### Video Transcript

The figure shows a body of mass
one-quarter kilograms before it started to slide along the surface. The two surfaces π΄π΅ and πΆπ· are
smooth. However, the horizontal plane π΅πΆ
is rough, and its coefficient of kinetic friction is seven-tenths. If the body started moving from
rest, find the distance that the body covered on π΅πΆ until it came to rest. Consider the acceleration due to
gravity to be π equals 9.8 meters per second squared.

We can add the labels of mass is
equal one-quarter kilogram and π, the coefficient of kinetic friction, is equal to
seven-tenths to the figure. We can apply the principle of
conservation of energy, where the total initial energy of the system is equal to the
total final energy of the system, to our problem.

Initially, at position π΄, the
object is at rest, so the only form of energy that the object has is gravitational
potential energy. The object comes to rest somewhere
along the plane of π΅πΆ. Therefore, the final energy does
not have any potential energy or kinetic energy, only the work done against
friction. We should recall that the
gravitational potential energy of an object is equal to the mass of the object times
acceleration due to gravity times the height of the object above the ground.

We can substitute in ππβ for the
potential energy in our formula. We should also recall that the work
done by friction is equal to the force of friction times the distance traveled,
which allows us to substitute in force of friction times distance for the work. The problem does not give us the
force of friction, but it does give us the coefficient of kinetic friction. So we must use the definition of
force of friction, which is the coefficient of kinetic friction times the normal
reaction force.

The normal reaction force is the
force that a surface puts on an object. When an object is traveling along a
horizontal surface as, it does in this problem, without any other vertical forces
besides the force of gravity and the normal reaction force acting on it, then we can
say the normal reaction force is equal to the force of gravity, where the force of
gravity is the mass of the object times the acceleration due to gravity.

Looking at our expanded equation,
we can see that thereβs an π and a π on both sides of the equation. Since these are both nonzero
numbers, we can cancel them out. Now we can plug in the values from
our problem, four for the height and seven-tenths for the coefficient of kinetic
friction. To isolate π, we can multiply both
sides of the equation by 10 over seven. This will cancel out the
seven-tenths on the right side. The left side of the equation
multiplies out to be 40 over seven. The distance the body covered on
π΅πΆ until it came to rest is 40 over seven meters.