Problem 55 - College Physics

Question

\(\bullet\) \(\bullet\) Four passengers with a combined mass of 250 \(\mathrm{kg}\) compress the springs of a car with wom-out shock absorbers by 4.00 \(\mathrm{cm}\) when they enter it. Model the car and passengers as a single body on a single ideal spring. If the loaded car has a period of vibration of \(1.08 \mathrm{s},\) what is the period of vibration of the empty car?

Step-by-Step Solution

Answer

Calculate the mass of the car and then substitute to find the period of the empty car using spring constant.

Step 1: Understand the Problem

We need to determine the period of vibration of the car without passengers (empty car) using given data including the mass of passengers and the period of vibration when the car is loaded.

Step 2: Define Given Quantities

- Combined mass of passengers, \( m = 250 \text{ kg} \)- Spring compression due to passengers, \( x = 4.00 \text{ cm} = 0.04 \text{ m} \)- Period of loaded car, \( T' = 1.08 \text{ s} \)

Step 3: Relation between Period and Spring Constant

The period \( T' \) for a simple harmonic oscillator is given by \( T' = 2\pi \sqrt{\frac{m+M}{k}} \), where \( m \) is the mass of the passengers, \( M \) is the mass of the car, and \( k \) is the spring constant.

Step 4: Calculate the Spring Constant

Using Hooke's Law, \( F = kx \), and the fact that the force is the weight of the passengers \( F = mg \), we have:\[k = \frac{mg}{x} = \frac{250 \cdot 9.81}{0.04} = 61250 \, \text{N/m}\]

Step 5: Express Period of Empty Car

The period \( T \) of the empty car is given by:\[T = 2\pi \sqrt{\frac{M}{k}}\]We don't have the mass \( M \) of the car directly, but we can find it from the loaded car period.

Step 6: Use Loaded Car Period to Find Car Mass

From the loaded car:\[1.08 = 2\pi \sqrt{\frac{m+M}{k}} \\Rightarrow \left(\frac{1.08}{2\pi}\right)^2 = \frac{250 + M}{61250} \M = 61250 \left(\frac{1.08}{2\pi}\right)^2 - 250\]

Key Concepts

Physics Problem Solving

Physics problems often require us to break down given information and employ mathematical relationships to find a solution. In scenarios involving harmonic motion, understanding key concepts and how they relate to each other is crucial.

The given problem involves calculating the period of a vibration for an empty car using the data from when it is loaded. This requires an understanding of how mass, force, and spring constants interact.

To tackle problems like these:

• First, identify what is known from the problem – variables such as mass, compression, and period.
• Second, understand the relationship between these variables. For oscillatory motion, formulas like the period of a harmonic oscillator are essential.
• Third, use relevant equations to connect the given information to what is unknown.

By methodically approaching each piece of the problem, complex physical concepts become more manageable.

Spring Constant Calculation

The spring constant, a crucial component in the context of harmonic oscillators, defines how resistant a spring is to being compressed or stretched. It's denoted by the symbol \(k\) and measured in Newtons per meter (N/m).

To find the spring constant in this problem, we apply Hooke's Law, which is formulated as \( F = kx \). Here:

• \(F\) represents the force applied to the spring, equating to the gravitational force in this context, \( F = mg \).
• \(k\) is what we need to find.
• \(x\) is the displacement the spring undergoes due to the force, 0.04 meters, in our case.

Therefore, our equation to find the spring constant becomes: \ \[ k = \frac{mg}{x} = \frac{250 \times 9.81}{0.04} = 61250 \, \text{N/m} \]

The calculation provides a clear picture of the spring's stiffness, indicating how much force is needed for a given compression or stretch.

Simple Harmonic Motion

Simple harmonic motion (SHM) describes a type of oscillatory motion where the restoring force is directly proportional to the displacement and acts in the direction opposite to that of displacement. Objects like pendulums or oscillating springs can exhibit this motion.

The period \( T \) of a simple harmonic oscillator such as the spring in our problem is given by the formula: \ \[ T = 2\pi \sqrt{\frac{M}{k}} \]where:

• \(T\) is the period.
• \(M\) is the system's mass (in our case, the car's mass).
• \(k\) is the spring constant.

In our scenario, while the period of the loaded car is already known, finding the period of the empty car involves recalculating using the known spring constant and the car's calculated mass minus the passenger's weight.

Thus, understanding SHM principles allows us to predict and calculate motions of oscillatory systems.