Problem 49 - A Graphical Approach to Precalculus with Limits

Question

Solve each problem. Hooke's Law If a 3 -pound weight stretches a spring 2.5 inches, how far will a 17 -pound weight stretch the spring?

Step-by-Step Solution

Answer

The 17-pound weight will stretch the spring 14.17 inches.

Step 1: Understand Hooke's Law

Hooke's Law states that the force required to extend or compress a spring by some distance is proportional to that distance. This can be expressed in the formula: \( F = kx \), where \( F \) is the force applied, \( k \) is the spring constant, and \( x \) is the distance that the spring is stretched or compressed.

Step 2: Set Up the Equation for Known Values

First, we identify the known values: a 3-pound weight stretches the spring by 2.5 inches. We can use these values to find the spring constant \( k \). Using the formula, set \( F = 3 \) pounds and \( x = 2.5 \) inches. Therefore, we have: \( 3 = k \times 2.5 \).

Step 3: Solve for the Spring Constant \( k \)

Rearrange the equation from Step 2 to solve for \( k \): \( k = \frac{3}{2.5} \). Perform the division to find \( k \).

Step 4: Use the Spring Constant to Find New Distance

Now, apply a 17-pound force and find the new stretch distance \( x \). Using the rearranged formula \( x = \frac{F}{k} \), substitute \( F = 17 \) and the calculated \( k \) from Step 3.

Step 5: Calculate the Distance \( x \) for a 17-Pound Weight

Substitute the values into the formula: \( x = \frac{17}{k} \). Use the value of \( k \) obtained earlier, perform the division, and find the new distance \( x \).

Key Concepts

Spring Constant

The spring constant, often denoted as \( k \), is a crucial factor in spring mechanics. It measures a spring's ability to resist deformation in response to an applied force. Simply put, it's a measure of a spring's stiffness. In Hooke's Law, the equation \( F = kx \) tells us that the force \( F \) applied to a spring is equal to the spring constant \( k \) multiplied by the displacement \( x \) from its original position.

• A larger \( k \) value means the spring is stiffer, requiring more force to stretch it the same distance compared to a spring with a smaller \( k \) value.
• Conversely, a smaller \( k \) means the spring is more flexible.

Calculating \( k \) involves rearranging the formula to \( k = \frac{F}{x} \). For instance, if a 3-pound force (\( F \)) stretches a spring by 2.5 inches (\( x \)), \( k \) would be \( \frac{3}{2.5} \). Compute this ratio to find the spring constant for use in subsequent calculations.

Force and Displacement

Force and displacement are central concepts in understanding Hooke's Law. When a force is applied to a spring, it causes the spring to extend or compress, known as the displacement. The amount of this extension or compression is directly related to both the applied force and the spring's constant, \( k \).

• The formula \( F = kx \) captures this relationship, where \( F \) is the applied force, and \( x \) is the displacement.
• If the force increases, the displacement also increases, assuming the spring constant remains unchanged.
• Conversely, reducing the force will decrease the displacement.

Understanding this relationship allows us to predict how a spring will behave under different forces. In practical terms, by knowing either the force or displacement and the spring constant, one can calculate the other value.

Proportionality

Proportionality is a key theme in Hooke's Law. The law is often expressed as \( F = kx \), highlighting that the force applied to a spring is directly proportional to its displacement. That means if you double the force, the stretch or compression of the spring doubles too, provided the spring operates within its elastic limits.

• This direct relationship makes it easier to predict spring behavior. If we know any two of the three variables \( F \), \( k \), or \( x \), we can find the third.
• This principle of proportionality applies as long as the spring does not reach its elastic limit, beyond which it may deform permanently.

When solving problems involving Hooke's Law, understanding proportionality helps confidently set up and manipulate equations to find unknown variables.

Physics Problem-Solving

Physics problem-solving often involves step-by-step analysis and application of established laws, like Hooke's Law, to determine unknowns. Here, we started with known values— a force of 3 pounds and a displacement of 2.5 inches to first find the spring constant \( k \).

• Once \( k \) was determined, it was used to calculate the new displacement caused by a 17-pound force.
• This analytical approach allows stepwise solving, ensuring each variable is accurately accounted for.

Such methodical problem-solving enhances comprehension of physical principles and hones skills critical in tackling real-world physics problems. The ability to break down complex situations into simpler mathematical equations is foundational in educating students in physics and other sciences.