- #1
Ummm phi is theta o ...and I changed the integral in part a to the integral in the attempt by substituting x = sin(theta/2) / sin ( theta o)kuruman said:This is all very confusing. What is ##\phi##? The radical in the first line of your development should be ##\sqrt{\sin^2(\theta_0/2)-\sin^2(\theta/2)}##. Also, the rest of the stuff in the integrand doesn't look right either. Please show your steps in more detail.
TSny said:I think you're OK so far (after realizing that ##\phi = \theta_0##). In the expression ##\frac{dx}{\cos \left(\theta / 2 \right)}##, express ##\cos \left(\theta / 2 \right)## in terms of ##x##.
EDIT: Did you mean to have (b/2) as the argument in sin2(b)?Clara Chung said:(I am using a as theta and b as theta o because I can't type them)
dx/cos(a/2) = dx / √(1-sin^2(x)sin^2(b))
So its approximation is
dx / {1-sin^2(x)sin^2(b)/2} ?
Another approach is to use the small angle approximation directly on ##\frac{1}{\cos \left( \theta /2 \right)}## rather than first expressing ##\cos \left(\theta /2 \right)## in terms of ##\sin \left(\theta /2 \right)##. But your method will work also with about the same amount of effort.Clara Chung said:(I am using a as theta and b as theta o because I can't type them)
dx/cos(a/2) = dx / √(1-sin^2(x)sin^2(b))
TSny said:EDIT: Did you mean to have (b/2) as the argument in sin2(b)?
You can continue to simplify this using the fact that b is small.
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Theta o onlyTSny said:In post #8, should ##\phi## stand for ##\theta_0## or ##\theta_0 / 2##?
Ahhh I understand it should be θ0 / 2Clara Chung said:Theta o only
A pendulum is a weight or mass suspended from a fixed point that is able to swing back and forth due to the force of gravity.
The motion of a pendulum is affected by its length, mass, and the force of gravity. Other factors that can affect its motion include air resistance and the angle at which it is released.
The formula for calculating the period of a pendulum is T = 2π√(L/g), where T is the period, L is the length of the pendulum, and g is the acceleration due to gravity.
The length of a pendulum is directly proportional to its period, meaning that as the length increases, the period also increases. This relationship is described by the formula T = 2π√(L/g).
In theory, a pendulum's motion can be perfectly periodic if there is no air resistance or friction present. However, in real-world situations, there will always be some external factors that can affect its motion, making it less than perfectly periodic.