How Is Tension Calculated in a Pendulum with Varying Angles?

[devanlevin]

how can i express tension in this case, using alpha, alpha max,g, m, and L ??

a case of a mass of m hanging from a ceiling on a cord with a length of L swinging at a maximum angle of "alpha max" on either side, and at any given moment and an angle of alpha(changes all the time)

how can i express T, the strings tension, using the parameters m,g,alpha,alpha max,and L??

what i did was say:
according to circular movement, looking at the radial acceleration ar
F=ma
T-mg*cos(alpha)=mar=m*v^2/L
T=m(v^2/L+g*cos(alpha))

now how can i express v using the parameters, what i think i need to do is look at the maximum angle, and say V=0 so mar=0 so T(point of max)=mgcos(alpha max) but i don't see how that cann further me at all,
again, in the end i need an expression T=? using only m,g,L,alph, alpha max

***no friction at all, mass will reach alpha max every time, no loss of energy

 

[devanlevin]

T-mg*cos(alpha)=(m/L)*v^2

V^2=(L/m)(T-mgcos(alpha))

from here using energy,
E(max point)=mgh=const
my point of reference for potential energy being the ceiling

h=cos(alpha)*L

E=0.5mv^2+mgh=-mgcos(alpha max)*L
v^2=(L/m)(T-mgcos(alpha))

0.5(m)(L/m)(T-mgcos(alpha))-mgcos(alpha)*L=-mgcos(alpha)*L

0.5(T-mgcos(alpha))-mgcos(alpha)=-mgcos(alpha)
T-mgcos(alpha)-2mgcos(alpha)=-2mgcos(alpha)

T=3mgcos(alpha)-2mgcos(alpha max)

is this correct??
the question asked me to use L in the expression, do i need it??

 

[Topher925]

I would start by determining the period of the pendulum which will be a function of length and gravity. You then have your alpha max which will determine your displacement. Period and displacement you can then equate to velocity.

EDIT: Bit of a time lapse with post #2, wasn't able to read it before this post. What happened to this L?

0.5(m)(L/m)(T-mgcos(alpha))-mgcos(alpha)*L=-mgcos(alpha)*L

 

[devanlevin]

i divided the whole equation by L