A cup of coffee is sitting on a table in an airplane that is flying horizontally. The coefficient of static friction between the cup and the table is 0.258. Suddenly, the plane accelerates horizontally. What is the maximum acceleration (in m/s2) that the plane can have without the cup sliding backward on the table?
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.
In this case, there are two main forces involved: the force of gravity acting vertically downward on the cup and the force of friction acting horizontally between the cup and the table. The force of friction opposes the motion and prevents the cup from sliding.
To find the maximum acceleration, we need to compare the maximum frictional force with the force of gravity. The maximum static frictional force can be calculated using the formula:
f_max = μ_s * N
where f_max is the maximum static frictional force, μ_s is the coefficient of static friction, and N is the normal force.
In this situation, the normal force exerted on the cup is equal to the force of gravity acting on it, since the cup is not accelerating vertically. The force of gravity can be calculated using the formula:
N = m * g
where m is the mass of the cup and g is the acceleration due to gravity (approximated as 9.8 m/s²).
Now we can substitute the values and solve for the maximum static frictional force:
f_max = 0.258 * m * g
To determine the maximum acceleration, we need to equate the maximum static frictional force to the force required to cause the cup to slide, which is given by:
f = m * a
where f is the force required to cause the cup to slide and a is the acceleration of the cup.
Setting f_max equal to f:
0.258 * m * g = m * a
Now, let's cancel out the mass (m) on both sides:
0.258 * g = a
Finally, we can calculate the maximum acceleration (a) by substituting the value of g:
a ≈ 0.258 * 9.8 (approximately)
a ≈ 2.53 m/s²
Therefore, the maximum acceleration that the plane can have without the cup sliding backward on the table is approximately 2.53 m/s².
a/g = .258
because
mu m g = m a