During periodic motion where is the velocity the greatest

A simple pendulum is defined to have an object that has a small mass, also known as the pendulum bob, which is suspended from a wire or string of negligible mass, such as shown in the illustrating figure. Exploring the simple pendulum a bit further, we can discover the conditions under which it performs simple harmonic motion, and we can derive an interesting expression for its period.

Pendulums : A brief introduction to pendulums both ideal and physical for calculus-based physics students from the standpoint of simple harmonic motion. We begin by defining the displacement to be the arc length s. Now, if we can show that the restoring force is directly proportional to the displacement, then we have a simple harmonic oscillator.

For the simple pendulum:. This result is interesting because of its simplicity. The only things that affect the period of a simple pendulum are its length and the acceleration due to gravity. The period is completely independent of other factors, such as mass. Even simple pendulum clocks can be finely adjusted and accurate.

Note the dependence of T on g. If the length of a pendulum is precisely known, it can actually be used to measure the acceleration due to gravity. In this case, the motion of a pendulum as a function of time can be modeled as:. The period of a physical pendulum depends upon its moment of inertia about its pivot point and the distance from its center of mass.

Recall that a simple pendulum consists of a mass suspended from a massless string or rod on a frictionless pivot. In that case, we are able to neglect any effect from the string or rod itself. In contrast, a physical pendulum sometimes called a compound pendulum may be suspended by a rod that is not massless or, more generally, may be an arbitrarily-shaped, rigid body swinging by a pivot see. Pendulums — Physical Pendulum : A brief introduction to pendulums both ideal and physical for calculus-based physics students from the standpoint of simple harmonic motion.

A Physical Pendulum : An example showing how forces act through center of mass. We can calculate the period of this pendulum by determining the moment of inertia of the object around the pivot point.

Gravity acts through the center of mass of the rigid body. Hence, the length of the pendulum used in equations is equal to the linear distance between the pivot and the center of mass h. In case we know the moment of inertia of the rigid body, we can evaluate the above expression of the period for the physical pendulum.

For illustration, let us consider a uniform rigid rod, pivoted from a frame as shown see. Uniform Rigid Rod : A rigid rod with uniform mass distribution hangs from a pivot point. This is another example of a physical pendulum.

However, we need to evaluate the moment of inertia about the pivot point, not the center of mass, so we apply the parallel axis theorem:. The important thing to note about this relation is that the period is still independent of the mass of the rigid body. However, it is not independent of the mass distribution of the rigid body.

A change in shape, size, or mass distribution will change the moment of inertia. This, in turn, will change the period. The total energy in a simple harmonic oscillator is the constant sum of the potential and kinetic energies. To study the energy of a simple harmonic oscillator, we first consider all the forms of energy it can have. Because a simple harmonic oscillator has no dissipative forces , the other important form of energy is kinetic energy KE.

Conservation of energy for these two forms is:. This statement of conservation of energy is valid for all simple harmonic oscillators, including ones where the gravitational force plays a role.

In the case of undamped, simple harmonic motion, the energy oscillates back and forth between kinetic and potential, going completely from one to the other as the system oscillates.

So for the simple example of an object on a frictionless surface attached to a spring, as shown again see , the motion starts with all of the energy stored in the spring.

As the object starts to move, the elastic potential energy is converted to kinetic energy, becoming entirely kinetic energy at the equilibrium position. It is then converted back into elastic potential energy by the spring, the velocity becomes zero when the kinetic energy is completely converted, and so on. This concept provides extra insight here and in later applications of simple harmonic motion, such as alternating current circuits. Energy in a Simple Harmonic Oscillator : The transformation of energy in simple harmonic motion is illustrated for an object attached to a spring on a frictionless surface.

All energy is potential energy. The conservation of energy principle can be used to derive an expression for velocity v. This total energy is constant and is shifted back and forth between kinetic energy and potential energy, at most times being shared by each.

The conservation of energy for this system in equation form is thus:. Notice that the maximum velocity depends on three factors. It is directly proportional to amplitude. As you might guess, the greater the maximum displacement, the greater the maximum velocity. It is also greater for stiffer systems because they exert greater force for the same displacement.

This observation is seen in the expression for v max ; it is proportional to the square root of the force constant k. Finally, the maximum velocity is smaller for objects that have larger masses, because the maximum velocity is inversely proportional to the square root of m. For a given force, objects that have large masses accelerate more slowly. Experience with a simple harmonic oscillator : A known mass is hung from a spring of known spring constant and allowed to oscillate.

