Question

An oscillating block-spring system has a mechanical energy of \(1.18 \mathrm{~J}\), an amplitude of \(9.84 \mathrm{~cm}\), and a maximum speed of \(1.22 \mathrm{~m} / \mathrm{s} .\) Find \((a)\) the force constant of the spring, \((b)\) the mass of the block, and \((c)\) the frequency of oscillation.

Short answer

The force constant of the spring is approximately 245 N/m, the mass of the block is approximately 1.59 kg, and the frequency of oscillation is approximately 2.24 Hz.

Step-by-step solution

Finding the force constant of the spring

We know, \[\text{{Total Mechanical Energy}} = \frac{1}{2}kA^2\]So,\[k = \frac{2 \times \text{{Total Mechanical Energy}}}{A^2}\]Substituting values, we have\[k= \frac{2 \times 1.18}{(9.84 \times 10^{-2})^2} \approx 245 \, \text{{N/m}}.\]

Finding the mass of the block

The maximum speed is at equilibrium when all energy is kinetic. Hence, we have :\[\frac{1}{2} m v_{\text{{max}}}^2 = E\]So,\[m = \frac{2E}{v_{\text{{max}}}^2}\]Substituting our known values in, we get\[m= \frac{2 \times 1.18}{1.22^2} \approx 1.59 \, \text{{kg}}.\]

Finding the frequency of oscillation

The frequency of oscillation can be found using the formula:\[f = \frac{1}{2 \pi } \sqrt{\frac{k}{m}}\]Substituting the known values, we get\[f = \frac{1}{2 \pi } \sqrt{\frac{245}{1.59}} \approx 2.24 \, \text{{Hz}}.\]

Key concepts

Mechanical Energy

Mechanical energy in an oscillating block-spring system is the sum of potential and kinetic energy at any point during its oscillation. In simple terms, it's the energy that enables the block to move back and forth.

For a spring-block oscillator at maximum compression or extension (its amplitude), all the mechanical energy is stored as potential energy in the spring, which can be expressed as \frac{1}{2}kA^2\, where \(k\) is the force constant of the spring, and \(A\) is the amplitude of oscillation. At this point, the kinetic energy is zero as the block momentarily stops before changing direction.

Conversely, at the equilibrium position, where the spring is neither compressed nor extended, the mechanical energy is all kinetic (\frac{1}{2}mv_{max}^2) as the block moves at its maximum speed \(v_{max}\). The remarkable fact about mechanical energy in this closed system, assuming no external forces like friction, is that it remains constant throughout the oscillation, transforming between kinetic and potential forms.

Force Constant of a Spring

The force constant of a spring, often denoted by \(k\), is a measure of the spring's stiffness. It essentially tells us how much force is needed to extend or compress the spring by a certain amount. The higher the force constant, the stiffer the spring, and the more force it takes to produce the same displacement.

In the context of our problem, we can determine the force constant by rearranging the formula for potential energy \(E_p = \frac{1}{2}kA^2\). This yields \(k = \frac{2E}{A^2}\), where \(E\) is the total mechanical energy when the block is at the amplitude of its motion, and \(A\) is the amplitude. This reveals that the force constant is directly proportional to the mechanical energy and inversely proportional to the square of the amplitude.

Mass of the Block

The mass of the block, denoted by \(m\), is a critical component in an oscillating system. It affects the system's inertia, or resistance to changes in motion. In our calculations, it also influences the relationship between kinetic energy and velocity.

To find the mass of the block in our example, we use the kinetic energy formula at the point of maximum speed: \(E_k = \frac{1}{2}mv_{max}^2\). Solving for mass gives us \(m = \frac{2E}{v_{max}^2}\). This formula shows us that the mass is proportional to the total mechanical energy and inversely proportional to the square of maximum speed. In simpler terms, a heavier block would need more energy to achieve the same maximum speed as a lighter one.

Frequency of Oscillation

The frequency of oscillation, represented as \(f\), is the number of complete back-and-forth cycles of an oscillating block-spring system per unit of time, measured in Hertz (Hz). The frequency formula for a simple harmonic oscillator like our block-spring system is \(f = \frac{1}{2 \text{{pi}}} \sqrt{\frac{k}{m}}\).

This formula illustrates a direct relationship between the frequency of oscillation and the force constant \(k\), while there is an inverse relationship with the mass \(m\). It implies that a stiffer spring (larger \(k\)) or a lighter block (smaller \(m\)) will result in a higher frequency of oscillation, meaning the system will oscillate more rapidly. It is crucial for students to understand that the frequency is an intrinsic property of the system which depends solely on its physical characteristics and not on external factors like amplitude or mechanical energy.