Problem 25 - University Physics with Modern Physics

Question

You are asked to design a spring that will give a \(1160-\mathrm{kg}\) satellite a speed of 2.50 \(\mathrm{m} / \mathrm{s}\) relative to an orbiting space shuttle. Your spring is to give the satellite a maximum acceleration of 5.00 \(\mathrm{g}\) . The spring's mass, the recoil kinetic energy of the shuttle, and changes in gravitational potential energy will be negligible. (a) What must the force constant of the spring be? (b) What distance must the spring be compressed?

Step-by-Step Solution

Answer

(a) The force constant of the spring is approximately 446584 N/m. (b) The spring must be compressed about 0.1274 m.

Step 1: Determine the Maximum Force

To find the maximum force exerted by the spring, use the maximum acceleration the satellite can have. The maximum acceleration is given by 5.00 g, where g is the acceleration due to gravity (9.81 m/s²).First, calculate the maximum acceleration:\[ a_{ ext{max}} = 5.00 \times 9.81 \, \text{m/s}^2 = 49.05 \, \text{m/s}^2 \]Next, use Newton's second law \( F = ma \) to find the maximum force:\[ F_{ ext{max}} = 1160 \, \text{kg} \times 49.05 \, \text{m/s}^2 = 56898 \, \text{N} \]

Step 2: Use Energy Conservation to Find Spring Constant

We need to find the spring constant \( k \). Use the energy conservation principle which involves the kinetic energy of the satellite and the potential energy stored in the spring.The kinetic energy \( KE \) of the satellite when it reaches a speed of 2.50 m/s is:\[ KE = \frac{1}{2} m v^2 = \frac{1}{2} \times 1160 \, \text{kg} \times (2.50 \, \text{m/s})^2 = 3625 \, \text{J} \]The potential energy stored in the compressed spring at maximum displacement is equal to the kinetic energy:\[ \frac{1}{2} k x^2 = 3625 \, \text{J} \]

Step 3: Relate Force and Displacement to Solve for Spring Constant

We know from Hooke's Law that \( F = kx \), where \( F = 56898 \, \text{N} \) and \( x \) is the displacement.From the previous step, we can use the relation for energy:\[ k x^2 = 2 \times 3625 \, \text{J} \to k = \frac{2 \times 3625}{x^2} \]Substitute \( F = kx \) into the equation:\[ x = \frac{F}{k} \to x = \frac{56898}{k} \]Plug this into the energy equation:\[ k \left( \frac{56898}{k} \right)^2 = 2 \times 3625 \]Solving gives:\[ 56898^2 = 2 \times 3625 \times k \Rightarrow k = \frac{56898^2}{2 \times 3625} \Rightarrow k \approx 446584 \, \text{N/m} \]

Step 4: Calculate Spring Compression Distance

Now we can find the distance \( x \) the spring needs to be compressed using \( F = kx \):\[ x = \frac{F}{k} = \frac{56898}{446584} \approx 0.1274 \, \text{m} \]

Key Concepts

Hooke's Law

Hooke's Law is a fundamental principle in physics that describes how the force required to extend or compress a spring is proportional to the distance it is stretched or compressed. This is expressed in the formula:

• \( F = kx \)

Where:

• \( F \) is the force applied to the spring (in Newtons, N)
• \( k \) is the spring constant (in Newtons per meter, N/m), characterizing the stiffness of the spring
• \( x \) is the displacement of the spring from its equilibrium position (in meters, m)

A larger spring constant \( k \) indicates a stiffer spring, which requires more force for a given displacement. In the context of the exercise, by understanding Hooke's Law, we calculated the maximum force the spring could exert on the satellite to determine how much it needed to be compressed.

Energy Conservation

Energy conservation is a vital principle governing physical interactions, stating that energy cannot be created or destroyed, only transformed from one form to another. In the exercise, energy conservation helps find the spring constant by equating the kinetic energy a satellite gains to the potential energy stored by the spring.The key concept is that when the spring is compressed, it stores potential energy described by:

• \( PE_{spring} = \frac{1}{2}kx^2 \)

When released, this potential energy converts into kinetic energy \( KE \) defined by:

• \( KE = \frac{1}{2}mv^2 \)

Ensuring energy conservation implies \( \frac{1}{2}kx^2 = \frac{1}{2}mv^2 \). This relationship allows us to calculate the spring constant \( k \) once the kinetic energy is known, confirming that no energy is lost in the transition from potential to kinetic.

Kinetic Energy

Kinetic energy is the energy of an object in motion and is indispensable for solving various physics problems, including those involving springs and accelerations. It allows scientists to predict how objects move and interact.For a moving satellite, its kinetic energy is calculated using:

• \( KE = \frac{1}{2}mv^2 \)

Where:

• \( m \) is the satellite's mass
• \( v \) is its velocity

In our exercise, knowing the satellite’s final velocity lets us determine its kinetic energy, which equates to the stored potential energy in the spring. This step is crucial to sizing the spring accurately to achieve the desired satellite speed upon release without any additional forces.

Newton's Second Law

Newton's second law of motion bridges the ideas of force, mass, and acceleration through the iconic equation:

• \( F = ma \)

This law is pivotal in calculating the force exerted by or on an object under uniform acceleration.In the context of this satellite problem, we first compute the maximum acceleration possible for the satellite (given as 5 times the acceleration due to gravity \( g \)). Substituting this acceleration into the equation allows us to calculate the maximum possible force exerted on the satellite. It embodies the essence of Newton's second law—how an unbalanced force acting on a mass always causes acceleration. Understanding and applying this law forms the foundational step for evaluating forces in mechanics, especially for systems involving springs like ours.