A box of mass m and initial speed v0 slides distance d across a horizontal floor before coming to rest. Use work and energy to find an expression for the coefficient of kinetic friction.Express your answer in terms of the variables v0, d, and free fall acceleration g.
Why did the box go to the party? Because it wanted to slide and have some fun! Now, let's get serious and solve this problem using work and energy.
The work done by the friction force can be found by taking the negative of the work done by the net force acting on the box. Since the box comes to rest, the net work done on it is equal to zero. So, we have:
Work done by friction = - (Work done by net force)
The work done by friction is equal to the product of the friction force and the distance over which it acts (d):
-W_friction * d = 0
Now, let's define the work done by the net force as the change in kinetic energy of the box:
Work done by net force = ΔKE
Since the box comes to rest, its initial kinetic energy (KE_initial) must be equal to zero. Therefore, we have:
ΔKE = KE_final - KE_initial
ΔKE = 0 - 0
ΔKE = 0
Now, using the work-energy principle, we can express the change in kinetic energy as the work done by the net force:
ΔKE = Work done by net force
0 = Work done by net force
Since the net force acting on the box is the sum of the friction force (f_friction) and the gravitational force (f_gravity), we can write:
0 = Work done by friction + Work done by gravity
The work done by the friction force is given by -W_friction * d, and the work done by the gravitational force is given by -m * g * d (negative because the forces act in opposite directions). Therefore, we have:
0 = -W_friction * d - m * g * d
Rearranging the equation, we get:
W_friction = -m * g
Since the work done by friction is defined as the product of the friction force and the distance over which it acts, we can write:
W_friction = f_friction * d
Substituting this into the equation, we get:
f_friction * d = -m * g
Finally, rearranging the equation to solve for the friction force:
f_friction = -m * g / d
So, the coefficient of kinetic friction (μ_k) can be defined as:
μ_k = f_friction / f_normal
The normal force (f_normal) is equal to the weight of the box (m * g). So, we can substitute this value into the equation to get:
μ_k = f_friction / (m * g)
Substituting the value of f_friction obtained earlier, we get:
μ_k = (-m * g / d) / (m * g)
Simplifying the equation, we find:
μ_k = -1 / d
And there you have it! The coefficient of kinetic friction can be expressed as -1 divided by the distance (d).
To find the expression for the coefficient of kinetic friction, we need to use the work-energy principle. According to the work-energy principle, the work done on an object is equal to the change in its kinetic energy.
The work done on the box by the friction force can be expressed as:
Work = force * distance
The force of friction can be calculated using Newton's second law, which states that the net force acting on an object is equal to its mass times its acceleration. Since the box comes to rest, the net force acting on it is given by:
Force = mass * acceleration
In this case, the acceleration is the acceleration due to gravity, g.
The distance the box slides, d, can be expressed using the equation of motion:
d = (v_f^2 - v_0^2) / 2a,
where v_f is the final velocity, and v_0 is the initial velocity.
Since the box comes to rest, the final velocity is zero (v_f = 0).
Now, we can substitute the equations and variables into the work expression:
Work = force * distance
= (mass * acceleration) * distance
= (mass * g) * distance
The work done on the box is equal to the change in its kinetic energy, which can be expressed as:
Work = ΔKE = 1/2 * mass * final velocity^2
Since the final velocity is zero (v_f = 0), the above equation simplifies to:
Work = ΔKE = 1/2 * mass * 0^2 = 0
Setting the expressions for work equal to each other, we can solve for the coefficient of kinetic friction:
(mass * g) * distance = 0
Simplifying the equation gives us:
distance = 0
Since the box slides a non-zero distance before coming to rest, this indicates that the coefficient of kinetic friction is:
μ_k = 0
Therefore, the expression for the coefficient of kinetic friction, in terms of the variables v0, d, and the acceleration due to gravity g, is μ_k = 0.
To find the expression for the coefficient of kinetic friction, let's start by analyzing the work-energy principle.
According to the work-energy principle, the work done on an object is equal to its change in kinetic energy. In this case, the work done on the box is the net work, which is given by the product of the net force acting on the box and the distance it moves. The net force can be broken down into the force of gravity (mg) and the force of kinetic friction (μk * mg), where μk is the coefficient of kinetic friction.
The work done by the net force can be written as:
Work = Force * Distance
Since the force of gravity and the force of kinetic friction are in the opposite direction to the motion of the box, we can express this as:
Work = (-mg - μk * mg) * d
Now, we need to find the change in kinetic energy. Initially, the box has kinetic energy due to its initial speed v0, and finally, it comes to rest, which means its final kinetic energy is zero.
Change in Kinetic Energy = Final Kinetic Energy - Initial Kinetic Energy
= 0 - (1/2) * m * v0^2
According to the work-energy principle, the work done on the box equals the change in its kinetic energy:
(-mg - μk * mg) * d = - (1/2) * m * v0^2
Now, let's rearrange the equation to solve for the coefficient of kinetic friction (μk):
μk * mg * d = (1/2) * m * v0^2 - mg * d
Dividing both sides by mg * d, we get:
μk = [(1/2) * m * v0^2 - mg * d] / (mg * d)
Simplifying further:
μk = (1/2) * v0^2 / (g * d) - 1
Therefore, the expression for the coefficient of kinetic friction (μk) is:
μk = (1/2) * v0^2 / (g * d) - 1
This expression shows how the coefficient of kinetic friction depends on the initial speed (v0), the distance traveled (d), and the acceleration due to gravity (g).
F = mu m g
work = F d = mu m g d
Ke lost = (1/2) m Vo^2
Vo^2 = 2 mu g d
mu = Vo^2/(2gd)