Problem 37 - College Physics

Question

\(\bullet\) In designing a machine part, you need a spring that is 8.50 \(\mathrm{cm}\) long when no forces act on it and that will store 15.0 \(\mathrm{J}\) of energy when it is compressed by 1.20 \(\mathrm{cm}\) from its equilibrium position. (a) What should be the force constant of this spring? (b) Can the spring store 850 \(\mathrm{J}\) by compression?

Step-by-Step Solution

Answer

(a) The force constant is 208333 N/m. (b) No, the spring cannot store 850 J by compression.

Step 1: Understand Hooke's Law and Energy Stored in a Spring

Hooke's law states that the force exerted by a spring is proportional to its displacement: \( F = kx \), where \( k \) is the spring constant and \( x \) is the displacement from the equilibrium position. The potential energy stored in the spring is given by \( U = \frac{1}{2} k x^2 \). In this exercise, we need to find \( k \) and check if the spring can store a certain amount of energy.

Step 2: Calculate the Spring Constant (Question a)

To find the spring constant \( k \), use the formula for potential energy: \( U = \frac{1}{2} k x^2 \). Rearranging the equation for \( k \), we have: \( k = \frac{2U}{x^2} \). Plug in the values: \( U = 15.0 \) J and \( x = 1.20 \) cm = 0.012 m. Calculate \( k \): \[ k = \frac{2 \times 15.0}{(0.012)^2} = \frac{30}{0.000144} = 208333.33 \, \text{N/m} \]. Thus, the force constant of the spring is approximately 208333 N/m.

Step 3: Analyze Energy Storage Capability (Question b)

To determine if the spring can store 850 J by compression, use the energy formula: \( U = \frac{1}{2} k x^2 \) and solve for \( x \):\[ x = \sqrt{\frac{2U}{k}} \]. Substitute \( U = 850 \) J and \( k = 208333 \) N/m: \[ x = \sqrt{\frac{2 \times 850}{208333}} \approx \sqrt{0.00815} \approx 0.090 \, \text{m} = 9.00 \, \text{cm} \].This displacement is greater than its original length (8.50 cm), indicating it can't be compressed that much. The spring cannot store 850 J by compression.

Key Concepts

Spring Constant

A spring constant, often denoted as \( k \), is a crucial component in understanding how springs behave under force. It serves as a measure of a spring's stiffness, effectively describing how much force is required to compress or elongate the spring by a unit of length. This reliance is captured succinctly in Hooke's Law, which states:

• \( F = kx \), where:
  • \( F \) is the force applied to the spring,
  • \( x \) is the displacement from the spring's equilibrium position,
  • \( k \) is the spring constant.

By using this formula, you can understand how much resistance a spring will offer when compressed or stretched by a certain amount. In our problem, the spring is compressed and asked to store a specific amount of energy. To find the spring constant, rearrange the potential energy equation into:

\[ k = \frac{2U}{x^2} \]where \( U \) is the potential energy, and \( x \) is the displacement.

Potential Energy in Springs

Potential energy in springs is the energy stored when a spring is either compressed or stretched. This energy is determined by both the spring constant and the displacement from its natural length. The formula to calculate this energy is:

\[ U = \frac{1}{2} k x^2 \]where:

• \( U \) is the potential energy stored in the spring,
• \( k \) is the spring constant,
• \( x \) is the distance the spring is compressed or stretched.

As we observed in the exercise, a spring compressed by 1.20 cm stores 15.0 J of energy. By knowing both the energy stored and the displacement, you can determine the spring constant, which enables a prediction of how much compression would be needed to store different amounts of energy.

However, storing large amounts of energy, like 850 J in our exercise, requires significantly more compression than might be possible, demonstrating the determining factor that potential energy has on understanding a spring's limitations.

Mechanical Energy

Mechanical energy in the context of springs encompasses both the kinetic and potential energies within a system. For springs, the main focus is often on potential energy, specifically how it changes as the spring is compressed or extended. Through the conservation of mechanical energy, we recognize that the total energy within an isolated system—such as a spring—remains constant, allowing us to predict outcomes when forces act on the spring.

In our case, this principle tells us that when a spring is compressed, the mechanical energy remains conserved, manifested as potential energy. However, the transformation of mechanical energy from its stored form can only reach specific limits defined by the system's parameters like the spring constant and maximum stretch or compression. Thus, even though a higher mechanical energy storage is theoretically possible, practical limitations—like the physical limits of the spring—often restrict it.

In summary, the interplay between potential energy storage and the spring constant governs the mechanics of springs, helping engineers determine how suitable a spring is for specific applications, such as the storage of mechanical energy in devices.