91Ó°ÊÓ

Imagine a spring as a fun and bouncy gadget! The spring constant, usually designated by the letter \( k \), is a value that tells us how stiff or stretchy the spring is. It's measured in newtons per meter (N/m), and it essentially tells us how much force is needed to stretch the spring by a certain amount.

When you have a large spring constant, the spring is quite stiff, meaning it requires more force to stretch it. Conversely, a smaller spring constant indicates a more flexible spring that stretches easily with less force. In the case of our bowstring, the spring constant is \( 480 \mathrm{N/m} \). This value helps the archer understand just how much force needs to be applied to achieve a certain stretch or displacement.

• High spring constant: Stiff spring, more force needed.
• Low spring constant: Flexible spring, less force needed.

With the spring constant known, we can apply Hooke's Law to find out how much the bowstring will move when the archer pulls with a given force.

Displacement is all about figuring out how much the bowstring stretches when a force is applied. In this context, displacement tells us the distance by which the bowstring is pulled back. To find this, we use the handy Formula from Hooke's Law: \[F = kx\]Where \( F \) is the force applied, \( k \) is the spring constant, and \( x \) is the displacement.

To solve for the displacement \( x \), we rearrange the formula:\[x = \frac{F}{k}\]

Using the values from the problem, where the force \( F \) is \( 240 \mathrm{N} \) and the spring constant \( k \) is \( 480 \mathrm{N/m} \), the calculation becomes:\[x = \frac{240}{480} = 0.5 \,\mathrm{m}\]

This means the bowstring is pulled back by 0.5 meters. It’s a simple division that gives us a clear understanding of how much movement occurs when a specific force is applied to an ideal spring system.

An ideal spring is a perfect simplification used in physics. It behaves in an absolutely predictable way according to Hooke's Law. But what does this actually mean? It implies that the force exerted by the spring is directly proportional to the displacement from its rest position.

For real-world applications, ideal springs provide a foundational model, even if real springs may not align perfectly due to factors like material fatigue or non-linear forces at extreme stretches. However, under normal conditions and within its elastic limit, many springs approximate this ideal behavior quite closely.

The bow in our exercise represents an ideal spring. This means it reliably follows Hooke's Law across the range of its motion, allowing straightforward calculations and predictions about how it will behave when the archer applies force. This ideality assumes no energy loss to friction or air resistance, creating a pure mathematical model that simplifies solving for displacement or force in practical situations.