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In the case of a pendulum with a point mass, the potential energy is generated by raising the mass against the gravitational field as the pendulum deviates by a given angle.

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Description

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A pendulum is described as a point mass $m$ hanging from a string that is attached to a pivot point and has a length $l$. It is called a mathematical pendulum because it is an abstraction of a physical pendulum, with the difference being that its mass is treated as a point mass.

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Description

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A pendulum is defined by a point mass $m$ hanging from a string attached to a pivot of length $l$. It is referred to as a mathematical pendulum because it is an abstraction of a physical pendulum, where the mass is considered to be concentrated at a single point.

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Parameters

$\omega_0$

omega_0

Angular Frequency of Mathematical Pendulum

rad/s

$g$

g

Gravitational Acceleration

m/s^2

$m_g$

m_g

Gravitational mass

kg

$m_i$

m_i

Inertial Mass

kg

$\theta_0$

theta_0

Initial Angle

rad

$K$

K

Kinetic energy of point mass

J

$L$

L

Pendulum Length

m

$\pi$

pi

Pi

rad

$V$

V

Potential Energy Pendulum, for small Angles

J

$\theta$

theta

Swing angle

rad

$E$

E

Total Energy

J

Variables

$\omega$

omega

Angular Speed

rad/s

$\nu$

nu

Frequency

Hz

$T$

T

Period

s

$t$

t

Time

s

Calculations

• Alternative method
• Traditional method
• Strategy
• 2D
• 3D

Equation

Number

First, select the equation: to , then, select the variable: to

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Calculations

Symbol

Equation

Solved

Translated

Calculations

Symbol

Equation

Solved

Translated

Variable Given Calculate Target : Equation To be used

Equations

ID:(15852, 0)

Equation

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The total energy corresponds to the sum of the total kinetic energy and the potential energy:

$E = K + V$

Conditions

Variables

$V$

$V$

Potential Energy Pendulum, for small Angles

$J$

6285

$E$

Total Energy

$J$

5290

$K$

$K$

Kinetic energy of point mass

$J$

6286

Relation

ID:(3687, 0)

Equation

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The kinetic energy of a rotating body is given by

where $I$ is the moment of inertia and $\omega$ is the angular velocity. For a point mass $m$ rotating at a distance $L$ from an axis, the moment of inertia is

hence,

$K =\displaystyle\frac{1}{2} m_i L ^2 \omega ^2$

Conditions

Variables

$\omega$

Angular Speed

$rad/s$

6068

$m_i$

Inertial Mass

$kg$

6290

$K$

Kinetic energy of point mass

$J$

6286

$L$

Pendulum Length

$m$

6282

Relation

ID:(4515, 0)

Equation

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The gravitational potential energy of a pendulum is

which for small angles can be approximated as:

$V =\displaystyle\frac{1}{2} m_g g L \theta ^2$

Conditions

Variables

$g$

Gravitational Acceleration

9.8

$m/s^2$

5310

$m_g$

Gravitational mass

$kg$

8762

$L$

Pendulum Length

$m$

6282

$V$

Potential Energy Pendulum, for small Angles

$J$

6285

$\theta$

Swing angle

$rad$

6283

Relation

Development

The gravitational potential energy of a pendulum with mass m, suspended from a string of length L and deflected by an angle \theta is given by

$U = m g L (1-\cos \theta )$

where g is the acceleration due to gravity.

For small angles, the cosine function can be approximated using a Taylor series expansion up to the second term

$\cos\theta\sim 1-\displaystyle\frac{1}{2}\theta^2$

This approximation leads to the simplification of the potential energy to

$V =\displaystyle\frac{1}{2} m_g g L \theta ^2$

It's important to note that the angle must be expressed in radians.

ID:(4514, 1)

Equation

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The gravitational potential energy of a pendulum is

which for small angles can be approximated as:

$E =\displaystyle\frac{1}{2} m_g g L \theta_0 ^2$

$V =\displaystyle\frac{1}{2} m_g g L \theta ^2$

Conditions

Variables

$g$

Gravitational Acceleration

9.8

$m/s^2$

5310

$m_g$

Gravitational mass

$kg$

8762

$L$

Pendulum Length

$m$

6282

$V$

$E$

Total Energy

$J$

5290

$\theta$

$\theta_0$

Initial Angle

$rad$

5296

Relation

Development

The gravitational potential energy of a pendulum with mass m, suspended from a string of length L and deflected by an angle \theta is given by

$U = m g L (1-\cos \theta )$

where g is the acceleration due to gravity.

For small angles, the cosine function can be approximated using a Taylor series expansion up to the second term

$\cos\theta\sim 1-\displaystyle\frac{1}{2}\theta^2$

This approximation leads to the simplification of the potential energy to

$V =\displaystyle\frac{1}{2} m_g g L \theta ^2$

It's important to note that the angle must be expressed in radians.

ID:(4514, 2)

Equation

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The masses that Newton used in his principles are related to the inertia of bodies, which leads to the concept of the inertial Mass ($m_i$).

Newton's law, which is linked to the force between bodies due to their masses, is related to gravity, hence known as the gravitational mass ($m_g$).

