Mary applies a force of 71 N to push a box with an acceleration of 0.57 m/s2. When she increases the pushing force to 83 N, the box's acceleration changes to 0.86 m/s2. There is a constant friction force present between the floor and the box.

a) what is the coefficient of the friction between the floor and the box?

37.7

To find the coefficient of friction between the floor and the box, we need to use the equation relating force, mass, and acceleration.

Let's assume the mass of the box is "m".

When Mary applies a force of 71 N, the net force acting on the box is given by:
Net force = Applied force - Frictional force
71 N - Frictional force = mass × acceleration (equation 1)

Similarly, when Mary applies a force of 83 N, the net force acting on the box is given by:
Net force = Applied force - Frictional force
83 N - Frictional force = mass × acceleration (equation 2)

Now, we can solve the two equations to determine the frictional force and the mass of the box.

Subtracting equation 1 from equation 2:

(83 N - Frictional force) - (71 N - Frictional force) = (mass × acceleration) - (mass × acceleration)
83 N - 71 N = mass × acceleration - mass × acceleration
12 N = 0

Since 12 N = 0 results in an impossible condition, it means there is something wrong with the problem statement or the provided values. Please check the given information and try again.

To find the coefficient of friction between the floor and the box, we can use the following equation:

F_net = m * a

Where:
- F_net is the net force acting on the box
- m is the mass of the box
- a is the acceleration of the box

Let's break down the problem step by step to find the coefficient of friction.

Step 1: Find the mass of the box
Since we have the force and acceleration values, we can rearrange the above equation to solve for mass:

m = F_net / a

For the first scenario:
F_net1 = 71 N (force applied by Mary)
a1 = 0.57 m/s^2 (acceleration of the box)
Substituting these values into the equation:

m1 = 71 N / 0.57 m/s^2

Step 2: Find the net force acting on the box
In this case, the net force will be equal to the force applied by Mary minus the force of friction:

F_net1 = F_applied1 - F_friction

Step 3: Find the force of friction
Rearranging the previous equation, we get:

F_friction = F_applied1 - F_net1

Step 4: Calculate the coefficient of friction
The force of friction can be written as:

F_friction = μ * N

Where:
- μ is the coefficient of friction
- N is the normal force (equal to the weight of the box, which is m * g, where g is the acceleration due to gravity, approximately 9.8 m/s^2)

Substituting the expression for F_friction in Step 3 and N in the equation above, we have:

μ * m * g = F_applied1 - F_net1

Now, let's repeat this process for the second scenario.

Step 5: Find the mass of the box for the second scenario
Use the same equation as in Step 1 but substitute the force and acceleration values for the second scenario:

m2 = F_net2 / a2

For the second scenario:
F_net2 = 83 N (increased force applied by Mary)
a2 = 0.86 m/s^2 (new acceleration of the box)

Step 6: Find the net force acting on the box in the second scenario
Similarly, the net force is given by:

F_net2 = F_applied2 - F_friction

Step 7: Calculate the coefficient of friction for the second scenario
As before, the force of friction can be expressed as:

F_friction = μ * m * g

Substituting this into the equation and rearranging, we get:

μ * m * g = F_applied2 - F_net2

With these equations, we can calculate the mass and net forces for both scenarios and then solve for the coefficient of friction μ.