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General Definitions

This section collects the laws and equations that will be used to build the instance models.

RefnameGD:velocityIX
LabelThe x-component of velocity of the pendulum
Unitsms
Equationvx=ωLrodcos(θp)
Description
  • vx is the x-component of velocity (ms)
  • ω is the angular velocity (rads)
  • Lrod is the length of the rod (m)
  • θp is the displacement angle of the pendulum (rad)
Source
RefBy

Detailed derivation of the x-component of velocity:

At a given point in time, velocity may be defined as

v(t)=dp(t)dt

We also know the horizontal position

px=Lrodsin(θp)

Applying this,

vx=dLrodsin(θp)dt

Lrod is constant with respect to time, so

vx=Lroddsin(θp)dt

Therefore, using the chain rule,

vx=ωLrodcos(θp)

RefnameGD:velocityIY
LabelThe y-component of velocity of the pendulum
Unitsms
Equationvy=ωLrodsin(θp)
Description
  • vy is the y-component of velocity (ms)
  • ω is the angular velocity (rads)
  • Lrod is the length of the rod (m)
  • θp is the displacement angle of the pendulum (rad)
Source
RefBy

Detailed derivation of the y-component of velocity:

At a given point in time, velocity may be defined as

v(t)=dp(t)dt

We also know the vertical position

py=Lrodcos(θp)

Applying this,

vy=(dLrodcos(θp)dt)

Lrod is constant with respect to time, so

vy=Lroddcos(θp)dt

Therefore, using the chain rule,

vy=ωLrodsin(θp)

RefnameGD:accelerationIX
LabelThe x-component of acceleration of the pendulum
Unitsms2
Equationax=ω2Lrodsin(θp)+αLrodcos(θp)
Description
  • ax is the x-component of acceleration (ms2)
  • ω is the angular velocity (rads)
  • Lrod is the length of the rod (m)
  • θp is the displacement angle of the pendulum (rad)
  • α is the angular acceleration (rads2)
Source
RefBy

Detailed derivation of the x-component of acceleration:

Our acceleration is:

a(t)=dv(t)dt

Earlier, we found the horizontal velocity to be

vx=ωLrodcos(θp)

Applying this to our equation for acceleration

ax=dωLrodcos(θp)dt

By the product and chain rules, we find

ax=dωdtLrodcos(θp)ωLrodsin(θp)dθpdt

Simplifying,

ax=ω2Lrodsin(θp)+αLrodcos(θp)

RefnameGD:accelerationIY
LabelThe y-component of acceleration of the pendulum
Unitsms2
Equationay=ω2Lrodcos(θp)+αLrodsin(θp)
Description
  • ay is the y-component of acceleration (ms2)
  • ω is the angular velocity (rads)
  • Lrod is the length of the rod (m)
  • θp is the displacement angle of the pendulum (rad)
  • α is the angular acceleration (rads2)
Source
RefBy

Detailed derivation of the y-component of acceleration:

Our acceleration is:

a(t)=dv(t)dt

Earlier, we found the vertical velocity to be

vy=ωLrodsin(θp)

Applying this to our equation for acceleration

ay=dωLrodsin(θp)dt

By the product and chain rules, we find

ay=dωdtLrodsin(θp)+ωLrodcos(θp)dθpdt

Simplifying,

ay=ω2Lrodcos(θp)+αLrodsin(θp)

RefnameGD:hForceOnPendulum
LabelHorizontal force on the pendulum
UnitsN
EquationF=max=Tsin(θp)
Description
  • F is the force (N)
  • m is the mass (kg)
  • ax is the x-component of acceleration (ms2)
  • T is the tension (N)
  • θp is the displacement angle of the pendulum (rad)
Source
RefBy

Detailed derivation of force on the pendulum:

F=max=Tsin(θp)

RefnameGD:vForceOnPendulum
LabelVertical force on the pendulum
UnitsN
EquationF=may=Tcos(θp)mg
Description
  • F is the force (N)
  • m is the mass (kg)
  • ay is the y-component of acceleration (ms2)
  • T is the tension (N)
  • θp is the displacement angle of the pendulum (rad)
  • g is the gravitational acceleration (ms2)
Source
RefBy

Detailed derivation of force on the pendulum:

F=may=Tcos(θp)mg

RefnameGD:angFrequencyGD
LabelThe angular frequency of the pendulum
Unitss
EquationΩ=gLrod
Description
  • Ω is the angular frequency (s)
  • g is the gravitational acceleration (ms2)
  • Lrod is the length of the rod (m)
Notes
Source
RefByGD:periodPend and IM:calOfAngularDisplacement

Detailed derivation of the angular frequency of the pendulum:

Consider the torque on a pendulum defined in TM:NewtonSecLawRotMot. The force providing the restoring torque is the component of weight of the pendulum bob that acts along the arc length. The torque is the length of the string Lrod multiplied by the component of the net force that is perpendicular to the radius of the arc. The minus sign indicates the torque acts in the opposite direction of the angular displacement:

τ=Lrodmgsin(θp)

So then

Iα=Lrodmgsin(θp)

Therefore,

Iddθpdtdt=Lrodmgsin(θp)

Substituting for I

mLrod2ddθpdtdt=Lrodmgsin(θp)

Crossing out m and Lrod we have

ddθpdtdt=(gLrod)sin(θp)

For small angles, we approximate sin θp to θp

ddθpdtdt=(gLrod)θp

Because this equation, has the same form as the equation for simple harmonic motion the solution is easy to find. The angular frequency

Ω=gLrod

RefnameGD:periodPend
LabelThe period of the pendulum
Unitss
EquationT=2πLrodg
Description
  • T is the period (s)
  • π is the ratio of circumference to diameter for any circle (Unitless)
  • Lrod is the length of the rod (m)
  • g is the gravitational acceleration (ms2)
Notes
  • The frequency and period are defined in the data definitions for frequency and period respectively
Source
RefBy

Detailed derivation of the period of the pendulum:

The period of the pendulum can be defined from the general definition for the equation of angular frequency

Ω=gLrod

Therefore from the data definition of the equation for angular frequency, we have

T=2πLrodg