Answer:
The block slides on the horizontal surface 25 m before coming to rest.
Explanation:
Hi there!
For this problem, we have to use the energy-conservation theorem. Initially, the block has only gravitational potential energy (PE) that can be calculated as follows:
PE = m · g · h
Where:
m = mass of the block.
g = acceleration due to gravity.
h = height at which the block is located.
As the block starts to slide down the track, its height diminishes as well as its potential energy. Due to the conservation of energy, energy can´t disappear, so the loss of potential energy is compensated by an increase of kinetic energy (KE). In other words, as the block slides, the potential energy is converted into kinetic energy. The equation of kinetic energy is the following:
KE = 1/2 · m · v²
Where:
m = mass of the block.
v = speed of the block.
Then, at the bottom of the ramp, the kinetic energy of the block will be equal to the potential energy that the block had at the top of the ramp.
Initial PE = KE at the bottom
When the block starts sliding horizontally, friction force does work to stop the block. According to the energy-work theorem, the change in the kinetic energy of an object is equal to the net work done on that object. In other words, the amount of work needed to stop the block is equal to its kinetic energy. Then, the work done by friction will be equal to the kinetic energy of the block at the bottom, that is equal to the potential energy of the block at the top of the track:
initial PE = KE at the bottom = work done by friction
The work done by friction is calculated as follows:
W = Fr · Δx
Where:
W = work
Fr = friction force.
Δx = traveled distance.
And the friction force is calculated as follows:
Fr = μ · N
Where:
μ = coefficient of friction.
N = normal force.
Since the block is not accelerated in the vertical direction, in this case, the normal force is equal to the weight (w) of the block:
Sum of vertical forces = ∑Fy = N - w = 0 ⇒N = w
And the weight is calculated as follows:
w = m · g
Where m is the mass of the block and g the acceleration due to gravity.
Then, the work done by friction can be expressed as follows:
W = μ · m · g · Δx
Using the equation:
intial PE = work done by friction
m · g · h = μ · m · g · Δx
Solving for Δx
h/μ = Δx
5.0 m / 0.20 = Δx
Δx = 25 m
The block slides on the horizontal surface 25 m before coming to rest.
