Final answer:
The length of the spring just as the box is ready to move is 50.352 cm. To find the length of the spring just as the box is ready to move, we need to consider the forces acting on the box. The static friction force between the box and the floor opposes the force applied by the spring. At the point just before the box starts moving, the static friction force reaches its maximum value, which is given by fs(max) = μsN where μs is the coefficient of static friction and N is the normal force.
Step-by-step explanation:
To find the length of the spring just as the box is ready to move, we need to consider the forces acting on the box. The static friction force between the box and the floor opposes the force applied by the spring. At the point just before the box starts moving, the static friction force reaches its maximum value, which is given by:
fs(max) = μsN
where μs is the coefficient of static friction and N is the normal force. The normal force is equal to the weight of the box, which is given by:
N = mg
where m is the mass of the box and g is the acceleration due to gravity. Since the box is at rest, the static friction force is equal to the force applied by the spring:
fs(max) = kx
where k is the spring constant and x is the displacement of the spring from its equilibrium position. Rearranging the equation, we can solve for x:
x = fs(max)/k
Substituting the values given in the question, we have:
x = (μsN)/k
= (0.65)(mg)/k
= (0.65)(18 kg)(9.8 m/s^2)/(2.5 N/cm)
= 50.352 cm.
Therefore, the length of the spring just as the box is ready to move is 50.352 cm.