Determing ing gravity by a pendulum (1 Viewer)

[henry08]

So gravity can be determined through the use of a pendulum, however why is it only able to be used for angles up to 10 degrees? An explanation would be much appreciated.

 

[lolokay]

I don't remember how the pendulum period formula is derived, but at some point you assume that sin[x]=x, which is only a good approximation for small angles

 

[ratcher0071]

g= 4pi² l / T²

Where;
g = acceleration due to gravity in m/s
l = length in m
T = period in s

 

[tommykins]

So gravity can be determined through the use of a pendulum, however why is it only able to be used for angles up to 10 degrees? An explanation would be much appreciated.

I recall doing this experiment in the beginnign of the year.
I think the angles don't matter as long as you make sure you drop the pendulum at the exact angle (wasn't humanely possible, a problem with the validity of the experiment) and record the time taken for 10 oscillations/the period.

 

[Kirjava]

I read some time ago that it's because T = 2pi(l/g)^1/2 is only an approximate solution to the differential equations governing a pendulum's motion (it only gives good results for small angles).

 

[lolokay]

a = -g sin@
and the horizontal component of the motion is l sin@
so if we make the assumption that the acceleration and the displacement are always acting in the same direction*, then we can take @=sqrt(g/l)*t to make it such that d²s/dt² holds when differentiating displacement w.r.t time.

*this assumption holds quite well for small angles, since the acceleration is mostly horizontal

we also get
dx = l dsin@ = l cos@ d@
dx/dt = l cos@ d@/dt = rt(g/l) cos@
dt = rt(l/g) d@
t = rt(l/g) @

for a complete oscillation we take @ = 2pi (not completely sure why)
so T = rt(l/g) 2pi

what I figured out so far. a bit sketchy I suppose, but just thought I'd share

 

[Continuum]

So gravity can be determined through the use of a pendulum, however why is it only able to be used for angles up to 10 degrees? An explanation would be much appreciated.

It should be because at small angles (less than 10 degrees), the period of the pendulum will remain constant. In other words, a pendulum performs simple harmonic motion when the initial angle is small. This is because of the lack of air resistance due to the fact that the linear velocity of the pendulum bob is small. If you had a large initial angle, air resistance will become a significant factor and prevent the pendulum bob from coming back to that initial angle. Since you're measuring 10 periods, it is essential that those periods are equal and this only occurs when the initial angle is less than 10 degrees. So for the experiment to be valid, the initial angle must be less than 10 degrees.

 

[shaon0]

this is exactly why.

F = -mg sin@ (component of weight tangent to arc)

for small angles sin@ ~ @

F ~ -mg@

sub arc length displacement: x = @l

F ~ - (mg/l) x

which means the pendulum undergoes simple harmonic motion (same form as hooke's law F = -kx)

Yeah. how did you know that? physics at high school?

 

[shaon0]

you will do simple harmonic motion in ext 1 math, you also cover it in first year physics at uni. dont remember it being mentioned in hsc physics which is a bit of a shame :/

yeah, the physics and chemistry syllabus is just stupid now. they should have kept 4unit physics and 4unit chemistry and i'd take them for sure

 

[henry08]

Thnaks for the explantion.

 

[Michaelmoo]

I don't remember how the pendulum period formula is derived, but at some point you assume that sin[x]=x, which is only a good approximation for small angles

The larger you make the angle, the more external factors are going to have an effect on the experiment, most notably the friction of the pendulum swinging.

Up to 10 degrees, these external factors are almost negligable...

 

[Forbidden.]

回复: Re: Determing ing gravity by a pendulum

this is exactly why.

F = -mg sin@ (component of weight tangent to arc)

for small angles sin@ ~ @

F ~ -mg@

sub arc length displacement: x = @l

F ~ - (mg/l) x

which means the pendulum undergoes simple harmonic motion (same form as hooke's law F = -kx)

i love how you always show these hsc physics kids the real physics out there

EDIT:

Perhaps a Free-Body Diagram will help your explanation

 

[youngminii]

this is exactly why.

F = -mg sin@ (component of weight tangent to arc)

for small angles sin@ ~ @

F ~ -mg@

sub arc length displacement: x = @l

F ~ - (mg/l) x

which means the pendulum undergoes simple harmonic motion (same form as hooke's law F = -kx)

Wait so if we used this as a reason for the whole validity thing, would we get marked up higher?
Or should we just stick to our Sylla-Bible and state the air resistance thingo
I'm getting assessed on this in less than 2 weeks ><