Problem 25 - University Physics with Modern Physics

Question

You are asked to design a spring that will give a 1160-kg satellite a speed of 2.50 m/s relative to an orbiting space shuttle. Your spring is to give the satellite a maximum acceleration of 5.00g. The spring's mass, the recoil kinetic energy of the shuttle, and changes in gravitational potential energy will all 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) k = 364260 N/m (b) x = 0.156 m

Step 1: Determine Maximum Force Required

We need to find the maximum force (\( F_{max} \)) that the spring must exert to provide the satellite with an acceleration of 5.00g. We know that \( g = 9.81 \ \text{m/s}^2 \), thus \( 5.00g = 5 imes 9.81 = 49.05 \ \text{m/s}^2 \). Therefore, the maximum force is given by \( F_{max} = m imes a_{max} = 1160 \times 49.05 = 56908 \ \text{N} \).

Step 2: Calculate the Force Constant of the Spring

The force constant \( k \) is found using Hooke's law, where \( F_{max} = k \times x_{max} \). We don't have \( x_{max} \) yet, so use the equation for energy \( \frac{1}{2}kx^2 = \frac{1}{2}mv^2 \) to substitute and solve for \( k \). Thus, \( kx_{max} = mv^2 \ \implies k = \frac{mv^2}{x_{max}^2} = \frac{1160 \times 2.50^2}{(x_{max})^2} = 7250/x_{max}^2 \).

Step 3: Express Compression Distance in Terms of Force Constant

From \( F_{max} = k \times x_{max} \), we find \( x_{max} = F_{max}/k \ \text{Thus,} \ k = \frac{56908^2}{mv^2} = \frac{56908^2}{1160 \times 2.50^2} \).Simplify to calculate \( k. \)

Key Concepts

Force Constant

The force constant is an essential parameter for springs, often represented by the symbol \( k \). It quantifies the stiffness of a spring. A higher force constant means the spring is stiffer, requiring a more significant force to compress or stretch it by a certain amount. When designing a spring to perform specific tasks—like launching a satellite—the force constant must be calculated accurately to ensure the spring provides the right amount of force.

• The unit of force constant is Newton per meter (N/m).
• A spring with a high force constant requires more force to compress.
• To find the force constant for a spring, we often use Hooke's Law: \( F = kx \).

In the given exercise, the force constant is determined by ensuring the spring imparting enough force to accelerate a satellite to a certain speed. This involves calculations using the given conditions of mass and acceleration, derived from the specified acceleration (which is expressed here as \( 5.00g \)) to find the force required.

Hooke's Law

Hooke's Law is a principle that states that the force exerted by a spring is directly proportional to its displacement. When you compress or stretch a spring, you feel resistance; this resistance is described by Hooke's Law.

• The formula for Hooke's Law is \( F = kx \), where \( F \) is the force applied by the spring, \( k \) is the force constant, and \( x \) is the displacement (compression or extension) from the spring's equilibrium position.
• This law is valid for the elastic region of a spring, meaning it holds true only when the spring is not permanently deformed.

For solving the problem, applying Hooke's Law helps us connect force, spring constant, and displacement, allowing us to understand how much the spring needs to be compressed to achieve the desired outcome without exceeding the system's limits.

Spring Compression

Spring compression refers to the extent a spring is shortened when force is applied. It is a critical factor in problems involving springs, as the compression will determine how much energy the spring can store and release.

• The displacement of the spring, often denoted as \( x \), is calculated from Hooke’s Law \( x = \frac{F}{k} \).
• The amount of compression determines the potential energy stored in the spring, given by the formula \( \frac{1}{2}kx^2 \).
• In the exercise, the compression of the spring is significant for ensuring it delivers the necessary kinetic energy to the satellite without exceeding its limit.

By manipulating the force constant and the maximum allowable force, we can calculate the precise compression required to achieve the desired satellite launch speed.

Kinetic Energy

Kinetic energy is the energy an object possesses due to its motion. When a spring is used to launch an object, it converts its stored potential energy into kinetic energy.

• The formula for kinetic energy is \( KE = \frac{1}{2}mv^2 \), where \( m \) is the mass and \( v \) is the velocity of the object.
• In spring-related problems, the stored energy in the spring, once released, becomes the kinetic energy of the object.
• For the exercise, it is crucial to ensure that the kinetic energy imparted to the satellite by the spring provides the required speed to complete its launch task.

By equating the potential energy in the compressed spring to the kinetic energy needed by the satellite, we can find the specific values like the force constant and compression distance needed for successful operation.