Problem 19 - Calculus Early Transcendentals

Question

Suppose a force of \(30 \mathrm{N}\) is required to stretch and hold a spring \(0.2 \mathrm{m}\) from its equilibrium position. a. Assuming the spring obeys Hooke's law, find the spring constant \(k\) b. How much work is required to compress the spring \(0.4 \mathrm{m}\) from its equilibrium position? c. How much work is required to stretch the spring \(0.3 \mathrm{m}\) from its equilibrium position? d. How much additional work is required to stretch the spring \(0.2 \mathrm{m}\) if it has already been stretched \(0.2 \mathrm{m}\) from its equilibrium position?

Step-by-Step Solution

Answer

Answer: The spring constant in this scenario is \(k = 150 \mathrm{N/m}\). The work required to compress the spring 0.4 m from its equilibrium position is 12 Joules, to stretch it 0.3 m from its equilibrium position is 6.75 Joules, and to stretch it an additional 0.2 m after it has already been stretched 0.2 m is 9 Joules.

Step 1: Write Hooke's Law equation

Write the formula for Hooke's Law: \(F = -kx\)

Step 2: Apply the given values to find k

We are given that a force of \(F = 30 \mathrm{N}\) is required to stretch the spring a distance of \(x = 0.2 \mathrm{m}\). Plug these values into the equation and solve for k: \(30 = -k(0.2)\) \(k = -\frac{30}{0.2}\) \(k = 150 \mathrm{N/m}\) #b. Work to compress the spring 0.4 m from its equilibrium position#

Step 3: Write the formula for work in terms of spring constant and displacement

The work done on the spring is given by the formula: \(W = \frac{1}{2}kx^2\)

Step 4: Apply the given values to find the work

We are asked to find the work required to compress the spring \(0.4 \mathrm{m}\). Since the spring is being compressed, the displacement will be negative, so \(x = -0.4 \mathrm{m}\). Plug this value into the equation and solve for the work: \(W = \frac{1}{2}(150)(-0.4)^2\) \(W = 12 \mathrm{J}\) (Joules) #c. Work to stretch the spring 0.3 m from its equilibrium position#

Step 5: Apply the given values to find the work for stretching 0.3 m

We are asked to find the work required to stretch the spring \(0.3 \mathrm{m}\). Since the spring is being stretched, the displacement will be positive, so \(x = 0.3 \mathrm{m}\). Plug this value into the equation and solve for the work: \(W = \frac{1}{2}(150)(0.3)^2\) \(W = 6.75 \mathrm{J}\) (Joules) #d. Additional work to stretch the spring 0.2 m if it has already been stretched 0.2 m from its equilibrium position#

Step 6: Calculate the initial and final work

We need to find the work required to stretch the spring an additional \(0.2 \mathrm{m}\) from a starting position of \(0.2 \mathrm{m}\) away from equilibrium. Calculate the initial work with \(x_1 = 0.2 \mathrm{m}\) and the final work with \(x_2 = 0.4 \mathrm{m}\): \(W_1 = \frac{1}{2}(150)(0.2)^2 = 3 \mathrm{J}\) \(W_2 = \frac{1}{2}(150)(0.4)^2 = 12 \mathrm{J}\)

Step 7: Calculate the additional work

Now find the additional work required by subtracting the initial work from the final work: \(W_\text{additional} = W_2 - W_1 = 12 \mathrm{J} - 3 \mathrm{J} = 9 \mathrm{J}\)

Key Concepts

Spring Constant

A spring constant, denoted by the letter \( k \), is a fundamental parameter in Hooke's Law. It measures the stiffness of a spring and tells us how much force is needed to stretch or compress the spring by a certain amount. In the formula \( F = -kx \), which is Hooke's Law, \( F \) represents the force applied to the spring, \( x \) is the displacement from the equilibrium position, and \( k \) is the spring constant.

The negative sign in the formula indicates that the force exerted by the spring is in the opposite direction of the displacement. A higher spring constant means a stiffer spring, requiring more force to achieve the same amount of displacement. For example, a spring constant of \( 150 \mathrm{N/m} \) suggests that 150 Newtons are required to stretch or compress the spring by one meter. Understanding the spring constant helps in predicting how springs behave under various loads.

Work Done on a Spring

The work done on a spring is the energy required to stretch or compress it. This is calculated using the formula \( W = \frac{1}{2}kx^2 \), where \( W \) represents work, \( k \) is the spring constant, and \( x \) is the displacement from the equilibrium position. The formula derives from the fact that the force required to move the spring changes linearly as the spring is displaced.

Since work is the energy transferred by a force over a distance, understanding how much work is done gives us insight into the energy changes within the spring as it is compressed or stretched. For example, stretching a spring with a constant of \( 150 \mathrm{N/m} \) by \( 0.3 \mathrm{m} \) requires \( 6.75 \mathrm{J} \) of work.

• This process emphasizes the concept that the required work increases with more displacement.
• To compress or stretch a spring, you need to exert energy, shown as work done on the spring.

Equilibrium Position

The equilibrium position of a spring is its natural, unstressed state, where it is neither stretched nor compressed. This is the point at which the net force on the spring is zero, meaning the spring is at rest. When you either stretch or compress a spring, you are moving it away from this equilibrium position.

In problems involving Hooke's Law, the equilibrium position serves as the reference point for measuring displacement. It is the baseline from which stretching or compressing is measured. Understanding the equilibrium position is crucial because, without it, you wouldn't have a starting point to measure how far a spring has been moved, consequently influencing how force and displacement are calculated.

Force and Displacement in Springs

The relationship between force and displacement in springs is described by Hooke's Law. This law establishes that the force required to either compress or stretch a spring is directly proportional to the displacement from its equilibrium position. The formula \( F = -kx \) captures this relationship, where \( F \) is the force, \( k \) is the spring constant, and \( x \) is the displacement.

A few important aspects to remember include:

• Displacement can be positive or negative, indicating stretching or compressing, respectively.
• The force needed increases linearly with more displacement, making it critical to know the spring constant to predict this force accurately.
• By understanding this relationship, you can predict how much force is required for a specified displacement and vice versa.

This linear relationship is fundamental to designing systems that involve springs, ensuring that they perform reliably under expected forces and displacements.