2.
To determine 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.
When the cup is at rest, the force of static friction acts between the cup and the table, preventing it from sliding. The maximum static frictional force can be calculated using the formula:
maximum static frictional force (F_friction) = coefficient of static friction (μ) * normal force (N)
The normal force (N) acting on the cup is equal to the weight of the cup, which can be calculated using the formula:
weight (W) = mass (m) * gravitational acceleration (g)
Since we are given the coefficient of static friction (μ = 0.100) and the value of gravitational acceleration (g = 9.81 m/s^2), we can substitute these values into the equations.
Now, when the plane accelerates forward, an additional force (due to the acceleration) acts on the cup in the opposite direction of the plane's movement. This force is equal to the mass of the cup (m) times the acceleration of the plane (a).
To maintain the cup's position on the table without sliding, the maximum static frictional force must be able to counteract this additional force. Therefore, the maximum static frictional force (F_friction) must be equal to or greater than the force due to the acceleration (m * a).
Substituting the equations and rearranging, we get:
μ * m * g ≥ m * a
Cancelling out the mass (m) on both sides, we have:
μ * g ≥ a
Since we want to find the maximum acceleration (a), we rearrange the equation to solve for a:
a ≤ μ * g
Plugging in the given values, we have:
a ≤ 0.100 * 9.81 m/s^2
Therefore, the maximum acceleration that the plane can have without the cup sliding backward on the table is:
a ≤ 0.981 m/s^2