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 0.420 . 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 only thing I could figure out was the static friction force which came out to be 4.12 N. I'm stuck trying to figure out how to find the acceleration. Please help!
The force needed to move the cup is Force Max(F)= Coefficient of friction(U)times Mass (M)times Gravity(G)and the force exerted by the plane is Mass(M) times Acceleration(A). Set these equal to each other since the force needed to move the cup is going to be the same as the force exerted by the plane and solve for (A).
UMG=MA
M cancels out leaving
UG=A
so the acceleration of the plane is equal to the coefficient of friction multiplied by gravity.
Well, I guess the cup of coffee is in for a wild ride!
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.
Since the cup is not sliding, the static friction force must be equal to the force pushing the cup forward, which is the product of the mass of the cup and the acceleration.
So we can write the equation as:
μs * N = m * a
where:
μs is the coefficient of static friction,
N is the normal force (equal to the weight of the cup),
m is the mass of the cup,
and a is the acceleration.
First, let's find the normal force N. The weight of the cup is given by:
Weight = mass * gravitational acceleration
N = m * g
Now we can substitute the value of N into the equation:
μs * (m * g) = m * a
Simplifying the equation, we find:
μs * g = a
Now we can plug in the values:
μs = 0.420 (given)
g = 9.81 m/s^2 (given)
a = 0.420 * 9.81
Calculating, we get:
a ≈ 4.09 m/s^2
So the maximum acceleration that the plane can have without the cup sliding backward on the table is approximately 4.09 m/s^2.
Keep in mind that this is a hypothetical scenario, and in reality, there might be other factors to consider. But hey, at least it gave us a chance to save some coffee spills!
To find the maximum acceleration at which the cup will not slide backward, we need to consider the forces acting on the cup and the maximum static friction force that can prevent it from sliding.
Let's analyze the problem step by step:
Step 1: Identify the forces acting on the cup.
- Weight (mg): The force due to gravity acting vertically downward with a magnitude of mg, where m is the mass of the cup and g is the acceleration due to gravity.
- Normal force (N): The force exerted by the table perpendicular to its surface. It is equal in magnitude and opposite in direction to the weight of the cup, N = mg.
- Static friction force (Ff): The force acting parallel to the surface of the table, opposing the motion, and preventing the cup from sliding backward.
Step 2: Determine the maximum static friction force.
The maximum static friction force can be found using the equation Ff(max) = μsN, where μs is the coefficient of static friction between the cup and the table, and N is the normal force. In this case, N = mg.
Therefore, Ff(max) = μs * mg.
Step 3: Equate the maximum static friction force to the force causing the cup to slide.
In this case, the force causing the cup to slide is the component of the force due to acceleration acting parallel to the table surface. Let's call it Fa.
We can equate Ff(max) to Fa, giving us:
μs * mg = Fa.
Step 4: Express Fa in terms of acceleration.
The force causing the cup to slide is given by Fa = ma, where m is the mass of the cup and a is the acceleration of the plane.
Step 5: Substitute back into the equation and solve for the maximum acceleration.
μs * mg = ma.
Rearrange the equation to solve for a:
a = (μs * mg) / m.
Given that μs = 0.420 and g = 9.81 m/s^2, you can substitute these values into the equation:
a = (0.420 * 9.81) / m.
Note: The mass of the cup (m) is not provided in the question. Therefore, you need to know the mass of the cup to calculate the maximum acceleration.
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.
Initially, when the plane is flying at a constant velocity, the cup is in equilibrium, and the static friction force between the cup and the table is equal to the force of gravity acting on the cup. We can calculate this by multiplying the mass of the cup (which we'll assume to be known) by the acceleration due to gravity (g = 9.81 m/s^2).
However, when the plane suddenly accelerates forward, an additional force acts on the cup in the same direction as the acceleration. This force is the force of inertia or the "pseudo-force" experienced by objects in the plane due to its acceleration.
If the acceleration of the plane is too high, the force of inertia will exceed the maximum static friction force, causing the cup to slide backward on the table.
To calculate the maximum acceleration, we can set up an equation equating the static friction force (Fs) to the force of inertia (Fi). The static friction force can be calculated as the product of the coefficient of static friction (μs) and the normal force (N) between the cup and the table. In this case, the normal force N is equal to the weight of the cup (mg), where m is the mass of the cup.
Therefore, the equation becomes:
Fs = Fi
μs * N = m * a
Where:
- μs is the coefficient of static friction (given as 0.420),
- N is the normal force (equal to mg),
- m is the mass of the cup (known),
- a is the acceleration of the plane (the unknown we want to find).
Substituting the expressions for N and Fs into the equation, we get:
μs * mg = m * a
Simplifying the equation, we find:
a = (μs * g)
Now we can substitute the given values into the equation to find the maximum acceleration:
a = (0.420) * (9.81 m/s^2)
a ≈ 4.11 m/s^2
Therefore, the maximum acceleration that the plane can have without the cup sliding backward on the table is approximately 4.11 m/s^2.