Problem 2 - Physics

Question

An archer, about to shoot an arrow, is applying a force of \(+240 \mathrm{~N}\) to a drawn bowstring. The bow behaves like an ideal spring whose spring constant is \(480 \mathrm{~N} / \mathrm{m}\). What is the displacement of the bowstring?

Step-by-Step Solution

Answer

The displacement is 0.5 meters.

Step 1: Understand the Problem

We need to determine the displacement of the bowstring when a force is applied to it. The bow behaves like an ideal spring, so we will use Hooke's Law to solve this problem.

Step 2: Hooke's Law Formula

Hooke's Law is expressed as \( F = kx \), where \( F \) is the force applied, \( k \) is the spring constant, and \( x \) is the displacement of the spring (or the bow).

Step 3: Rearrange the Formula

To find the displacement \( x \), we need to rearrange the formula to \( x = \frac{F}{k} \).

Step 4: Substitute Given Values

We substitute the given values into the formula: \( F = 240 \text{ N} \) and \( k = 480 \text{ N/m} \). So we have \( x = \frac{240}{480} \).

Step 5: Calculate the Displacement

Now, perform the division: \( x = \frac{240}{480} = 0.5 \). Thus, the displacement of the bowstring is 0.5 meters.

Key Concepts

Understanding the Spring Constant

The spring constant, denoted as \( k \), is a fundamental concept when dealing with springs, particularly in systems like bows that mimic spring behavior. It represents the stiffness or rigidity of a spring. A higher spring constant indicates a stiffer spring, which requires more force to achieve the same displacement compared to a spring with a lower constant.
To visualize this, imagine trying to stretch a rubber band versus a steel spring. The steel spring would have a higher spring constant because it is much harder to stretch. In our exercise, the bow's spring constant is given as \( 480 \text{ N/m} \), suggesting it is relatively stiff and requires a substantial force to achieve significant displacement.

• Measured in Newtons per meter (N/m).
• Directly proportional to the force applied.
• Determines how much force is needed for a unit of displacement.

Understanding the spring constant helps in predicting how the bowstring will respond when force is applied. It's crucial for designing systems that require precise control over force and movement.

Force and Displacement Connection

A pivotal concept in physics, the link between force and displacement is beautifully illustrated through Hooke's Law. When a force \( (F) \) is applied to a spring or a spring-like object, it causes the object to move, or displace. This displacement \( (x) \) is directly proportional to the force applied and inversely proportional to the spring constant \( (k) \).
According to Hooke's Law, the formula \( F = kx \) elegantly encapsulates this relationship. In our scenario:

• \( F = 240 \text{ N} \) is the force exerted by the archer pulling the bowstring.
• \( k = 480 \text{ N/m} \) is the spring constant of the bow modeled as a spring.
• \( x \) is the displacement we want to calculate.

Rearranging the formula to \( x = \frac{F}{k} \) allows us to directly calculate how far the bowstring is pulled back. It's essential as understanding this relationship aids in predicting how different forces will alter the position of the bowstring, impacting the arrow's potential energy and trajectory.

Characteristics of an Ideal Spring

The term "ideal spring" in physics represents a theoretical model of a spring that obeys Hooke's Law perfectly. In this model, a spring's force and displacement are linearly related without any deformation or change in its properties, irrespective of how much stress is applied within its elastic limit.
An ideal spring is characterized by:

• No energy loss due to heat.
• No permanent deformation.
• Response solely based on Hooke's Law \( F = kx \).

Our exercise uses the analogy of an ideal spring to describe the behavior of the bow. This simplifies the calculation as it assumes linear elasticity, meaning the relationship between force and displacement remains constant throughout the range of motion. Although no physical spring or bow can be truly ideal, this concept helps in understanding and simplifying complex physical systems, such as the bow being treated like an ideal spring for ease of calculation.