Problem 7 - AP Physics 1 with 2 Practice Test 2nd

Question

A \(2-\mathrm{kg}\) mass is oscillating horizontally on a frictionless surface when attached to a spring. The total energy of the system is observed to be \(10 \mathrm{~J}\). If the mass is replaced by a 4 -kg mass, but the amplitude of oscillations and the spring remain the same, the total energy of the system will be (A) \(10 \mathrm{~J}\) (B) \(5 \mathrm{~J}\) (C) \(20 \mathrm{~J}\) (D) \(3.3 \mathrm{~J}\)

Step-by-Step Solution

Answer

The total energy of the system remains 10 J (Option A).

Step 1 - Understand the Total Energy in a Spring-Mass System

The total energy in a spring-mass system is given by the formula: \( E = \frac{1}{2} k A^2 \), where \( E \) is the total energy, \( k \) is the spring constant, and \( A \) is the amplitude of the oscillations. Thus, the total energy depends only on the spring constant and the amplitude of the oscillations.

Step 2 - Analyze Given Information

Given that a 2-kg mass has a total energy of 10 J, this means \( E = 10 \text{ J} \). According to our formula, this energy is determined by the spring constant \( k \) and the amplitude \( A \).

Step 3 - Replace Mass with Same Amplitude

When the mass is replaced with a 4-kg mass, we are asked to find the total energy. Importantly, the formula \( E = \frac{1}{2} k A^2 \) shows that the total energy depends solely on the spring constant \( k \) and amplitude \( A \), and not on the mass.

Step 4 - Conclude Total Energy with New Mass

Since the amplitude of oscillations \( A \) and spring constant \( k \) remain unchanged, the total energy of the system will also remain unchanged despite the mass being replaced. Therefore, the total energy will remain the same: 10 J.

Key Concepts

Oscillations

Oscillations are repetitive back and forth movements around an equilibrium position. In a spring-mass system, these movements are caused by the restoring force of the spring, trying to bring the mass back to its equilibrium position. When the mass is displaced, the spring force works to return it, creating an oscillatory motion.
For a horizontal spring-mass system on a frictionless surface, this means that the mass will continue to move back and forth indefinitely. The frequency and period of these oscillations depend on the properties of the system, such as the mass and the spring constant.
Understanding the nature of oscillations helps us predict how the system behaves over time. This knowledge is crucial because the total energy in the system remains constant for simple harmonic motion, assuming no energy is lost due to friction or other forces.

Spring Constant

The spring constant, denoted as \( k \), is a measure of the stiffness of a spring. It determines how much force is needed to stretch or compress a spring by a unit length. This is given by Hooke's Law: \[ F = -k x \]
Where:

• \(F\) is the restoring force exerted by the spring.
• \(k\) is the spring constant.
• \(x\) is the displacement from the equilibrium position.

The spring constant plays a significant role in determining the total energy in a spring-mass system. In the formula \( E = \frac{1}{2} k A^2 \), \( k \) directly affects the amount of energy stored in the system for a given amplitude \( A \).
A larger spring constant indicates a stiffer spring, which requires more force to achieve the same displacement, thus storing more energy for the same amplitude. Understanding the spring constant allows us to comprehend how different springs will respond under various forces.

Amplitude

Amplitude refers to the maximum displacement of the mass from its equilibrium position during oscillations. It is represented by \( A \) in the total energy formula \( E = \frac{1}{2} k A^2 \).
The amplitude is a crucial factor in determining the total energy of the system. Because the energy is proportional to the square of the amplitude, even small changes in amplitude can lead to significant changes in energy.
In our example, if the amplitude remains the same, the total energy does not change, even if the mass is replaced. This is because the energy depends solely on the spring constant and amplitude, not on the mass attached to the spring.
Understanding amplitude helps us predict the energy requirements and behavior of oscillatory systems. It's key to note that larger amplitudes result in higher energy and more vigorous oscillations.