Why do springs stop oscillating

If at , the amplitude was , then suppose at the amplitude is half that,. What happens after another minute, at?

Well we expect that it should halve again, and be. After another minute it should halve again. This is describes an exponential decay of the amplitude. Instead of the amplitude being constant, it's decaying with time. Here's a plot of of an example of such a function. The green line is. It is the envelope of the oscillation.

Obviously depending on the rate of decay of the amplitude, and the frequency, you'll get a different picture. But qualitatively you'll see an oscillating function whose amplitude decays away to zero.

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 simple harmonic motion. Find two identical wooden or plastic rulers. Tape one end of each ruler firmly to the edge of a table so that the length of each ruler that protrudes from the table is the same. On the free end of one ruler tape a heavy object such as a few large coins. Pluck the ends of the rulers at the same time and observe which one undergoes more cycles in a time period, and measure the period of oscillation of each of the rulers.

If the shock absorbers in a car go bad, then the car will oscillate at the least provocation, such as when going over bumps in the road and after stopping See Figure 2. Figure 2. The bouncing car makes a wavelike motion. The wave is the trace produced by the headlight as the car moves to the right.

The mass and the force constant are both given. The values of T and f both seem about right for a bouncing car. You can observe these oscillations if you push down hard on the end of a car and let go. Figure 3. The vertical position of an object bouncing on a spring is recorded on a strip of moving paper, leaving a sine wave.

If a time-exposure photograph of the bouncing car were taken as it drove by, the headlight would make a wavelike streak, as shown in Figure 2. Both waves are sine functions. All simple harmonic motion is intimately related to sine and cosine waves.

Furthermore, from this expression for x , the velocity v as a function of time is given by. The minus sign in the first equation for v t gives the correct direction for the velocity. Just after the start of the motion, for instance, the velocity is negative because the system is moving back toward the equilibrium point. So, a t is also a cosine function:. Hence, a t is directly proportional to and in the opposite direction to a t. Figure 4 shows the simple harmonic motion of an object on a spring and presents graphs of x t , v t , and a t versus time.

Figure 4. Graphs of and versus t for the motion of an object on a spring. Note that the initial position has the vertical displacement at its maximum value X ; v is initially zero and then negative as the object moves down; and the initial acceleration is negative, back toward the equilibrium position and becomes zero at that point. The most important point here is that these equations are mathematically straightforward and are valid for all simple harmonic motion.

They are very useful in visualizing waves associated with simple harmonic motion, including visualizing how waves add with one another. Suppose you pluck a banjo string. You hear a single note that starts out loud and slowly quiets over time. If the mass is pushed up a distance A and then released, it oscillates above and below that equilibrium level.

Distance A , that is the maximum deviation from equilibrium , is called the " amplitude " of oscillations. This formula is a result of the solution to a 2 nd order linear differential equation with constant coefficients.

The differential equation is set up very easily as follows. This may be written as :. Procedure :. The value of k , the spring constant , may be measured in two ways. One method is to use Hooke's law.

The other method is to measure the period T of oscillations of a mass-spring system. The values of k determined by the two methods may then be compared and used as a verification of the validity of the theories involved.

The Hooke's Law Method :. The mass-spring system acts similar to a spring scale. It has a vertical ruler that measures the spring's elongation. Measure the mass of the hanger without the spring. Attach the spring and hanger to the support. Zero the system by sliding the ruler against the needle. The ruler slides easily once its collar or slider at the back of the ruler is squeezed with two fingers. By zeroing the system with its small weight hanger attached, you do not have to take its mass into account for this part of the experiment.

It is better to use two 5 0 g slotted masses instead of a single 10 0 g mass.