# Calculating g with pendulum (1 Viewer)

#### QZP

##### Well-Known Member
When calculating the value of g with the motion of a pendulum, why must I consider the the arc angle (the angle string makes with vertical when released from rest)?
Sources say to keep it within ~30 degrees but can someone tell me why?

#### studybuddy101

##### Active Member
an angle too large will introduce experimental error since the string will start to experience tension. keeping the angle small ensures gravity is the only force acting

#### seanieg89

##### Well-Known Member
an angle too large will introduce experimental error since the string will start to experience tension. keeping the angle small ensures gravity is the only force acting
Uhh no, the string will always "experience tension". If gravity was the only force acting, then the weight at the end of the string would fall straight down, not swing.

The reason we keep the angle small is that when we solve the differential equation for the motion of a pendulum in terms of the unknown constant g, we approximate sin(x) as x. This is a linear approximation that is accurate for small angles. If we do not make this approximation, we cannot find such a simple expression for the period in terms of g (which we then invert to tell us g from the observed period).

#### studybuddy101

##### Active Member
Uhh no, the string will always "experience tension". If gravity was the only force acting, then the weight at the end of the string would fall straight down, not swing.

The reason we keep the angle small is that when we solve the differential equation for the motion of a pendulum in terms of the unknown constant g, we approximate sin(x) as x. This is a linear approximation that is accurate for small angles. If we do not make this approximation, we cannot find such a simple expression for the period in terms of g (which we then invert to tell us g from the observed period).
lol my bad :L week off from hsc and already starting to lose it

#### AnimeX

##### Member
When calculating the value of g with the motion of a pendulum, why must I consider the the arc angle (the angle string makes with vertical when released from rest)?
Sources say to keep it within ~30 degrees but can someone tell me why?
it's recommended 5-10 degrees to keep the pendulum undergoing simple harmonic motion, so in order for the experiment to be valid do it at small angles.

#### QZP

##### Well-Known Member
What happens physically to the pendulum though that invalidates the experiment if we exceed small angles? I want to write it as part of my discussion

#### seanieg89

##### Well-Known Member
What happens physically to the pendulum though that invalidates the experiment if we exceed small angles? I want to write it as part of my discussion
Nothing, it still moves from side to side, but simple harmonic motion is no longer an accurate description, and we have no easy way of calculating period in terms of g.

#### QZP

##### Well-Known Member
Nothing, it still moves from side to side, but simple harmonic motion is no longer an accurate description, and we have no easy way of calculating period in terms of g.
Ah, I see. Thank you!!

#### seanieg89

##### Well-Known Member
Also, if the pendulum has enough initial momentum that it's angle will at some stage exceed pi/2 then the analysis is even more complicated.

#### probjer

##### Member
Ok mate, you do realise that seanieg89 is trolling you right? He is giving you mathematical explanations that are beyond the HSC physics syllabus/ level. The reason why we want the swing angle small is to minimise experimental error. Common sense: a small swing angle will allow the pendulum to swing smoothly and not bobble around, hence making our results more accurate.

#### Carrotsticks

##### Retired
Ok mate, you do realise that seanieg89 is trolling you right? He is giving you mathematical explanations that are beyond the HSC physics syllabus/ level. The reason why we want the swing angle small is to minimise experimental error. Common sense: a small swing angle will allow the pendulum to swing smoothly and not bobble around, hence making our results more accurate.
Your reasoning is correct, but it is not the only reason.

Mathematically, using a smaller angle also minimises experimental error due to the fact that the common expression studied for pendulums are linear, when it should in fact be in terms of the sine function. But the margin of accuracy decreases as the swing angle gets larger because as the angle gets larger, the linear function no longer approximates the sine function very well.

#### Fizzy_Cyst

##### Well-Known Member
Ok mate, you do realise that seanieg89 is trolling you right? He is giving you mathematical explanations that are beyond the HSC physics syllabus/ level. The reason why we want the swing angle small is to minimise experimental error. Common sense: a small swing angle will allow the pendulum to swing smoothly and not bobble around, hence making our results more accurate.
Shame on you Seanieg!

Using mathematics to explain HSC Physics! Tsk tsk.

#### seanieg89

##### Well-Known Member
Ok mate, you do realise that seanieg89 is trolling you right? He is giving you mathematical explanations that are beyond the HSC physics syllabus/ level. The reason why we want the swing angle small is to minimise experimental error. Common sense: a small swing angle will allow the pendulum to swing smoothly and not bobble around, hence making our results more accurate.
How was I trolling? My explanation was beyond syllabus because the motion of a large amplitude pendulum is beyond syllabus.

A properly designed pendulum would not "bobble around" markedly if we increase angle size, and there is neglible difference in the experimental error between an angle of 50 degrees and an angle of 30 degrees. But no matter how accurate the measurement equipment we use and how perfect our pendulum is, the fact remains that the motion of a pendulum is NOT simple harmonic, and gets "less" simple harmonic the larger our angle of oscillation. We will be able to measure period as accurately as we like, but what can we do with this number if we do not have a simple way of relating it to g?

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#### seanieg89

##### Well-Known Member
Shame on you Seanieg!

Using mathematics to explain HSC Physics! Tsk tsk.
Huh?

Sarcasm mate ;-)

#### seanieg89

##### Well-Known Member
Sarcasm mate ;-)
Haha I thought so, just can never be entirely sure on the internet.