When a pendulum bob is displaced from its equilibrium position, hanging vertically,

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This can also be described using frequency per oscillation, which measures how many oscillations it does per second. Frequency is inverse of the period, f = 1/T. Also the period is inverse of the frequency, T = 1/f. We know that the horizontal component of the force is Tsinθ = -ma and the vertical component of the force is Tcosθ = mg. Therefore we can say that:

Tsinθ/Tcosθ= (-ma)/mg tanθ= (-a)/g

In our investigation we are going to keep the angle θ of the pendulum very small therefore we can say that the length L is going to be approximately the same for T. Also tanθ will approximately equal to sinθ. sinθ=(-a)/g x/l= (-a)/g ∴a=(-x)/l g

Using the laws of circular motion we can calculate the time period using the values for a we found above. w=2πf a=-(〖w)〗^2 x

Where w is the angular speed and f is the frequency. Using the equation above for a we can set it equal to the one we found previously to find T using f from angular speed.

(-x)/g=-(〖2πf)〗^2 x g/l=(〖2πf)〗^2 √(g/l)=2πf f=√(g/l)/2πf T=2π√(l/g)

We can also solve for g to find the acceleration due to gravity from our result by rearranging the equation.

g=(4π^2