Solve each problem. Hooke's Law for a SpringHooke's law for an elastic spring states that the distance a spring stretches varies directly as the force applied. If a force of 15 lb stretches a certain spring 8 in., how much will a force of 30 lb stretch the spring?

Verified step by step guidance

1. Identify the relationship described by Hooke's Law: the distance a spring stretches (d) varies directly as the force applied (F). This can be expressed as d = kF, where k is the constant of proportionality.
2. Use the given information to find the constant of proportionality (k). You know that a force of 15 lb stretches the spring 8 in., so substitute these values into the equation: 8 = k * 15.
3. Solve for k by dividing both sides of the equation by 15: k = 8/15.
4. Now, use the value of k to find the new distance the spring stretches when a force of 30 lb is applied. Substitute F = 30 and k = 8/15 into the equation: d = (8/15) * 30.
5. Simplify the expression to find the new distance d.

Key Concepts

Hooke's Law

Hooke's Law states that the force exerted by a spring is directly proportional to the distance it is stretched or compressed from its rest position. Mathematically, it is expressed as F = kx, where F is the force applied, k is the spring constant, and x is the displacement from the equilibrium position. This principle is fundamental in understanding how springs behave under various forces.

Direct Variation

Direct variation describes a relationship between two variables where an increase in one variable results in a proportional increase in the other. In the context of Hooke's Law, as the force applied to the spring increases, the distance the spring stretches also increases proportionally. This concept is crucial for solving problems involving linear relationships.

Proportional Relationships

Proportional relationships are mathematical relationships where two quantities maintain a constant ratio. In the case of the spring problem, the ratio of force to distance stretched remains constant. Understanding this concept allows for the calculation of unknown values when one variable is known, facilitating problem-solving in various algebraic contexts.