Question

A mass \(m=5.00 \mathrm{~kg}\) is suspended from a spring and oscillates according to the equation of motion \(x(t)=0.5 \cos (5 t+\pi / 4) .\) What is the spring constant?

Short answer

Answer: The spring constant for the system is 125 N/m.

Step-by-step solution

Identify the equation of motion parameters

In this problem, we are given the equation of motion as \(x(t) = 0.5 \cos (5 t + \pi/4)\). This is the equation for simple harmonic motion in the form of \(x(t) = A \cos (\omega t + \phi)\), where \(A\) is the amplitude, \(\omega\) is the angular frequency, and \(\phi\) is the phase angle. By comparing the given equation, we can identify the following parameters: Amplitude (A): \(0.5 \mathrm{~m}\) Angular Frequency (\(\omega\)): \(5 \mathrm{~s^{-1}}\) Phase Angle (\(\phi\)): \(\pi/4 \mathrm{~rad}\)

Calculate the frequency and the period

Next, we will find the frequency (f) and the period (T) of the oscillation, since they will be useful in determining the spring constant. We can use the following relationships between angular frequency, frequency, and period: \(\omega = 2 \pi f\) and \(T = \dfrac{1}{f}\) From the given equation, we have an angular frequency (\(\omega\)) of 5 : \(f = \dfrac{\omega}{2 \pi} = \dfrac{5}{2 \pi} \approx 0.796 \mathrm{~Hz}\) Now we can calculate the period (T): \(T = \dfrac{1}{f} = \dfrac{1}{0.796} \approx 1.256 \mathrm{~s}\)

Determine the spring constant using Hooke's Law

To find the spring constant, we will use Hooke's Law and the relationship between the period and mass on a spring: Hooke's Law: \(F = -kx\) Relationship between period and mass: \(T = 2\pi \sqrt{\frac{m}{k}}\) The force acting on the mass due to the spring (F) is the product of mass (m) and acceleration (a). In simple harmonic motion, acceleration is given by \(a = -\omega^{2} x\). Thus, \(F = -ma = m\omega^{2}x\) Substituting Hooke's Law into this equation, we get: \(kx = m\omega^{2}x\) Now, we will solve for the spring constant (k). Since x ≠ 0, we can divide both sides by x: \(k = m\omega^{2}\) Finally, we will plug in the given mass (m = 5.00 kg) and the angular frequency (\(\omega\) = 5 s⁻¹): \(k = (5.00 \mathrm{~kg}) (5 \mathrm{~s^{-1}})^{2} = 5 \cdot 25 = 125 \mathrm{~N/m}\) The spring constant is 125 N/m.

Key concepts

Simple Harmonic Motion

Simple Harmonic Motion (SHM) is a type of periodic motion, like the swing of a pendulum or the vibration of a guitar string, where the restoring force is directly proportional to the displacement and acts in the opposite direction. It's characterized by oscillations about an equilibrium position. The motion is 'simple' because it can be described by simple equations and 'harmonic' because it is periodic and follows a harmonic function, typically sinusoidal. In the given exercise, a mass attached to a spring demonstrates SHM as it moves back and forth, where the displacement from the equilibrium position follows a cosine function of time.

Hooke's Law

Hooke's Law is a principle that states the force required to extend or compress a spring by some distance (x) is proportional to that distance. Mathematically, it can be expressed as:

\(F = -kx\)

Here, \(F\) is the force applied, \(k\) is the spring constant, and \(x\) is the displacement from the original length of the spring. The negative sign indicates that the force exerted by the spring opposes the displacement. This law is crucial in determining the behavior of the spring and the motion of the mass attached to it. In the context of the exercise, Hooke's Law is instrumental in finding the spring constant by relating force to displacement.

Angular Frequency

Angular frequency (\(\omega\)) is a measure of how fast something is oscillating in SHM and is related to the number of oscillations an object completes in a unit of time (frequency). It's represented in radians per second and is given by the equation:

\(\omega = 2\pi f\)

where \(f\) is the frequency in cycles per second, or Hertz (Hz), and \(2\pi\) is the factor needed to convert frequency to angular frequency (since there are \(2\pi\) radians in a full cycle). Angular frequency is crucial to understanding the timing of the oscillations, which in turn helps in calculating other important quantities like the spring constant.

Oscillation Equation

The oscillation equation in the context of SHM provides a way to calculate the position of the oscillating object at any given time (t), which is particularly helpful in applications like the design of springs and pendulums. It can be represented as:

\(x(t) = A \cos (\omega t + \phi)\)

In this equation, \(A\) represents the amplitude, \(\omega\) is the angular frequency, \(\phi\) is the phase constant determining the initial angle, and \(t\) is the time variable. This equation gives us a full description of the motion of the object in time, and in the exercise, it allowed us to determine the mass's motion on the spring to find the spring constant.