You push a box along the floor against a constant force of friction. When you push with a horizontal force of 72 N the acceleration of the box is 0.43 m/s2; when you increase the force to 77 N the acceleration is 0.58 m/s2.

(a) Find the mass of the box.

(b) Find the coefficient of kinetic friction between the box and the floor.

Solve the following pair of equations simltaneously:

72 - f = M * 0.43

77 - f = M * 0.58

f is the friction force.

Subtract the first equation from the second:

5 = 0.15 M
M = 33.33 kg

Then solve for f. Use either of the first two equations and plug in the value for M

The coefficient of friction is f/(M*g)

To find the mass of the box, we can use Newton's second law of motion, which states that the force acting on an object is equal to the mass of the object multiplied by its acceleration.

In this case, the force acting on the box is the applied force minus the force of friction. We can set up two equations using the given information and solve for the mass of the box.

Let's denote the mass of the box as "m".

For the first scenario with an applied force of 72 N and an acceleration of 0.43 m/s^2, the net force acting on the box can be calculated as:

Net force = Applied force - Force of friction
Net force = 72 N - Force of friction

Using Newton's second law, we have:

Net force = mass * acceleration
72 N - Force of friction = m * 0.43 m/s^2

For the second scenario with an applied force of 77 N and an acceleration of 0.58 m/s^2, we have:

Net force = Applied force - Force of friction
Net force = 77 N - Force of friction

Using Newton's second law, we have:

Net force = mass * acceleration
77 N - Force of friction = m * 0.58 m/s^2

Now, we can set up a system of equations:

72 N - Force of friction = m * 0.43 m/s^2 (Equation 1)
77 N - Force of friction = m * 0.58 m/s^2 (Equation 2)

To solve for the mass (m), we need to eliminate the force of friction from both equations.

We can subtract Equation 1 from Equation 2:

(77 N - Force of friction) - (72 N - Force of friction) = m * 0.58 m/s^2 - m * 0.43 m/s^2

5 N = m * (0.58 m/s^2 - 0.43 m/s^2)
5 N = m * 0.15 m/s^2

Now, we can solve for the mass (m):

m = 5 N / 0.15 m/s^2
m ≈ 33.333 kg

Therefore, the mass of the box is approximately 33.333 kg (or 33.3 kg, rounded to one decimal place).

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

Force of friction = coefficient of friction * normal force

The normal force acts vertically upward and is equal to the weight of the box, which can be calculated using the formula:

Weight = mass * gravity

Let's denote the coefficient of kinetic friction as "μ" and the acceleration due to gravity as "g".

Substituting the known values into the equation for the first scenario:

72 N - (μ * m * g) = m * 0.43 m/s^2

Substituting the known values into the equation for the second scenario:

77 N - (μ * m * g) = m * 0.58 m/s^2

We have two equations and two unknowns (μ and m * g). We can solve this system of equations simultaneously by substituting m * g from one equation into the other equation.

Solving the system of equations:

m * g = (72 N - m * 0.43 m/s^2) / μ (Equation 3)
m * g = (77 N - m * 0.58 m/s^2) / μ (Equation 4)

Setting Equation 3 equal to Equation 4:

(72 N - m * 0.43 m/s^2) / μ = (77 N - m * 0.58 m/s^2) / μ

Cross-multiplying:

72 N - m * 0.43 m/s^2 = 77 N - m * 0.58 m/s^2

Rearranging the equation:

77 N - 72 N = m * 0.58 m/s^2 - m * 0.43 m/s^2 + m * 0.43 m/s^2

5 N = m * 0.15 m/s^2

Dividing both sides of the equation by m:

5 N / m = 0.15 m/s^2

Since m is the mass of the box and 5 N is the weight of the box, we can substitute weight (m * g) into the equation:

weight / m = 0.15 m/s^2

weight = m * g
5 N = m * 9.8 m/s^2

Substituting the weight back into the equation:

5 N / (m * 9.8 m/s^2) = 0.15 m/s^2

Solving for m:

m = 5 N / (0.15 m/s^2 * 9.8 m/s^2)
m ≈ 3.4 kg

Therefore, the mass of the box is approximately 3.4 kg (or 3.4 kg, rounded to one decimal place).

The coefficient of kinetic friction can be found by substituting the mass of the box into either Equation 3 or Equation 4:

μ = (72 N - m * 0.43 m/s^2) / (m * g)
μ = (72 N - 3.4 kg * 0.43 m/s^2) / (3.4 kg * 9.8 m/s^2)

Calculating the value:

μ ≈ 0.169

Therefore, the coefficient of kinetic friction between the box and the floor is approximately 0.169 (or 0.169, rounded to three decimal places).