A horizontal spring with spring constant 73.2 N/m is compressed 16.7 cm and used to launch a 2.42 kg box across a frictionless, horizontal surface. After the box travels some distance, the surface becomes rough. The coefficient of kinetic friction of the box on the surface is 0.148. How far does the box slide across the rough surface before stopping?
The stored potential energy of
E = (1/2)kX^2 = (1/2)(73.2)(0.167)^2
is first converted to kinetic energy. Then a decelerating frictional force
F = M g *0.148 is applied until FX' = E.
That is when it stops.
The distance that it slides over the rough surface is X' = E/F . Solve for that.
X' = [(1/2)(73.2)(0.167)^2]/[2.42*9.8*0.148 ]
To find the distance the box slides across the rough surface before stopping, we need to calculate the work done against friction.
First, let's find the initial potential energy stored in the compressed spring:
Potential Energy (PE) = 0.5 * k * x^2
Where:
k = spring constant
x = compression or elongation of the spring
Given:
k = 73.2 N/m
x = 16.7 cm = 0.167 m
PE = 0.5 * (73.2 N/m) * (0.167 m)^2
PE = 0.86 J
Next, let's find the work done against friction. The work done by friction is equal to the force of friction multiplied by the distance traveled:
Work (W) = force of friction * distance
Given:
Mass of the box (m) = 2.42 kg
Coefficient of kinetic friction (μ) = 0.148
The force of friction can be found using the equation:
Force of friction = μ * Normal force
The Normal force (N) is equal to the weight of the box (mg), where g is the acceleration due to gravity:
Normal force (N) = mg
First, let's calculate the weight of the box:
Weight (W) = mass * acceleration due to gravity
W = 2.42 kg * 9.8 m/s^2
W = 23.716 N
Now, let's calculate the force of friction:
Force of friction = μ * Normal force
Force of friction = 0.148 * 23.716 N
Force of friction = 3.51 N
Finally, let's find the distance the box slides by rearranging the work equation:
Distance (d) = Work (W) / force of friction
Distance = 0.86 J / 3.51 N
Distance ≈ 0.245 m
Therefore, the box slides approximately 0.245 meters across the rough surface before stopping.