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Chapter 22: Problem 41

If the block-and-spring model is subjected to the periodic force \(F=F_{0} \cos \omega t\), show that the differential equation of motion is \(\ddot{x}+(k / m) x=\left(F_{0} / m\right) \cos \omega t\), where \(x\) is measured from the equilibrium position of the block. What is the general solution of this equation?

Short Answer

Expert verified
The differential equation of motion is \( \ddot{x}+(k / m) x=\left(F_{0} / m\right) \cos \omega t \). The general solution to this equation is \( x(t) = X \cos(\omega t - \delta) + C \cos \sqrt{\frac{k}{m}}t + D \sin \sqrt{\frac{k}{m}}t \).

Step by step solution

01

Derive Differential Equation of Motion

Considering a block and spring system with applied force \( F = F_{0} \cos \omega t \), the total force acting on the block can be expressed as the sum of the applied force and the spring force (which follows Hooke's law). According to Newton's second law, the total force is also equal to the mass times acceleration. Hence: \( m\ddot{x} = -kx + F_{0} \cos \omega t \). Now divide this equation by 'm', which gives: \( \ddot{x} + \frac{k}{m}x = \frac{F_{0}}{m} \cos \omega t \)
02

Find the General Solution

This is a driven harmonic oscillator differential equation, which has a general solution given as: \( x(t) = X \cos(\omega t - \delta) + C \cos \sqrt{\frac{k}{m}}t + D \sin \sqrt{\frac{k}{m}}t \), where 'X' and '\delta' are the amplitude and phase shift of the force, and 'C' and 'D' are constants determined by initial conditions.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Driven Harmonic Oscillator
A driven harmonic oscillator refers to a system where an external force is applied, causing the system to oscillate. In the context of a block-and-spring model, this force is periodic, as represented by \( F = F_0 \cos \omega t \). The differential equation of motion for such a system includes terms relating to both the natural oscillations due to the spring force and the external driving force.
  • The term \( \ddot{x} + \frac{k}{m}x \) describes the natural spring motion.
  • The term \( \frac{F_0}{m} \cos \omega t \) represents the driving force.
The solution combines these influences, resulting in a complex oscillatory motion characterized by the amplitude and frequency of both the natural and driven components. It's important to understand how these elements interact to predict the oscillator's behavior under external forces.
Newton's Second Law
Newton's Second Law is a fundamental principle in physics that dictates how the motion of an object is influenced by forces. For the block-and-spring system, it states that the net force acting on the object equals the mass of the object times its acceleration, or \( F_{net} = m\ddot{x} \). This relationship is pivotal in formulating the equation of motion for the system.
When we apply this law:
  • The spring force, calculated as \(-kx\) from Hooke's Law, opposes the block's displacement.
  • The external force, \( F_0 \cos \omega t \), drives the system, adding energy and causing additional motion.
By balancing these forces, Newton's Second Law provides a framework for constructing the differential equation that describes the system's dynamics.
Hooke's Law
Hooke's Law is a principle that states the force exerted by a spring is proportional to its displacement from equilibrium. Mathematically, this is expressed as \( F = -kx \), where \( k \) is the spring constant, and \( x \) is the displacement.
This law is at the heart of understanding the spring force component in the driven harmonic oscillator model. In the model:
  • The negative sign indicates the spring force acts in the opposite direction of displacement, thus restoring equilibrium.
  • It defines how the spring will compress or extend depending on the force applied to it, crucial for determining the system's natural frequency.
Hooke's Law thus provides insight into how the spring responds to various forces and vibrations, key to solving the dynamics of the oscillator.

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Most popular questions from this chapter

The 200-lb electric motor is fastened to the midpoint of the simply supported beam. It is found that the beam deflects 2 in. when the motor is not running. The motor turns an eccentric flywheel which is equivalent to an unbalanced weight of 1 lb located 5 in. from the axis of rotation. If the motor is turning at \(100 \mathrm{rpm}\), determine the amplitude of steady-state vibration. The damping factor is \(c / c_{c}=0.20 .\) Neglect the mass of the beam.

A spring has a stiffness of \(800 \mathrm{~N} / \mathrm{m}\). If a \(2-\mathrm{kg}\) block is attached to the spring, pushed \(50 \mathrm{~mm}\) above its equilibrium position, and released from rest, determine the equation that describes the block's motion. Assume that positive displacement is downward.

The spring system is connected to a crosshead that oscillates vertically when the wheel rotates with a constant angular velocity of \(\omega=5 \mathrm{rad} / \mathrm{s}\). If the amplitude of the steady-state vibration is observed to be \(400 \mathrm{~mm}\), determine the two possible values of the stiffness \(k\) of the springs. The block has a mass of \(50 \mathrm{~kg}\).

A platform, having an unknown mass, is supported by four springs, each having the same stiffness \(k\). When nothing is on the platform, the period of vertical vibration is measured as \(2.35 \mathrm{~s}\); whereas if a \(3-\mathrm{kg}\) block is supported on the platform, the period of vertical vibration is \(5.23 \mathrm{~s}\). Determine the mass of a block placed on the (empty) platform which causes the platform to vibrate vertically with a period of \(5.62 \mathrm{~s}\). What is the stiffness \(k\) of each of the springs?

A 6-lb weight is suspended from a spring having a stiffness \(k=3 \mathrm{lb} / \mathrm{in} .\) If the weight is given an upward velocity of \(20 \mathrm{ft} / \mathrm{s}\) when it is \(2 \mathrm{in}\). above its equilibrium position, determine the equation which describes the motion and the maximum upward displacement of the weight, measured from the equilibrium position. Assume positive displacement is downward.
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