A 4kg box is compress 50 cm on a spring (k=100 N/m) and then slides across a horizontal floor. If (u=0.4) between the box and the floor
1) How much work is done by friction as the box comes to a stop?
2) Assuming the frictional force is constant, how far will the box slide before it comes to a stop?
1) The work done by friction will equal the initial potential energy of compression in the spring, but will be of the opposite sign. Work is done against friction, not by it.
2)
(1/2)kd^2 = M*g*u*X
d is the initial spring compression.
g is the acceleration of gravity
M is the box's mass
k is the spring constant
u is the kinetic coefficient of friction
Solve for X, the distance that the box slides.
To find the answers to these questions, we need to first understand the concepts of work and mechanical energy.
1) For the first question, we need to calculate the work done by friction as the box comes to a stop. Work is defined as the product of force and displacement. In this case, the force is the frictional force, and the displacement is the compression of the spring.
The work done by friction is given by the equation: Work = Force × Distance.
The force of friction can be calculated using the equation: Force of friction = μ × Normal force.
The normal force in this case is equal to the weight of the box, which can be calculated using the equation: Weight = Mass × Gravitational acceleration.
Given:
Mass of the box (m) = 4 kg.
Compression of the spring (d) = 50 cm = 0.5 m.
Coefficient of friction (μ) = 0.4.
Gravitational acceleration (g) = 9.8 m/s^2.
First, calculate the normal force:
Weight = 4 kg × 9.8 m/s^2 = 39.2 N.
Then, calculate the force of friction:
Force of friction = 0.4 × 39.2 N = 15.68 N.
Finally, calculate the work done by friction:
Work = Force × Distance = 15.68 N × 0.5 m = 7.84 Joules.
Therefore, the work done by friction as the box comes to a stop is 7.84 Joules.
2) For the second question, we need to determine how far the box will slide before coming to a stop. To do this, we need to calculate the work done against friction, which will be equal to the initial mechanical energy of the box.
The initial mechanical energy is given by the equation: Initial mechanical energy = Potential energy of the spring.
The potential energy of the spring is given by the equation: Potential energy = 0.5 × k × (Compression)^2.
Given:
Spring constant (k) = 100 N/m.
Compression of the spring (d) = 50 cm = 0.5 m.
First, calculate the potential energy of the spring:
Potential energy = 0.5 × 100 N/m × (0.5 m)^2 = 6.25 Joules.
Since the initial mechanical energy is equal to the work done against friction, which is equal to 7.84 Joules (as calculated in the first question), the box will slide until its mechanical energy is depleted.
Therefore, the box will slide until its mechanical energy is depleted, which is 6.25 Joules.