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

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Description

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Unlike the mathematical pendulum, the physical pendulum deals with a real, non-point mass. While the length $l$ is defined as the distance between the pivot point and the center of mass of the body, the potential energy of both pendulums is the same. However, the kinetic energy can no longer be approximated using expressions that depend solely on $l$ and $m$; instead, you need to know the actual moment of inertia of the body.

ID:(7097, 0)

Description

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Unlike the mathematical pendulum, the physical pendulum deals with a real mass, not a point mass. As we define the length $l$ as the distance between the pivot and the center of mass of the body, the potential energy of both pendulums coincides. However, the kinetic energy can no longer be approximated by an expression that depends solely on $l$ and $m$; instead, it must incorporate the actual moment of inertia of the body.

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Parameters

$\omega_0$

omega_0

Angular Frequency of Physical Pendulum

rad/s

$g$

g

Gravitational Acceleration

m/s^2

$m_g$

m_g

Gravitational mass

kg

$\theta_0$

theta_0

Initial Angle

rad

$K_r$

K_r

Kinetic energy of rotation

J

$I$

I

Moment of inertia for axis that does not pass through the CM

kg m^2

$L$

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:(15853, 0)

Equation

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

$E = K_r + V$

$E = K + V$

Conditions

Variables

$V$

$V$

Potential Energy Pendulum, for small Angles

$J$

6285

$E$

Total Energy

$J$

5290

$K$

$K_r$

Kinetic energy of rotation

$J$

5289

Relation

ID:(3687, 0)

Equation

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In the case being studied of translational motion, the definition of energy

is applied to Newton's second law

resulting in the expression

$K_r =\displaystyle\frac{1}{2} I \omega ^2$

Conditions

Variables

$\omega$

Angular Speed

$rad/s$

6068

$K_r$

Kinetic energy of rotation

$J$

5289

$I$

Moment of inertia for axis that does not pass through the CM

$kg m^2$

5315

Relation

Development

The energy required for an object to change its angular velocity from $\omega_1$ to $\omega_2$ can be calculated using the definition

$\Delta W = T \Delta\theta$

Applying Newton's second law, this expression can be rewritten as

$\Delta W=I \alpha \Delta\theta=I\displaystyle\frac{\Delta\omega}{\Delta t}\Delta\theta$

Using the definition of angular velocity

$\bar{\omega} \equiv\displaystyle\frac{ \Delta\theta }{ \Delta t }$

we get

$\Delta W=I\displaystyle\frac{\Delta\omega}{\Delta t}\Delta\theta=I \omega \Delta\omega$

The difference in angular velocities is

$\Delta\omega=\omega_2-\omega_1$

On the other hand, angular velocity itself can be approximated with the average angular velocity

$\omega=\displaystyle\frac{\omega_1+\omega_2}{2}$

Using both expressions, we obtain the equation

$\Delta W=I \omega \Delta \omega=I(\omega_2-\omega_1)\displaystyle\frac{(\omega_1+\omega_2)}{2}=\displaystyle\frac{I}{2}(\omega_2^2-\omega_1^2)$

Thus, the change in energy is given by

$\Delta W=\displaystyle\frac{I}{2}\omega_2^2-\displaystyle\frac{I}{2}\omega_1^2$

This allows us to define kinetic energy as

$K_r =\displaystyle\frac{1}{2} I \omega ^2$

ID:(3255, 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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Regarding the physical pendulum:

The energy is given by:

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

As a result, the angular frequency is:

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

Conditions

Variables

$\omega_0$

Angular Frequency of Physical Pendulum

$rad/s$

6288

$g$

Gravitational Acceleration

9.8

$m/s^2$

5310

$m_g$

Gravitational mass

$kg$

8762

$I$

Moment of inertia for axis that does not pass through the CM

$kg m^2$

5315

$L$

Pendulum Length

$m$

6282

Relation

Development

Given that the kinetic energy of the physical pendulum with moment of inertia $I$ and angular velocity $\omega$ is represented by

$K_r =\displaystyle\frac{1}{2} I \omega ^2$

and the gravitational potential energy is given by

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

where $m$ is mass, $l$ is string length, $\theta$ is the angle, and $g$ is angular acceleration, the energy equation can be expressed as

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

As the period is defined as

$T=2\pi\sqrt{\displaystyle\frac{I}{mgl}}$

we can determine the angular frequency as

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

ID:(4517, 0)

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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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 Physical Pendulum

$rad/s$

6288

$\nu$

Frequency

$Hz$

5077

$\pi$

Pi

3.1415927

$rad$

5057

Relation

ID:(12338, 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 Physical Pendulum

$rad/s$

6288

$T$

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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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 Physical Pendulum

$rad/s$

6288

$t$

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 Physical Pendulum

$rad/s$

6288

$x_0$

$\theta_0$

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)