Question

A \(5.22-\mathrm{kg}\) object is attached to the bottom of a vertical spring and set vibrating. The maximum speed of the object is \(15.3 \mathrm{~cm} / \mathrm{s}\) and the period is \(645 \mathrm{~ms}\). Find \((a)\) the force constant of the spring, (b) the amplitude of the motion, and \((c)\) the frequency of oscillation.

Short answer

(a) The force constant of the spring can be calculated using the mass of the object and the period of oscillation. (b) The amplitude of the motion can be found using the maximum speed and period. (c) The frequency of oscillation can be found as the reciprocal of the period. Detailed calculations need to be carried out for the exact numerical values.

Step-by-step solution

Convert Units

First, standard SI units must be used to simplify calculations and avoid mistakes. Convert period from milliseconds to seconds and maximum speed from cm/s to m/s. So, the period \(T = 645ms = 0.645s\) and maximum speed \(v_{max} = 15.3cm/s = 0.153m/s\).

Calculate Force Constant

The force constant \(k\) of the spring can be determined using the formula for the period of oscillation \(T = 2 \pi \sqrt{\frac{m}{k}}\), where m is the mass of the object. Rearranging for k, we get \(k=\frac {4\pi ^2m}{T^2}\). Substituting \(m = 5.22kg\) and \(T = 0.645s\) into the formula, we calculate \(k\).

Calculate Amplitude

The amplitude \(A\) of the motion can be calculated using the maximum speed formula \(v_{max} = \omega A\), where \(\omega = \frac {2\pi}{T}\) is the angular frequency. Rearranging for A, we get \(A = \frac {v_{max}}{\omega}\). Substituting \(\omega\) and \(v_{max} = 0.153m/s\) into the formula, we calculate \(A\).

Calculate Frequency of Oscillation

The frequency \(f\) of the oscillation is the reciprocal of the period \(T\), \(f = \frac {1}{T}\). Substituting \(T = 0.645s\) into the formula, we calculate \(f\).

Key concepts

Spring Force Constant

The spring force constant, symbolized as 'k', quantifies the stiffness of a spring. In other words, it measures how much force is required to stretch or compress the spring by a certain amount. The higher the force constant, the stiffer the spring, and the more force it takes to deform it.

In the context of simple harmonic motion, the force constant relates to the mass of the object attached to the spring and the period of oscillation through the formula: \[k = \frac{4 \pi^2 m}{T^2}\] Given the mass 'm' of an object and the period of oscillation 'T', we can determine 'k' and understand how responsive the spring is to the forces applied to it. Students should remember to use SI units when calculating the force constant to ensure accuracy. If they encounter different units, converting them to kilograms for mass and seconds for time is essential, as demonstrated in the textbook solution.

Oscillation Amplitude

Oscillation amplitude, represented by the letter 'A', defines the maximum displacement of an object from its equilibrium position during harmonic motion. It's a measure of how far the object moves during its vibration and is a key factor in determining the energy involved in the oscillation.

In our textbook example, we use the maximum speed of the object and the angular frequency to find the amplitude. The relationship is given by the equation: \[A = \frac{v_{max}}{\omega}\] Here, 'v_max' is the maximum speed, and 'omega' (\(\omega\)) is the angular frequency which can be obtained from the period 'T'. A common student error is neglecting the conversion of units which can entirely skew the results. Always ensure the speed is in meters per second (m/s) for consistency with SI units. Amplitude is crucial as it affects the potential and kinetic energy of the system at different points in the oscillation.

Oscillation Frequency

Oscillation frequency, labeled as 'f', refers to the number of oscillations an object completes in one second. It's inversely proportional to the period 'T', meaning that a shorter period yields a higher frequency. It's important to understand that frequency is about how often something happens within a specific timeframe.

We can get the frequency from the period using the equation: \[f = \frac{1}{T}\]For the example exercise, once we've converted the period from milliseconds to seconds, we calculate the frequency to find out how many oscillations occur every second. Students should be aware that frequency is measured in hertz (Hz), which is equivalent to oscillations per second. Understanding frequency is vital, as it affects the energy and the motion characteristics of the system being analyzed.

Period of Oscillation

The period of oscillation (T) signifies the time it takes to complete one full cycle of motion—returning to the starting point. It's a fundamental concept in understanding oscillatory motion, as it links directly to how fast something vibrates. In simple harmonic motion, all cycles take the same amount of time, emphasizing the 'periodic' nature of these systems.

As seen in our example, the period is given as 645 ms, which we must convert to seconds before using in any further calculations to adhere to SI units. With the period, we can determine other properties of the motion, such as the force constant and the frequency: \[k = \frac{4 \pi^2 m}{T^2}\] and \[f = \frac{1}{T}\]One common challenge for students is not correctly converting time units, leading to incorrect calculations of both the force constant and the frequency. Always ensure to convert milliseconds to seconds or vice versa when necessary to maintain accuracy.