Problem 8 - University Physics with Modern Physics

Question

Object \(A\) is four times heavier than object B. Fach object is attached to a spring, and the springs have equal spring constants. The two objects are then pulled from their equilibrium positions and released from rest. What is the ratio of the periods of the two oscillators if the amplitude of \(A\) is half that of \(B ?\) a) \(T_{A}: T_{\mathrm{g}}=1: 4\) c) \(T_{A}: T_{\mathrm{B}}=2\) : b) \(T_{A}: T_{B}=4: 1\) d) \(T_{A}: T_{B}=1: 2\)

Step-by-Step Solution

Answer

Answer: The ratio of the periods is 2 : 1.

Step 1: Identify the formula for the period of a simple harmonic oscillator

For a simple harmonic oscillator, the period (T) is given by the formula: \(T = 2\pi\sqrt{\frac{m}{k}}\), where \(m\) is the mass of the object, and \(k\) is the spring constant.

Step 2: Set up the variables for objects A and B

Let \(m_A = 4m_B\) be the mass of object A, and \(m_B\) be the mass of object B since object A is four times heavier than object B. The spring constants are equal, so let \(k_A = k_B = k\).

Step 3: Calculate the periods for objects A and B

Using the period formula, we will calculate the periods for objects A and B: \(T_A = 2\pi\sqrt{\frac{m_A}{k}} = 2\pi\sqrt{\frac{4m_B}{k}}\) \(T_B = 2\pi\sqrt{\frac{m_B}{k}}\)

Step 4: Determine the ratio of the periods

Now we will find the ratio of the periods: \(\frac{T_A}{T_B} = \frac{2\pi\sqrt{\frac{4m_B}{k}}}{2\pi\sqrt{\frac{m_B}{k}}}\) Cancel out the common factors and simplify: \(\frac{T_A}{T_B} = \frac{\sqrt{4m_B}}{\sqrt{m_B}} = \frac{2\sqrt{m_B}}{\sqrt{m_B}} = 2\) So the ratio of the periods is \(T_A : T_B = 2 : 1\), which corresponds to answer choice (c).

Key Concepts

Oscillation Period

The oscillation period, often denoted as T, is a fundamental concept when exploring simple harmonic motion. It describes the time it takes for an object to complete one full cycle of motion – from its starting position, through its maximum displacement, to its opposite extreme, and then back again to the initial position. For a mass-spring system, the period T is determined by using the formula:

\[\begin{equation}T = 2\text{\pi}\sqrt{\frac{m}{k}}\text{\end{equation}\]}
Where:

• m represents the mass of the object attached to the spring
• k is the spring constant, which is a measure of the stiffness of the spring

Importantly, the period depends on the square root of the mass and is inversely proportional to the square root of the spring constant, meaning if the mass increases, the period will also increase, and if the spring constant increases, the period decreases. This concept is crucial for understanding how different factors affect the motion of oscillating systems.

Spring Constant

Another core component of simple harmonic motion is the spring constant, denoted by the symbol k. This constant is a measure of the stiffness of a spring and dictates how resistant it is to being compressed or stretched. Mathematically, it's defined by Hooke's Law, which states that the force F exerted by a spring is directly proportional to the displacement x that is applied to it:

\[F = -kx\]
In this context, the negative sign indicates that the force exerted by the spring is in the opposite direction of the displacement. The higher the spring constant, the more force is needed to stretch or compress it by a given amount. This characteristic directly influences the oscillation period of mass-spring systems, as mentioned in the previous section.

Mass-Spring System

When diving into the dynamics of a mass-spring system, one must carefully consider both the mass of the object and the spring constant of the spring involved. A mass-spring system consists of a mass (m) that is attached to a spring with a certain spring constant (k). This type of system can undergo simple harmonic motion when displaced from its equilibrium position.

The motion of such a system is predictable and repeatable, characterized by a back-and-forth oscillation about the equilibrium position. A fundamental property of the mass-spring system is that its motion is determined solely by m and k, regardless of the amplitude of oscillation – which means how far the mass is pulled or pushed from the equilibrium position does not affect the system's period, a fact reflected in the period formula and the exercise. This concept paves the way for a wide range of applications, including timekeeping in watches and seismology, where the principles governing mass-spring systems are utilized to interpret the vibrations of the Earth.