Problem 77 - College Physics

Question

the task of designing an accelerometer to be used inside a rocket ship in outer space. Your equipment consists of a very light spring that is \(15.0 \mathrm{~cm}\) long when no forces act to stretch or compress it, \(\begin{array}{lllll}\text { plus } & \text { a } & 1.10 & \text { kg } & \text { weight. }\end{array}\) be attached to a friction- free tabletop, while the \(1.10 \mathrm{~kg}\) weight is attached to the other end, as shown in Figure 5.67 . (Such a spring-type accelerometer system was actually used in the ill-fated Genesis Mission, which collected particles of the solar wind. Unfortunately, because it was installed backward, it did not measure the acceleration correctly during the craft's descent to earth. As a result, the parachute failed to open and the capsule crashed on Sept. \(8,2004 .)\) (a) What should be the force constant of the spring so that it will stretch by \(1.10 \mathrm{~cm}\) when the rocket accelerates forward at \(2.50 \mathrm{~m} / \mathrm{s}^{2}\) ? Start with a free-body diagram of the weight. (b) What is the acceleration (magnitude and direction) of the rocket if the spring is compressed by \(2.30 \mathrm{~cm} ?\)

Step-by-Step Solution

Answer

(a) The force constant \\( k \\) is 250 N/m. (b) The rocket's acceleration is 5.23 m/s^2 in the opposite direction.

Step 1: Analyze the Free-Body Diagram

To find the force constant of the spring, first consider the free-body diagram of the 1.10 kg weight. When the rocket accelerates, the only horizontal force on the weight is the spring force, which must be equal to the mass times the acceleration of the rocket.

Step 2: Apply Newton's Second Law

Newton's second law states that force equals mass times acceleration. Thus, the force exerted by the spring is equal to the product of the mass of the weight (1.10 kg) and the rocket's acceleration (2.50 m/s^2).\[ F = ma = 1.10 ext{ kg} \times 2.50 ext{ m/s}^2 = 2.75 ext{ N} \]

Step 3: Use Hooke's Law to Find Force Constant

Hooke's Law relates the force exerted by a spring to its displacement, given by \( F = kx \), where \( k \) is the spring constant and \( x \) is the displacement. Substitute the known values:\[ 2.75 ext{ N} = k \times 0.0110 ext{ m} \] Solving for \( k \), we get:\[ k = \frac{2.75}{0.0110} = 250 ext{ N/m} \]

Step 4: Determine Rocket's Acceleration for Compressed Spring

When the spring is compressed by 2.30 cm, we again use Hooke's Law to determine the restoring force:\[ F = kx = 250 ext{ N/m} \times 0.0230 ext{ m} = 5.75 ext{ N} \] Using Newton’s second law, the new acceleration \( a \) of the rocket is given by:\[ F = ma \Rightarrow 5.75 ext{ N} = 1.10 ext{ kg} \times a \] Solving for \( a \), we have:\[ a = \frac{5.75}{1.10} = 5.23 ext{ m/s}^2 \]

Step 5: Assess Acceleration Direction

A compressed spring exerts a force opposite to its displacement. In this scenario, the compression indicates the rocket is decelerating. Hence, the acceleration of the rocket is 5.23 m/s^2 in the opposite direction to the initial acceleration.

Key Concepts

Newton's Second Law

Newton's Second Law is fundamental in understanding the behavior of objects when subjected to forces. It tells us that an object's acceleration is directly proportional to the net force acting on it and inversely proportional to its mass. The law is mathematically expressed as \( F = ma \), where \( F \) represents the force, \( m \) is the mass, and \( a \) is the acceleration.
For example, in the spring accelerometer design exercise, the force exerted by the spring is calculated using this law. Here, the weight of the spring, which is 1.10 kg, experiences an acceleration due to the motion of the rocket. When the rocket accelerates forward, the spring force equals the product of mass and acceleration, providing a clear application of Newton's second law.
This direct relation ensures that we can predict how much force will be needed for a known mass and desired acceleration, a crucial component for designing systems to measure or respond to forces, such as an accelerometer in a rocket. It allows us to monitor changes in motion and understand dynamic systems effectively.

Free-Body Diagram

A free-body diagram is a simple, insightful tool that helps visualize the forces acting on an object. It's a crucial step in problem-solving in physics because it isolates an object and all the forces working on it. By representing these forces as vectors, we can better understand how they interact and affect the object’s motion.
In the context of the spring accelerometer, the free-body diagram shows the forces on the 1.10 kg weight. There's a horizontal spring force acting on the weight when the rocket accelerates. Since the tabletop is frictionless, it simplifies the analysis by ensuring the spring force is the only horizontal force.
This visualization helps in systematically applying Newton’s Second Law. We see how the spring force opposes or aids the motion and how it relates to the changes in acceleration of the weight, providing insights into the mechanics of the entire system.

Hooke's Law

Hooke's Law is a principle of physics that connects the force exerted by a spring to its displacement. Expressed with the formula \( F = kx \), where \( k \) is the spring constant and \( x \) is the displacement, it describes how springs behave under applied forces.
In the accelerometer design, Hooke's Law is essential for determining the spring constant \( k \), needed for specific displacement behavior. When calculating what spring constant would cause a 1.10 cm stretch at an acceleration of 2.50 m/s², Hooke's Law provides the framework. The earlier calculation of the spring force using Newton’s second law, 2.75 N, allows us to rearrange and solve \( k = \frac{F}{x} \). This process lets us tailor the spring to the application's specific needs, ensuring accurate measurements.
Similarly, when the spring is compressed, Hooke’s Law helps find the new force and consequently adjust to new conditions, essential for controlled environments like space missions.

Rocket Acceleration Analysis

Analyzing rocket acceleration involves understanding how forces affect motion. In this context, it means using known physical principles to determine how a spring's behavior can reveal changes in a rocket's speed.
When designing the accelerometer for a rocket, we examined both scenarios of the spring stretching and compressing. The direction and magnitude of these deformations indicate the rocket's motion status. If the spring stretches, it suggests forward acceleration, while compression hints at deceleration.
The exercise's initial calculation shows that a 1.10 cm displacement indicates acceleration at 2.50 m/s². Meanwhile, a 2.30 cm compression results in a calculation of 5.23 m/s². This methodical approach allows engineers to monitor a rocket's performance and adapt as necessary. This understanding of acceleration is crucial for safely navigating and controlling the rocket on its journey, emphasizing why tools like accelerometers are indispensable in aerospace technology.