The time for one oscillation period is measured. When comparing these two vibrating objects - the 1. Observe that the description of the two objects uses the terms frequently and infrequently. The terms fast and slow are not used since physics types reserve the words fast and slow to refer to an object's speed. Here in this description we are referring to the frequency, not the speed.

An object can be in periodic motion and have a low frequency and a high speed. As an example, consider the periodic motion of the moon in orbit about the earth. The moon moves very fast; its orbit is highly infrequent.

Yet it makes a complete cycle about the earth once every Objects like the piano string that have a relatively short period i. Frequency is another quantity that can be used to quantitatively describe the motion of an object is periodic motion.

The frequency is defined as the number of complete cycles occurring per period of time. The unit Hertz is used in honor of Heinrich Rudolf Hertz, a 19th century physicist who expanded our understanding of the electromagnetic theory of light waves.

The concept and quantity frequency is best understood if you attach it to the everyday English meaning of the word. Frequency is a word we often use to describe how often something occurs. You might say that you frequently check your email or you frequently talk to a friend or you frequently wash your hands when working with chemicals. Used in this context, you mean that you do these activities often. To say that you frequently check your email means that you do it several times a day - you do it often.

In physics, frequency is used with the same meaning - it indicates how often a repeated event occurs. High frequency events that are periodic occur often, with little time in between each occurrence - like the back and forth vibrations of the tines of a tuning fork. The vibrations are so frequent that they can't be seen with the naked eye. A Hz tuning fork has tines that make complete back and forth vibrations each second.

At this frequency, it only takes the tines about 0. A Hz tuning fork has an even higher frequency. Its vibrations occur more frequently; the time for a full cycle to be completed is 0. In comparing these two tuning forks, it is obvious that the tuning fork with the highest frequency has the lowest period. The two quantities frequency and period are inversely related to each other. In fact, they are mathematical reciprocals of each other. The frequency is the reciprocal of the period and the period is the reciprocal of the frequency.

This reciprocal relationship is easy to understand. After all, the two quantities are conceptual reciprocals a phrase I made up. Consider their definitions as restated below:. Even the definitions have a reciprocal ring to them. To understand the distinction between period and frequency, consider the following statement:.

In this problem, the event that is repeating itself is the clapping of hands; one hand clap is equivalent to a cycle. The final measurable quantity that describes a vibrating object is the amplitude.

The amplitude is defined as the maximum displacement of an object from its resting position. The resting position is that position assumed by the object when not vibrating. Once vibrating, the object oscillates about this fixed position.

The data in Figure can still be modeled with a periodic function, like a cosine function, but the function is shifted to the right. This shift is known as a phase shift and is usually represented by the Greek letter phi. This is the generalized equation for SHM where t is the time measured in seconds,. 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:. Here, A is the amplitude of the motion, T is the period,. The spring can be compressed or extended.

The equilibrium position is marked as. Once the angular frequency is found, we can determine the maximum velocity and maximum acceleration. The angular frequency can be found and used to find the maximum velocity and maximum acceleration:. The position, velocity, and acceleration can be found for any time.

It is important to remember that when using these equations, your calculator must be in radians mode. 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 angular frequency is defined as. 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. Two forces act on the block: the weight and the force of the spring. The weight is constant and the force of the spring changes as the length of the spring changes. When the block reaches the equilibrium position, as seen in Figure , the force of the spring equals the weight of the block,. If the block is displaced and released, it will oscillate around the new equilibrium position.

As shown in Figure , if the position of the block is recorded as a function of time, the recording is a periodic function. If the block is displaced to a position y , the net force becomes.

This is just what we found previously for a horizontally sliding mass on a spring. The constant force of gravity only served to shift the equilibrium location of the mass. Therefore, the solution should be the same form as for a block on a horizontal spring,.

The equations for the velocity and the acceleration also have the same form as for the horizontal case. Note that the inclusion of the phase shift means that the motion can actually be modeled using either a cosine or a sine function, since these two functions only differ by a phase shift. Simple harmonic motion SHM is oscillatory motion for a system where the restoring force is proportional to the displacement and acts in the direction opposite to the displacement. Maximum displacement is the amplitude A.

The angular frequency. 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. Describe what happens to the sound waves in terms of period, frequency and amplitude as the sound decreases in volume. A babysitter is pushing a child on a swing.