Pendulum question?

Question

Length of simple pendulum = 0.760 m
Pendulum bob = 0.365 kg
Theta = 12.0 degrees
a) Find the frequency (Don't bother with this part)
b) What is the pendulum bob's speed when it passes thru the lowest point of the swing? (equilibrium)
c) What is the total energy stored in this oscillation, assuming no loss? (use answer from #2)
**assume simple harmonic motion for all questions

-For part 2, I've tried 3 different approaches, and I don't know which is correct:
1) centripetal force
mv^2/r = Tension- mg...but how to find tension?
2) definition of v:
v = x/t
x = 2* L * theta
1/t = frequency
v = 2*l*theta*frequency (this is what I think works)
3) Find total energy first; Total energy = (1/2)kA^2
k = mg/L for a simple pendulum
A = L * theta
So, the total energy is: (1/2)(mg/L)(L*theta)^2
Then, this amount equals (1/2)mv^2
so solve for v:
root((g)(theta)^2*L) = v
So, my best tries are:
v = root(gL*theta^2)
v = 2L*theta *frequency
which is correct?

Answer 1

b) Use energy! At the lowest point, PE = 0. At the highest point, KE = 0. Due to conservation of energy, the KE at the lowest point must equal the PE at the highest point.

KEf = PEi

½m v² = mgh

m's drop off

v = sqrt(2gh) = sqrt(2 * 9.8 * 0.76 * [1 - cos(12º)])
v = 0.571 m/s

How did I get that deal with the cos? Draw a picture of the situation (a crude one appears below)

|--\
|12\
|-----\
|------\
|-------\
|
|

The vertical leg of the triangle is 0.76 * cos(12º). But that's not the height of the bob. Remember that the string is 0.76, so the hypotenuse of the triangle is 0.76. The full length of the string minus this distance is the height of the bob. Draw a picture and stare at it, you'll get it eventually :)

Your first method won't work because, as you pointed out, you don't know the tension. You can use conservation of angular momentum (torque) to determine this, but you won't get anywhere useful.

Your second method won't work because the velocity is not constant. v = x/t only applies for constant velocity. If it is uniformly acclerating, then we must use vf = vi + at... However, the pendulum is NOT uniformly accelearting, so this won't even work!!!

Your 3rd method would work if we were dealing with a spring... pendulums do not have spring constants, lol.

c) This one is easy now that you know v.

E = ½mv² = 0.0594 J

Answer 2

Use the potential energy at the point where the motion is zero. Do this by using the theta of 12 degrees to figure the height above the bottom. Then at the bottom the energy will be all kinetic and you can figure the speed.

Frequency would not be part of this calculation.