Example 2: A linear spring has a length of Find a its constant, and b its free no load length. The spring force F s is always opposite to the applied force. As the following figures indicate,. When x is positive, F s is negative and vise versa.
This fact is reflected by the - sign in the formula. Note that F s is not the applied force , it is the force that the spring exerts. Simple Harmonic Motion:. Note: for a review on radian, the Metric unit for angles, refer to the end of the chapter. Now, picture mass M is performing a uniform circular motion in a vertical plane as shown.
Its shadow on the x-axis performs a back-and-forth motion that is called simple harmonic motion. To understand the following figure, visualize that mass M moves slowly and counterclockwise on the circle of radius A , and at different positions, picture its shadow on the floor. The graph of x vs. Note that the farthest distance Point K can go from Point O is as much as length A , the radius of the circle. A, the maximum deviation from the equilibrium position, is called the " Amplitude " of oscillations.
Example 3: A bicycle wheel of radius The shadow of a bump on its edge performs an oscillatory motion on the floor. Note that your calculator must be in radians mode for the last calculation. As it was mentioned, when mass M attached to a linear spring is pulled and released, its up-and-down motion above and below the equilibrium level is called " simple harmonic motion. It is for this reason that the motion is called harmonic. Figure a below shows a spring that is not loaded. Figure b shows the same spring but loaded and stretched a distance - h , and Figure c shows the loaded spring stretched further a distance - A and released.
Example 5: A gram mass hung from a weak spring has stretched it by 3. Example 6: The graph of x the distance from the equilibrium position versus time t for the oscillations of a mass-spring system is given below :.
For such oscillations, find a the amplitude, b the period, c the frequency, d the angular speed frequency , e the spring constant k if the mass of the object is The given graph is a sine function. If an object is oscillating up and down, for example, it is easy to see that its velocity becomes zero at the extreme points, i. This is simply because it has to come to stop at those points in order to change direction. It is also easy to see that velocity gains its maximum magnitude at the midpoint or the equilibrium level.
We may therefore state that:. Acceleration has a different story. At the extreme ends, when a spring is at its maximum stretch or compress, the spring force is at its maximum magnitude, and therefore the acceleration it gives to the attached mass is maximum. Using Calculus, if the equation for x is.
V max and a max become:. We may disregard the - signs if only the magnitudes are to be calculated. Using the given equation, substituting for t , and putting the calculator in Radians Mode, we get:. Chapter 15 Test Yourself 1: For answers, click here. N, it has a constant of a Is you calculator in radians mode? In cos 0. If in every minute 5.
Based on the equation. Problem: Refer to the figure of Example 6. There are 12 segments time intervals on the t- axis. Use the equation of motion found under f to calculate the following. Compare your calculations with the vertical line segments x values to see if they make sense. Make sure to perform all calculations with you calculator in the correct mode. For an oscillating mass-spring system, for example, the reason is that a the velocity at ed points is zero.
Oscillations Part 2 : Calculus-based Approach. Doing this, results in. It was discussed that a max occurs at the end points. For periodic motion, frequency is the number of oscillations per unit time. The relationship between frequency and period is.
The SI unit for frequency is the hertz Hz and is defined as one cycle per second :. A cycle is one complete oscillation. A very common type of periodic motion is called simple harmonic motion SHM. A system that oscillates with SHM is called a simple harmonic oscillator. In simple harmonic motion, the acceleration of the system, and therefore the net force, is proportional to the displacement and acts in the opposite direction of the displacement.
A good example of SHM is an object with mass m attached to a spring on a frictionless surface, as shown in Figure The object oscillates around the equilibrium position, and the net force on the object is equal to the force provided by the spring. The maximum displacement from equilibrium is called the amplitude A. The units for amplitude and displacement are the same but depend on the type of oscillation.
For the object on the spring, the units of amplitude and displacement are meters. What is so significant about SHM? For one thing, the period T and frequency f of a simple harmonic oscillator are independent of amplitude. The string of a guitar, for example, oscillates with the same frequency whether plucked gently or hard. Two important factors do affect the period of a simple harmonic oscillator. The period is related to how stiff the system is.
A very stiff object has a large force constant k , which causes the system to have a smaller period. Period also depends on the mass of the oscillating system.
The more massive the system is, the longer the period. For example, a heavy person on a diving board bounces up and down more slowly than a light one.
In fact, the mass m and the force constant k are the only factors that affect the period and frequency of SHM. To derive an equation for the period and the frequency, we must first define and analyze the equations of motion. Note that the force constant is sometimes referred to as the spring constant. Consider a block attached to a spring on a frictionless table Figure At the equilibrium position, the net force is zero.
The maximum x -position A is called the amplitude of the motion. The period is the time for one oscillation. Figure When the position is plotted versus time, it is clear that the data can be modeled by a cosine function with an amplitude A and a period T. The maximum of the cosine function is one, so it is necessary to multiply the cosine function by the amplitude A.
Consider 10 seconds of data collected by a student in lab, shown in Figure The data in Figure The equation of the position as a function of time for a block on a spring becomes. It should be noted that because sine and cosine functions differ only by a phase shift, this motion could be modeled using either the cosine or sine function.
The velocity of the mass on a spring, oscillating in SHM, can be found by taking the derivative of the position equation:. The acceleration of the mass on the spring can be found by taking the time derivative of the velocity:. In summary, the oscillatory motion of a block on a spring can be modeled with the following equations of motion:.
The period of the motion is 1. Determine the equations of motion. One interesting characteristic of the SHM of an object attached to a spring is that the angular frequency, and therefore the period and frequency of the motion, depend on only the mass and the force constant, and not on other factors such as the amplitude of the motion.
Consider the block on a spring on a frictionless surface. There are three forces on the mass: the weight, the normal force, and the force due to the spring. The only two forces that act perpendicular to the surface are the weight and the normal force, which have equal magnitudes and opposite directions, and thus sum to zero.
The only force that acts parallel to the surface is the force due to the spring, so the net force must be equal to the force of the spring:. Substituting the equations of motion for x and a gives us. The angular frequency depends only on the force constant and the mass, and not the amplitude. The period also depends only on the mass and the force constant. The greater the mass, the longer the period. The stiffer the spring, the shorter the period.
The frequency is. When a spring is hung vertically and a block is attached and set in motion, the block oscillates in SHM. In this case, there is no normal force, and the net effect of the force of gravity is to change the equilibrium position. Consider Figure
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