Empirically, it has been concluded that both masses are equivalent, and therefore, we define

$m_g = m_i$

Conditions

Variables

$m_g$

Gravitational mass

$kg$

8762

$m_i$

Inertial Mass

$kg$

6290

Relation

Einstein was the one who questioned this equality and, from that doubt, understood why both 'appear' equal in his theory of gravity. In his argument, Einstein explained that masses deform space, and this deformation of space causes a change in the behavior of bodies. Thus, masses turn out to be equivalent. The revolutionary concept of space curvature implies that even light, which lacks mass, is affected by celestial bodies, contradicting Newton's theory of gravitation. This was experimentally demonstrated by studying the behavior of light during a solar eclipse. In this situation, light beams are deflected due to the presence of the sun, allowing stars behind it to be observed.

ID:(12552, 0)

Equation

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In the case of the mathematical pendulum

the energy can be expressed as

$E=\displaystyle\frac{1}{2}ml^2\omega^2+\displaystyle\frac{1}{2}mgl\theta^2$

and from this expression, we can obtain the angular frequency

$\omega_0 ^2=\displaystyle\frac{ g }{ L }$

Conditions

Variables

$\omega_0$

Angular Frequency of Mathematical Pendulum

$rad/s$

6287

$g$

Gravitational Acceleration

9.8

$m/s^2$

5310

$L$

Pendulum Length

$m$

6282

Relation

Development

The kinetic energy of the mathematical pendulum with mass $m$, string length $r$, and angular velocity $\omega$ is

$K =\displaystyle\frac{1}{2} m_i L ^2 \omega ^2$

and the gravitational potential energy is

$V =\displaystyle\frac{1}{2} m_g g L \theta ^2$

With $\theta$ representing the angle and $g$ the angular acceleration, the equation for the total energy is expressed as

$E=\frac{1}{2}m r^2 \omega^2 + \frac{1}{2}m g r \theta^2$

Given that the period is equal to

$T=2\pi\sqrt{\frac{m r^2}{m g r}}=2\pi\sqrt{\frac{r}{g}}$

we can relate the angular frequency as

$\omega_0 ^2=\displaystyle\frac{ g }{ L }$

ID:(4516, 0)

Equation

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The angular frequency ($\omega$) is with the period ($T$) equal to

$\omega_0 = \displaystyle\frac{2 \pi }{ T }$

$\omega = \displaystyle\frac{2 \pi }{ T }$

Conditions

Variables

$\omega$

$\omega_0$

Angular Frequency of Mathematical Pendulum

$rad/s$

6287

$T$

Period

$s$

5078

$\pi$

Pi

3.1415927

$rad$

5057

Relation

ID:(12335, 0)

Equation

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The frequency ($\nu$) corresponds to the number of times an oscillation occurs within one second. The period ($T$) represents the time it takes for one oscillation to occur. Therefore, the number of oscillations per second is:

$\nu =\displaystyle\frac{1}{ T }$

Conditions

Variables

$\nu$

Frequency

$Hz$

5077

$T$

Period

$s$

5078

Relation

Frequency is indicated in Hertz (Hz).

ID:(4427, 0)

Equation

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Como la frecuencia angular es con angular frequency $rad/s$, period $s$ and pi $rad$ igual a

y la frecuencia con frequency $Hz$ and period $s$ igual a

se tiene que con frequency $Hz$ and period $s$ igual a

$\omega_0 = 2 \pi \nu$

$\omega = 2 \pi \nu$

Conditions

Variables

$\omega$

$\omega_0$

Angular Frequency of Mathematical Pendulum

$rad/s$

6287

$\nu$

Frequency

$Hz$

5077

$\pi$

Pi

3.1415927

$rad$

5057

Relation

ID:(12338, 0)

Equation

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With the description of the oscillation using

the real part corresponds to the temporal evolution of the amplitude

$\theta = \theta_0 \cos \omega_0 t$

$x = x_0 \cos \omega_0 t$

Conditions

Variables

$x_0$

$\theta_0$

Initial Angle

$m$

5296

$x$

$\theta$

Swing angle

$m$

6283

$\omega_0$

$\omega_0$

Angular Frequency of Mathematical Pendulum

$rad/s$

6287

$t$

Time

$s$

5264

Relation

ID:(14074, 0)

Equation

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When we extract the real part of the derivative of the complex number representing the oscillation

whose real part corresponds to the velocity

$\omega = - \theta_0 \omega_0 \sin \omega_0 t$

$v = - x_0 \omega_0 \sin \omega_0 t$

Conditions

Variables

$\omega_0$

$\omega_0$

Angular Frequency of Mathematical Pendulum

$rad/s$

6287

$x_0$

$\theta_0$

Initial Angle

$m$

5296

$v$

$\omega$

Angular Speed

$m/s$

6068

$t$

Time

$s$

5264

Relation

Development

Using the complex number

$z = x_0 \cos \omega_0 t + i x_0 \sin \omega_0 t$

introduced in

$\dot{z} = i \omega_0 z$

we obtain

$\dot{z} = i\omega_0 z = i \omega_0 x_0 \cos \omega_0 t - \omega_0 x_0 \sin \omega_0 t$

thus, the velocity is obtained as the real part

$v = - x_0 \omega_0 \sin \omega_0 t$

ID:(14076, 0)