You push a box along the floor against a constant force of friction. When you push with a horizontal force of 75 N the acceleration of the box is 0.55 m/s2; when you increase the force to 85 N the acceleration is 0.78 m/s2.
(a) Find the mass of the box. kg
(b) Find the coefficient of kinetic friction between the box and the floor.
Net force= m*a
75-friction=m*.55
85-friction=m*.78
subtract the first equation from the second
10=m(.23)
m= 10/.23
then solve for friction in either equation, finally, mu=frictionforce/mg
To solve this problem, we can use Newton's second law of motion, which states that the net force acting on an object is equal to the mass of the object multiplied by its acceleration. We can also apply the equation for the force of friction, which is equal to the coefficient of kinetic friction multiplied by the normal force.
(a) To find the mass of the box, we can first calculate the net force acting on the box for each case.
First, when a force of 75 N is applied, the net force can be calculated as follows:
Net force = applied force - force of friction
Net force = 75 N - force of friction
Second, when a force of 85 N is applied, the net force can be calculated as follows:
Net force = applied force - force of friction
Net force = 85 N - force of friction
We know the accelerations for each case, so we can use the equations:
Net force = mass x acceleration
For the first case:
75 N - force of friction = mass x 0.55 m/s²
For the second case:
85 N - force of friction = mass x 0.78 m/s²
Since the acceleration is caused by the net force, we can equate the two equations:
75 N - force of friction = 85 N - force of friction
Simplifying, we find:
75 N = 85 N
This equation shows that the force of friction is the same for both cases. Therefore, we can omit it from our calculations.
So, we have:
mass x 0.55 m/s² = mass x 0.78 m/s²
To find the mass of the box, we can cancel out the mass on both sides of the equation:
0.55 m/s² = 0.78 m/s²
The mass cancels out, and we are left with:
0.55 = 0.78
This equation is not true, which means there is an error in the problem or the given information.
(b) Since we couldn't find the mass of the box, we cannot find the coefficient of kinetic friction without it.
To find the mass of the box and the coefficient of kinetic friction, we can use Newton's second law of motion and the equation for the force of friction.
(a) Finding the mass of the box:
According to Newton's second law of motion, the force applied to an object is equal to the product of its mass and acceleration.
F = m * a
Let's use the values given:
Force, F = 75 N
Acceleration, a = 0.55 m/s^2
Substituting these values into the equation, we can solve for the mass, m:
75 N = m * 0.55 m/s^2
m = 75 N / 0.55 m/s^2
m = 136.36 kg (rounded to two decimal places)
Therefore, the mass of the box is approximately 136.36 kg.
(b) Finding the coefficient of kinetic friction:
The force of friction can be calculated using the equation:
Frictional force (Ff) = coefficient of kinetic friction (μ) * normal force (N)
The normal force is the force exerted by the surface on the box, which is equal to the weight of the box (mg).
Since the force applied when the acceleration is 0.55 m/s^2 (75 N) is less than the force applied when the acceleration is 0.78 m/s^2 (85 N), the difference in the applied force must be equal to the force of friction.
Let's calculate the force of friction using the two sets of given values:
Force difference = 85 N - 75 N = 10 N
Using this force difference and the mass of the box we calculated earlier (136.36 kg), we can determine the force of friction:
Frictional force (Ff) = Force difference
μ * mg = 10 N
Now, substitute the known values:
μ * (136.36 kg) * (9.8 m/s^2) = 10 N
Cancelling out the common factors and solving for μ gives:
μ ≈ 0.0074 (rounded to four decimal places)
Therefore, the coefficient of kinetic friction between the box and the floor is approximately 0.0074.