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Wize AP Physics C: Mechanics Textbook > Unit 6: Oscillations (6-14%)

Simple Pendulums

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Simple Pendulums


Simple pendulums are pendulums for which the mass attached to the string is considered as a point. The motion of these pendulums could be described as a simple harmonic motion. We usually focus on the condition in which the angle pendulum makes respect to the vertical line is very small.




Let's look at a schematic diagram of a pendulum to understand it better:







Wize Tip
Note that the mass moves on part of a circular path when pendulum is oscillating. The only force along this motion is the component of gravitational force.

The driving force acting on the mass is mgsin⁡θmg\sin\thetaand since it is restoring (Always pushes the mass toward equilibrium which is the lowest point), we have:
Fθ=−mgsin⁡θF_{\theta}=-mg\sin\theta


For θ\thetasmall: sin⁡θ≈θ\sin\theta\approx\theta. So: Fθ=−mgθF_{\theta}=-mg\theta




The distance travelled by the mass is the arc length corresponding to angle θ\theta:


x=lθx=l\theta

Fθ=−mglx\boxed{F_\theta=-\dfrac{mg}{l}x}
where llin the length of pendulum.


By comparing with the general form of SHM forces (F=−kxF=-kx) :


k =mgl\boxed{k\ =\frac{mg}{l}}

This is called the “effective spring constant” of a simple pendulum.

So, it is a Simple Harmonic Motion and its angular frequency is:


ω=gl\boxed{\omega=\sqrt{\dfrac{g}{l}}}

Watch Out!
Note that the equations found above for sprig constant and angular frequency are only valid when angleθ\thetais very small!


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Example: Simple Pendulum Length


A simple pendulum is set to oscillate with an amplitude of 21 cm. If it oscillates with a frequency of 0.50 Hz, determine the length of the pendulum and its average speed.

Solution:

We can find the length of the pendulum by remembering the equation for the angular frequency of a simple pendulum:

ω=2πf=gl\omega=2\pi f=\sqrt{\frac{g}{l}}

f= 12πglf=\ \frac{1}{2\pi}\sqrt{\frac{g}{l}}

(2πf)2=gl\left(2\pi f\right)^2=\frac{g}{l}

l=g(2πf)2l=\frac{g}{\left(2\pi f\right)^2}

l = 0.99m      l\ =\ 0.99m\ \ \ \ \ \

To find the average speed, we need to find the distance travelled in one cycle of oscillation and divide it by the time of one cycle of oscillation which is basically the period:

T =1f=10.5=2  sT\ =\frac{1}{f}=\frac{1}{0.5}=2\ \ s

vave=distancetime=4(0.21)T=4(0.21)2=0.42  msv_{ave}=\frac{\text{distance}}{\text{time}}=\frac{4\left(0.21\right)}{T}=\frac{4\left(0.21\right)}{2}=0.42\ \ \frac{m}{s}


You pull a simple pendulum to the side and release the mass at an angle of 2.23°2.23\degree.

If the length of the pendulum is 25 cm, how much time does it take the mass to reach its highest speed? (in s)