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Magazine Chapter 3: Problem 8
A mass \(m\) is suspended from a spring of stiffness \(4000 \mathrm{~N} / \mathrm{m}\) and is subjected to a harmonic force having an amplitude of \(100 \mathrm{~N}\) and a frequency of \(5 \mathrm{~Hz}\). The amplitude of the forced motion of the mass is observed to be \(20 \mathrm{~mm}\). Find the value of \(\mathrm{m}\).
Short Answer
Expert verified
The mass \(m\) in the given mass-spring system is approximately \(1.01 \mathrm{kg}\).
Step by step solution
01
Write down the equation of motion for a mass-spring system
The equation of motion for a mass-spring system subjected to an external force F(t) is given by:
\(m\frac{d^2x}{dt^2} + kx = F(t)\)
where m is the mass, k is the spring stiffness, x is the displacement, and F(t) is the external force.
02
Write the equation for the given external force
The given external force is a harmonic force of amplitude 100 N and frequency 5 Hz:
\(F(t) = F_0\sin(\omega t)\), where \(F_0 = 100 \mathrm{N}\) and \(\omega = 2\pi f = 2\pi(5 \mathrm{Hz}) = 10\pi \mathrm{rad/s}\)
03
Write the equation for the steady-state solution
The steady-state solution for the displacement of the mass x(t) due to a harmonic force is given by:
\(x(t) = A\sin(\omega t + \phi)\), where A is the amplitude of the forced motion, and 蠁 is the phase angle.
Since we are given the amplitude A = 20 mm, we can write the steady-state solution as:
\(x(t) = 0.02 \mathrm{m}\sin(10\pi t + \phi)\)
04
Find the equation relating the amplitude and the mass
By differentiating the steady-state solution twice with respect to time, we get the acceleration of the mass:
\(\frac{d^2x}{dt^2} = -(10\pi)^2\cdot0.02 \mathrm{m}\sin(10\pi t + \phi)\)
Substituting this acceleration and the given external force into the equation of motion, we get:
\(-m(10\pi)^2\cdot0.02 \mathrm{m}\sin(10\pi t + \phi) + k\cdot0.02 \mathrm{m}\sin(10\pi t + \phi) = 100 \mathrm{N}\sin(10\pi t)\)
Since the displacement and force are both sinusoidal functions with the same frequency, the sine terms on both sides of the equation will be equal. Thus, we can write:
\(-m(10\pi)^2\cdot0.02 \mathrm{m} + k\cdot0.02 \mathrm{m} = 100 \mathrm{N}\)
05
Solve for the mass
Now we can solve for the mass, m. First we plug in the given values for the spring stiffness, k = 4000 N/m:
\(-m(10\pi)^2\cdot0.02 \mathrm{m} + (4000\mathrm{N/m})\cdot0.02 \mathrm{m} = 100 \mathrm{N}\)
Rearranging the terms and dividing both sides by 0.02 m, we get:
\(-m(10\pi)^2 + 4000 \mathrm{N/m} = \frac{100 \mathrm{N}}{0.02 \mathrm{m}}\)
Now, calculate the value on the right-hand side:
\(\frac{100 \mathrm{N}}{0.02 \mathrm{m}} = 5000 \mathrm{N/m}\)
Finally, solve for the mass, m:
\(-m(10\pi)^2 = 5000 \mathrm{N/m} - 4000 \mathrm{N/m}\)
\(m(10\pi)^2 = 1000 \mathrm{N/m}\)
\(m = \frac{1000 \mathrm{N/m}}{(10\pi)^2}\)
Approximating the value for m, we get:
\(m \approx 1.01 \mathrm{kg}\)
Hence, the value of the mass, m, is approximately 1.01 kg.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Mass-Spring System
A mass-spring system is a fundamental concept in physics that models the behavior of a mass attached to a spring. Imagine a small object, like a weight, being hung from a spring. This setup allows us to study how the mass moves when forces are applied. The spring tries to return to its original shape by exerting a force on the mass. This force is proportional to the displacement of the mass from its rest position. The governing equation for this system is
- \( m\frac{d^2x}{dt^2} + kx = F(t) \)
- \( m \) is the mass attached to the spring,
- \( k \) is the spring stiffness, and
- \( F(t) \) is any additional external force applied.
Harmonic Force
A harmonic force is a type of force that acts in a periodic manner, like the oscillations of a sine wave. When we introduce a harmonic force to a mass-spring system, this force can make the system vibrate at the same frequency as the force itself. In our exercise:
- The force amplitude is 100 N.
- The force frequency is 5 Hz, or equivalently \( \omega = 10\pi \) rad/s.
Steady-State Solution
The steady-state solution refers to the behavior of a system after initial transients have died out, leaving a consistent oscillation. When analyzing a system subject to a harmonic force, it's important to understand the long-term movement. For our problem, the steady-state displacement is given by:
- \( x(t) = A\sin(\omega t + \phi) \)
- \( A \) is the amplitude of motion.
- \( \phi \) is the phase angle.
Spring Stiffness
Spring stiffness (also known as spring constant) is a measure of how resistant a spring is to being compressed or stretched. It is represented by \( k \) in equations. In our scenario, the spring stiffness is 4000 N/m. A stiffer spring means that more force is required to deform it, which results in less displacement for the same force compared to a less stiff spring.
Spring stiffness directly influences the natural frequency and response to external forces of the mass-spring system. This affects the amplitude and frequency at which the system will resonate when subjected to a harmonic force.
Understanding the exact spring stiffness is crucial for designing systems that need to perform specific tasks, like shock absorbers in cars or precision instruments in laboratories.
Spring stiffness directly influences the natural frequency and response to external forces of the mass-spring system. This affects the amplitude and frequency at which the system will resonate when subjected to a harmonic force.
Understanding the exact spring stiffness is crucial for designing systems that need to perform specific tasks, like shock absorbers in cars or precision instruments in laboratories.
