Problem 31 - University Physics with Modern Physics

Question

A block with mass \(m\) is attached to an ideal spring that has force constant \(k .\) (a) The block moves from \(x_{1}\) to \(x_{2},\) where \(x_{2}>x_{1} .\) How much work does the spring force do during this displacement? (b) The block moves from \(x_{1}\) to \(x_{2}\) and then from \(x_{2}\) to \(x_{1}\) . How much work does the spring force do during the displacement from \(x_{2}\) to \(x_{1} ?\) What is the total work done by the spring during the entire \(x_{1} \rightarrow x_{2} \rightarrow x_{1}\) displacement? Explain why you got the answer you did. (c) The block moves from \(x_{1}\) to \(x_{3},\) where \(x_{3}>x_{2}\) . How much work does the spring force do during this displacement? The block then moves from \(x_{3}\) to \(x_{2}\) . How much work does the spring force do during this displacement? What is the total work done by the spring force during the \(x_{1} \rightarrow x_{3} \rightarrow x_{2}\) displacement? Compare your answer to the answer in part (a), where the starting and ending points are the same but the path is different.

Step-by-Step Solution

Answer

(a) Work: \( \frac{1}{2} k x_1^2 - \frac{1}{2} k x_2^2 \). (b) Total work: 0. (c) Total work: \( \frac{1}{2} k x_1^2 - \frac{1}{2} k x_2^2 \), same as (a).

Step 1: Understanding the Problem

We have a block attached to a spring and need to calculate work done by the spring force as the block moves through different points. The work done by a spring is calculated using the spring force and Hooke's Law.

Step 1: Calculate Work from x1 to x2

The work done by a spring force can be calculated using the formula: \[ W = \frac{1}{2} k x_1^2 - \frac{1}{2} k x_2^2 \]for the displacement from \( x_1 \) to \( x_2 \). This formula comes from the change in potential energy of the spring.

Step 2: Work for x2 to x1 and Total for x1 to x2 to x1

For the block moving back from \( x_2 \) to \( x_1 \), the work done by the spring is the negative of the work done from \( x_1 \) to \( x_2 \): \[ W = \frac{1}{2} k x_2^2 - \frac{1}{2} k x_1^2 \]Therefore, the total work done over the path \( x_1 \rightarrow x_2 \rightarrow x_1 \) is zero because the work cancels out, confirming energy conservation in the system.

Step 3: Calculate Work from x1 to x3

For displacement from \( x_1 \) to \( x_3 \), the work done is: \[ W = \frac{1}{2} k x_1^2 - \frac{1}{2} k x_3^2 \]as calculated similarly to previous displacement.

Step 4: Work for x3 to x2 and Total for x1 to x3 to x2

From \( x_3 \) to \( x_2 \), the work done is:\[ W = \frac{1}{2} k x_3^2 - \frac{1}{2} k x_2^2 \]The total work done for the path \( x_1 \rightarrow x_3 \rightarrow x_2 \) is:\[ \left(\frac{1}{2} k x_1^2 - \frac{1}{2} k x_3^2\right) + \left(\frac{1}{2} k x_3^2 - \frac{1}{2} k x_2^2\right) = \frac{1}{2} k x_1^2 - \frac{1}{2} k x_2^2 \]This result is the same as in part (a) because the path taken does not affect the initial and final positions of the system, and potential energy is state-dependent.

Key Concepts

Hooke's Law

Hooke's Law is a fundamental principle in physics that describes how springs behave when they are compressed or stretched. The law states that the force exerted by a spring is directly proportional to the displacement from its equilibrium position. Mathematically, it can be expressed as:\[ F = -kx \]where:

• \( F \) is the spring force,
• \( k \) is the spring constant, which measures the stiffness of the spring,
• \( x \) is the displacement from the equilibrium position.

This relation is foundational because it helps us understand how springs store potential energy and how they exert forces. The negative sign indicates that the force exerted by the spring is always in the opposite direction of the displacement, thus restoring the object to its equilibrium position. This law is key when calculating work done by spring forces in various problems.

Spring Potential Energy

Spring potential energy refers to the energy stored in a spring when it is either compressed or stretched. According to Hooke's Law, the work done by the spring force during a displacement involves a change in this potential energy. The formula for spring potential energy is given by:\[ PE_{spring} = \frac{1}{2} k x^2 \]where:

• \( PE_{spring} \) is the potential energy stored in the spring,
• \( k \) is the spring constant,
• \( x \) is the displacement from the equilibrium position.

This formula indicates that the energy stored in the spring is proportional to the square of the displacement, meaning the further you stretch or compress the spring, the more energy is stored. Spring potential energy is essential for understanding how energy is conserved and transferred within systems that involve springs, which in turn influences how work is calculated between different states.

Conservation of Energy

The principle of conservation of energy is a cornerstone of physics, stating that in a closed system, the total energy remains constant. In the context of spring systems, conservation of energy plays a crucial role in explaining how energy is transferred between kinetic energy and potential energy without any net loss.When the block attached to a spring moves from one position to another, the work done is associated with changes in potential energy, as calculated with the formula:\[ W = \frac{1}{2} k x_1^2 - \frac{1}{2} k x_2^2 \]This reflects the shift in potential energy as the spring changes. When a spring returns to its original position after being displaced, the work done over the full cycle tends to zero, showing that energy has been conserved throughout the motion. This assurance of energy conservation helps us predict outcomes in mechanical problems, especially those involving cyclic or repetitive spring motions.