Answer:
The block travels 25 m on the horizontal surface before stopping.
Explanation:
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To solve this problem, we apply the principle of energy conservation. Initially, the block has gravitational potential energy (PE), which can be calculated as follows:
PE = m · g · h
Where:
m = block's mass.
g = gravitational acceleration.
h = height at which the block is situated.
As the block descends the track, its height reduces and, consequently, its potential energy diminishes. Due to the conservation of energy, the potential energy lost translates into increased kinetic energy (KE). In simpler terms, as the block slides, potential energy changes into kinetic energy. The kinetic energy formula is given as:
KE = 1/2 · m · v²
Where:
m = block's mass.
v = block's velocity.
Thus, at the bottom of the ramp, the block's kinetic energy corresponds to what it had as potential energy at the top.
Initial PE = KE at the base
Once the block slides on a level surface, friction acts to halt its motion. According to the energy-work theorem, the kinetic energy change of an object equals the total work performed on it. Therefore, to bring the block to rest, the work associated with friction equals the kinetic energy it possesses at the base, which is also the potential energy from the top:
initial PE = KE at the base = work exerted by friction
The work done by friction can be calculated like this:
W = Fr · Δx
Where:
W = work
Fr = frictional force.
Δx = distance covered.
Furthermore, the frictional force can be found as follows:
Fr = μ · N
Where:
μ = friction coefficient.
N = normal force.
Since the block is not moving vertically, the normal force is identical to the block's weight:
Sum of vertical forces = ∑Fy = N - w = 0 ⇒N = w
And the block's weight can be determined by:
w = m · g
Where m is the mass and g denotes the gravitational acceleration.
So, the work performed by friction can be expressed as:
W = μ · m · g · Δx
Utilizing the equation:
initial PE = work done by friction
m · g · h = μ · m · g · Δx
Solving for Δx, we have:
h/μ = Δx
5.0 m / 0.20 = Δx
Δx = 25 m
Thus, the block slides a total of 25 m on the level surface before it comes to a halt.
