Maybe you’ve already seen one at an amusement park ship swing seen as an attraction or you drove on it yourself. Did you know that you can physically describe such a swing to calculate the speed at which the swing rushes by? The model will help you with this thread pendulum.
Yarn pendulum definition
The thread pendulum, often also called mathematical pendulum referred to, consists of a a thread of length l at the one body of mass m hangs. The pendulum begins to oscillate as a result of a deflection x0 from the equilibrium position.
That thread pendulum consists of a freely suspended thread the length l at the end one mass w is attached.
The calculations of the filament pendulum are all made under the assumption that the weight of the filament can be neglected. Likewise, the expansion of the thread due to the weight is not taken into account.
Furthermore, it should be noted: The length l of the thread is from the suspension until main emphasis of mass m measured. So you should be careful not only to consider the length to the surface of the mass m.
How can you now carry out calculations with the basic properties of the thread pendulum?
pendulum formulas
Now it’s a matter of describing various mathematical quantities on the thread pendulum. You are looking at here small deflections, to keep the calculations simple. The formulas are based on a linear deflection. Small deflections can be approximated as being straight rather than circular as in reality. When doing the calculations, you can usually use the Ignore friction losses. The thread pendulum then swings with a harmonic vibration.
The most important variables of a harmonic oscillation are above all the amplitude A and the period or oscillation duration T. However, the speed of the mass body can also be determined.
With the thread pendulum, the lowest point and the point of maximum deflection are the most important points in the course of the thread pendulum.
Yarn pendulum deflection
the deflection x is measured from the middle point of the pendulum. where x0 is the point of the maximum deflectionwhich also determines the deflection at the beginning.
The deflection x can be determined depending on the time t. You can neglect any friction of the pendulum, such as air resistance, which could lead to energy loss.
the deflection x a thread pendulum the length l to the time t and the angular frequency ω0 you can calculate it like this:
where x0 is the point of the maximum deflection and g the location factor for the thread pendulum.
Note that you do not need the mass m to calculate the deflection, so the deflection is independent of the mass.
You can use this formula to determine the deflection x of the pendulum at any point in time t.
ω0 describes the angular frequency with which the pendulum swings. This is discussed in more detail in the section on the period of oscillation.
What is the amplitude of the thread pendulum if you already know the deflection?
Yarn pendulum amplitude
the amplitude of an oscillation is also called the maximum deflection. It describes how far a swinging body maximum from its equilibrium position can be removed.
The following applies to the thread pendulum: The amplitude is the maximum distance x between the rest position () and the maximum deflection x0. Since the thread pendulum cannot increase its deflection without additional force, the initial deflection x0 is equal to the amplitude of the pendulum in our considerations.
In the case of the thread pendulum, this corresponds to amplitude the initial deflection x0. This is the maximum distance the pendulum can travel from its equilibrium position.
If the thread pendulum were to experience friction, such as B. air resistance, the deflection x would continue to decrease with each period. For example, you would start with a deflection x0, but by the time the pendulum has swung to the other side, the deflection would already be less than x0.
How long does such an oscillation last in the thread pendulum?
Period duration Oscillation duration thread pendulum
The thread pendulum swings with a certain period duration. Often you will also find the name period of oscillation. It describes the duration that the thread pendulum needs for one oscillation or one period.
An oscillation or a period is the time interval in which the thread pendulum swings from the deflection x back to this same position x with the same direction of movement as before.
the period duration T of a thread pendulum of length l describes the time that the pendulum needs for a complete swing.
For the calculation you need the location factor g.
Alternatively, you can also angular frequency, with which the thread pendulum oscillates. In contrast to the frequency, however, it does not indicate the number of oscillation periods in a period of time, but the swept phase angle of vibration per period of time, i.e. how large the angle covered per second is.
the angular frequency of a thread pendulum you calculate with the length l and the location factor g:
As you can see, you only need the length l of the thread and the gravitational acceleration g to calculate the period. The period is thus independently of the Dimensions m of the oscillating body and also of the amplitude x0.
So as long as you carry out your experiment in the same place, a varying filament length is the only thing that causes a change in the period duration.
yarn shuttle speed
To calculate the speed of the thread pendulum, you can take a look at the Energies in the thread pendulum throw. Depending on when you look at the pendulum, it has different amounts of kinetic and potential energy. The two energies are converted into each other.
