A box of unknown mass is sliding with an initial speed
vi = 5.60 m/s
across a horizontal frictionless warehouse floor when it encounters a rough section of flooring
d = 3.40 m
long. The coefficient of kinetic friction between the rough section of flooring and the box is 0.100. Using energy considerations, determine the final speed of the box after sliding across the rough section of flooring.
To determine the final speed of the box after sliding across the rough section of flooring using energy considerations, we need to consider the conservation of mechanical energy.
The mechanical energy of the box has two components: kinetic energy (KE) and gravitational potential energy (PE).
Initially, the box has only kinetic energy given by:
KE_initial = (1/2) * mass * vi^2
where vi is the initial speed and mass is the unknown mass of the box.
When the box slides across the rough section of flooring, friction acts in the direction opposite to the motion. This friction force does work and converts some of the initial kinetic energy into heat due to the roughness of the surface.
The work done by friction can be calculated using:
work_friction = friction_force * distance
where the friction_force is given by:
friction_force = coefficient_of_friction * normal_force
and the normal_force is equal to the weight of the box acting vertically downward (since the floor is rough and horizontal).
The work done by friction is equal to the change in kinetic energy:
work_friction = KE_initial - KE_final
where KE_final is the final kinetic energy of the box after sliding across the rough section.
Now, we can write the equation for the work done by friction:
coefficient_of_friction * normal_force * distance = (1/2) * mass * vi^2 - (1/2) * mass * vf^2
where vf is the final speed of the box.
Since the floor is horizontal, the normal_force is equal to the weight of the box:
normal_force = mass * g
where g is the acceleration due to gravity.
Substituting the expressions for normal_force and friction_force into the equation, we get:
coefficient_of_friction * mass * g * distance = (1/2) * mass * vi^2 - (1/2) * mass * vf^2
Simplifying the equation and solving for vf, we have:
vf^2 = vi^2 - 2 * coefficient_of_friction * g * distance
Taking the square root of both sides, we get:
vf = sqrt(vi^2 - 2 * coefficient_of_friction * g * distance)
Now we can substitute the given values:
vi = 5.60 m/s
coefficient_of_friction = 0.100
g = 9.8 m/s^2 (acceleration due to gravity)
distance = 3.40 m
Plugging these values into the equation, we find:
vf = sqrt((5.6^2) - 2 * (0.1) * (9.8) * (3.4))
vf = sqrt(31.36 - 6.828)
vf = sqrt(24.532)
vf = 4.95 m/s (approximately)
Therefore, the final speed of the box after sliding across the rough section of flooring is approximately 4.95 m/s.
To determine the final speed of the box, we can consider the conservation of mechanical energy during its motion across the rough section of flooring.
Let's break down the problem step by step:
1. Calculate the initial kinetic energy (KEi) of the box:
The formula for kinetic energy is KE = 0.5 * mass * velocity^2. However, we don't know the mass of the box, so let's denote it as 'm'.
KEi = 0.5 * m * (vi)^2
2. Calculate the work done by friction (Wfriction):
The work done by friction can be calculated using the equation W = force * distance. The force of friction can be determined by multiplying the coefficient of kinetic friction (μ) with the normal force acting on the box. In this case, since the box is on a horizontal surface with no vertical acceleration, the normal force is equal to the weight of the box (mg).
Wfriction = force of friction * distance
Wfriction = μ * m * g * d
3. Calculate the final kinetic energy (KEf) of the box:
Since energy is conserved, the initial kinetic energy minus the work done by friction should be equal to the final kinetic energy.
KEf = KEi - Wfriction
4. Solve for the final velocity (vf):
The final velocity can be determined by rearranging the equation for kinetic energy:
vf = sqrt(2 * KEf / m)
By following these steps, we can determine the final velocity of the sliding box after encountering the rough section of flooring.
KE₂-KE₁=W(fr)
mv₂²/2 -mv₁²/2 = F(fr)d cosα =
=μ•m•g•d•cos180º
v₂²/2 - v₁²/2 =μ•g•d• (-1)
v₂²/2= v₁²/2- μ•g•d
v₂= sqrt(v₁²- 2μ•g•d} = …