Problem 67 - Physics

Question

The shock absorbers in the suspension system of a car are in such bad shape that they have no effect on the behavior of the springs attached to the axles. Each of the identical springs attached to the front axle supports \(320 \mathrm{~kg}\). A person pushes down on the middle of the front end of the car and notices that it vibrates through five cycles in \(3.0 \mathrm{~s}\). Find the spring constant of either spring.

Step-by-Step Solution

Answer

The spring constant for either spring is approximately \(35100\, \text{N/m}\).

Step 1: Understand the Problem

The problem gives us that each spring supports a mass of \(320\, \text{kg}\). When the car is pushed down, it vibrates through five cycles in \(3.0\, \text{s}\). We need to determine the spring constant \(k\) for one of these springs.

Step 2: Find the Frequency of the System

The frequency of oscillation can be found using the formula \( f = \frac{n}{t} \), where \(n\) is the number of cycles, and \(t\) is the time. Here \( n = 5 \) cycles and \( t = 3.0 \) seconds, so \( f = \frac{5}{3.0} \approx 1.67\, \text{Hz} \).

Step 3: Relate Frequency to Angular Frequency

The angular frequency \( \omega \) is related to the frequency \( f \) by the formula \( \omega = 2 \pi f \). Substituting \( f = 1.67\, \text{Hz} \) gives \( \omega = 2 \pi (1.67) \approx 10.47 \text{ rad/s} \).

Step 4: Use Angular Frequency to Find the Spring Constant

The angular frequency \( \omega \) is also related to the spring constant \( k \) and mass \( m \) by the formula \( \omega = \sqrt{\frac{k}{m}} \). Solving for \( k \), we get \( k = m \omega^2 \).

Step 5: Calculate the Spring Constant

Since each spring supports \( m = 320 \text{ kg} \) and \( \omega \approx 10.47 \text{ rad/s} \), substitute these into the formula: \( k = 320 \times (10.47)^2 \approx 35100\, \text{N/m} \). Thus, the spring constant of either spring is \( 35100 \text{ N/m} \).

Key Concepts

Oscillations

Oscillations occur in many physical systems and are characterized by a repetitive back and forth motion around a central point or equilibrium. In the context of a car's suspension system, oscillations are seen when the car vibrates up and down after it has been disturbed.
The complete motion from starting point, through its extremes, and back to starting point, forms one cycle of oscillation. Understanding oscillations is crucial for analyzing how the suspension system in a car reacts to different forces. This information helps in determining the damping and spring constants needed to ensure comfort during a ride.
When you push down on the car and it vibrates, those vibrations are essentially oscillations governed by the properties of the system, such as mass and spring constant. The number of oscillations or cycles in a given time frame provides useful information about the dynamics of the system.

Angular Frequency

Angular frequency, often denoted by \(\omega\), is a measurement of how quickly something oscillates and is related to the concept of rotational movement.
Though frequency \(f\) tells us how many cycles occur per second (measured in hertz), angular frequency gives us a similar concept but measured in radians per second. The relationship between the two is \( \omega = 2 \pi f \).
In our car suspension example, knowing the angular frequency helps us determine how fast the suspension system is oscillating. This in turn helps us find the spring constant of the system using the formula: \(\omega = \sqrt{\frac{k}{m}}\).
With this, we can determine the spring constant, which plays a critical role in designing a suspension system that effectively absorbs shocks and provides a smooth ride.

Suspension System

A suspension system in a car is designed to absorb shocks and ensure a smooth ride by managing the vibrations caused by uneven road surfaces. It involves various components such as springs and dampers (shock absorbers) working in unison.
In this scenario, the shock absorbers are ineffective, so the springs are solely responsible for managing these vibrations. The interplay between mass, spring constant, and damping force dictates how well a suspension system operates.
Key components of a suspension system include:

• Springs: Support the weight and absorb energy from vertical impacts.
• Shock absorbers: Dampen the energy to reduce oscillations and improve ride quality (though absent in this case).
• Control arms: Connects the wheels to the suspension, guiding motion and alignment.

Understanding these elements and how they contribute to the car's ride quality is essential when diagnosing issues or improvements within the suspension system.

Spring Mass System

A spring-mass system is a simple mechanical model used to describe the dynamics of many practical systems, such as a suspension in a vehicle. It typically consists of a mass attached to a spring that can oscillate about an equilibrium position.
In our problem, the mass is 320 kg, which is supported by a spring. The characteristics of this spring, specifically its spring constant, determine how the mass will oscillate when disturbed.
The spring constant \(k\) describes the stiffness of the spring, indicating how much force is needed to extend or compress the spring by a unit of distance. The relation is governed by Hooke's law: \( F = kx \). In the spring-mass model, we apply this to find out how the system responds to external forces, such as the weight of a car or an external push.
This concept is crucial for understanding how devices from car suspensions to building supports manage forces and motion dynamics, enabling the design of systems that are safe, stable, and perform as intended.