You push a box along the floor against a constant force of friction. When you push with a horizontal force of 78 N the acceleration of the box is 0.46 m/s2; when you increase the force to 86 N the acceleration is 0.76 m/s2. (a) Find the mass of the box.
(b) Find the coefficient of kinetic friction between the box and the floor.
78-mu(mass)g=mass*.46
86-mu(mass)g=mass*.76
subtract the first equation from the second
8=mass(.30)
solve for mass.
b. in either equation,put in mass, g, then solve for mu.
To find the mass of the box, we can use Newton's second law of motion, which states that the net force applied to an object is equal to the mass of the object multiplied by its acceleration.
Let's use the first scenario, where the force applied is 78 N and the acceleration is 0.46 m/s^2.
According to Newton's second law:
Net force = mass x acceleration
The net force in this case is the applied force (78 N) minus the force of friction. Let's call the force of friction 'f':
Net force = 78 N - f
Substituting the values into the equation:
78 N - f = mass x 0.46 m/s^2
Now, let's consider the second scenario, where the force applied is 86 N and the acceleration is 0.76 m/s^2.
Using the same equation:
86 N - f = mass x 0.76 m/s^2
We now have a system of two equations with two unknowns (mass and friction force). We can solve these equations simultaneously to find the mass and force of friction.
Subtracting the first equation from the second equation, we can eliminate the force of friction (f):
(86 N - f) - (78 N - f) = mass x (0.76 m/s^2 - 0.46 m/s^2)
86 N - f - 78 N + f = mass x 0.3 m/s^2
The friction forces cancel out, and we are left with:
8 N = mass x 0.3 m/s^2
Now, we can solve for the mass:
mass = 8 N / 0.3 m/s^2
mass = 26.67 kg
Therefore, the mass of the box is approximately 26.67 kg.
Now, to find the coefficient of kinetic friction between the box and the floor, we can use the formula:
Force of friction = coefficient of kinetic friction x normal force
From the first scenario, the force of friction can be calculated as follows:
f = 78 N - (mass x 0.46 m/s^2)
f = 78 N - (26.67 kg x 0.46 m/s^2)
f = 78 N - 12.27 N
f = 65.73 N
The normal force is equal to the weight of the box, which is given by:
Normal force = mass x gravity
Normal force = 26.67 kg x 9.8 m/s^2
Normal force = 261.666 N
Now we can find the coefficient of kinetic friction:
coefficient of kinetic friction = force of friction / normal force
coefficient of kinetic friction = 65.73 N / 261.666 N
coefficient of kinetic friction ≈ 0.251
Therefore, the coefficient of kinetic friction between the box and the floor is approximately 0.251.