General Definitions
This section collects the laws and equations that will be used to build the instance models.
Refname | GD:velocityIX |
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Label | The x-component of velocity of the pendulum |
Units | ms |
Equation | vx=ωLrodcos(θp) |
Description |
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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)
Refname | GD:velocityIY |
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Label | The y-component of velocity of the pendulum |
Units | ms |
Equation | vy=ωLrodsin(θp) |
Description |
|
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)
Refname | GD:accelerationIX |
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Label | The x-component of acceleration of the pendulum |
Units | ms2 |
Equation | ax=−ω2Lrodsin(θp)+αLrodcos(θp) |
Description |
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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)
Refname | GD:accelerationIY |
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Label | The y-component of acceleration of the pendulum |
Units | ms2 |
Equation | ay=ω2Lrodcos(θp)+αLrodsin(θp) |
Description |
|
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)
Refname | GD:hForceOnPendulum |
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Label | Horizontal force on the pendulum |
Units | N |
Equation | F=max=−Tsin(θp) |
Description |
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Source | – |
RefBy |
Detailed derivation of force on the pendulum:
F=max=−Tsin(θp)
Refname | GD:vForceOnPendulum |
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Label | Vertical force on the pendulum |
Units | N |
Equation | F=may=Tcos(θp)−mg |
Description |
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Source | – |
RefBy |
Detailed derivation of force on the pendulum:
F=may=Tcos(θp)−mg
Refname | GD:angFrequencyGD |
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Label | The angular frequency of the pendulum |
Units | s |
Equation | Ω=√gLrod |
Description |
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Notes |
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Source | – |
RefBy | GD: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
Refname | GD:periodPend |
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Label | The period of the pendulum |
Units | s |
Equation | T=2π√Lrodg |
Description |
|
Notes | |
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=2π√Lrodg