A 16 kg box slides across the floor with an initial velocity of +7.8 m/s. It comes to a stop after sliding +14.6 m. How much friction was present?
To determine the amount of friction present, we need to use the concept of kinetic friction.
The equation for frictional force is given by:
Frictional force (f) = coefficient of friction (μ) * normal force (N)
Since the box is sliding across the floor, the normal force is equal to the weight of the box.
Normal force (N) = mass (m) * acceleration due to gravity (g)
First, let's calculate the normal force:
m = 16 kg (given)
g = 9.8 m/s^2 (acceleration due to gravity)
N = m * g
N = 16 kg * 9.8 m/s^2
N = 156.8 N
Now, we need to find the coefficient of friction (μ). This depends on the nature of the surfaces in contact. We can assume it is constant.
Next, we can calculate the initial kinetic energy of the box using the formula:
Initial kinetic energy (KE_initial) = (1/2) * m * v^2
v = +7.8 m/s (given)
KE_initial = (1/2) * 16 kg * (7.8 m/s)^2
KE_initial = 499.68 J (joules)
As the box comes to a stop, all of its initial kinetic energy is converted into work done against friction.
The work done (W) against friction is equal to the product of the frictional force (f) and the distance (d) over which the force is applied:
W = f * d
Since the box comes to a stop, its final velocity is zero (v = 0 m/s). Therefore, we can use the work-energy principle to relate the initial kinetic energy to the work done against friction:
KE_initial = W
Now, we can determine the work done against friction:
W = KE_initial
W = 499.68 J
And since W = f * d, where d = +14.6 m (given), we have:
f * (+14.6 m) = 499.68 J
Simplifying:
f = 499.68 J / 14.6 m
f = 34.24 N
Therefore, the amount of friction present is approximately 34.24 N.