With the oscillation of the thread pendulum there is two special points at the ratio between potential and kinetic energy:
Reaches the pendulum its greatest deflection x0, own it there only potential energy epot. The pendulum is h above the rest position at . The kinetic energy Ekin at this point is zero because the pendulum stands still for a brief moment while changing direction.
At the lowest point, with of deflection , the potential energy is zero and the kinetic energy of the pendulum is maximum. Here the pendulum moves with the maximum speed v.
Below you can see the whole thing defined in more detail:
Reaches the pendulum its greatest deflection x0, own it there only potential energy Epot.
At the lowest point the potential energy is zero and the kinetic energy Ekin of the pendulum is maximum.
The two energies are constantly converted into one another in the course of the pendulum process.
The following applies:
In Figure 4 you can see the principle of energy conversion in the thread pendulum once again graphically. Between the two times shown, the pendulum has both kinetic and potential energy.
But how can you now determine the maximum speed of the pendulum?
By connecting the two energies with the help of the conservation of energy merge
You can determine the height of the pendulum at maximum deflection and use it to calculate the maximum potential energy.
For the deflection all potential energy is converted into kinetic energy. That is, here is the kinetic energy and therefore also the maximum speed.
To now this maximum speed To calculate v, you equate the potential energy Epot and the kinetic energy Ekin. Due to the law of conservation of energy, the maximum potential energy is equal to the maximum kinetic energy if there is only and complete conversion between these two forms of energy.
Next you solve the equation for the desired speed v. where h is the height of the pendulum at position x0.
This gives you the formula for calculating the maximum speed.
The maximum speed of the pendulum depends solely on the height of the pendulum.
you can maximum speed vs of the pendulum with the height h of the pendulum in position x0 and with the location factor g to calculate:
The speed you calculate is the maximum speed of the pendulum when it is. However, what is the velocity in the rest of the oscillation?
Filament pendulum physics history
The relationships between the energetic quantities become clear when considering the course of time. Figure 5 shows Ekin in blue, Epot in turquoise and the deflection in red.
Figure 5: Temporal progression of the energies in the thread pendulum
You can see it in the picture process of energy conversion between kinetic and potential energy. Whenever one of the two energies has reached its maximum amount, the other energy is zero.
You can also see when which energy reaches its maximum. The kinetic energy is maximum when the displacement is zero, the potential energy is maximum when the magnitude of the displacement is maximum.
The kinetic energy corresponds to the speed of the thread pendulum. You could also read the course of the speed compared to the deflection here.
This is exactly how the ride on the swing boat feels: At the points of maximum deflection you are furthest up, so you have the maximum potential energy.
When driving down, you accelerate at maximum speed until you race past the lowest point. After that, the ship slows down until you’ve reached the top on the other side and the whole thing starts over again.
Compute pendulum experiment
Here you will find another task on the subject of thread pendulums, with which you can apply your knowledge.
task
A body of mass hangs on a thread pendulum of length .
a) Calculate the period T of the pendulum.
b) How large must the initial deflection x0 be if the deflection is exactly at the point in time?
solution
a) Insert the two statements from the task into the equation for the period.
You have now calculated periods of about one second.
b) To determine the initial deflection of the pendulum, you can look at the general oscillation equation of a pendulum.
For the thread pendulum, the frequency is . Therefore you can also write the oscillation equation of the thread pendulum as:
Now change the formula according to the amplitude x0 and insert the information about the time (deflection).
The necessary initial deflection is therefore 2.3 cm.
Thread pendulum – the most important thing
- That thread pendulum consists of a freely suspended thread the length l, at the end one mass w is attached.
- the deflection x a thread pendulum the length l to the time t you can calculate it like this:
-
the period duration T of a thread pendulum of length l describes the time that the pendulum needs for a complete oscillation.
-
the angular frequency of a thread pendulum you calculate with the length l and the location factor g:
-
you can speed vs of the pendulum with the height h of the pendulum in position x0 and dem location factor g to calculate:
-
Reaches the pendulum its greatest deflection x0 , owns it there only potential energy epot. At the lowest…