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Wize University Physics Textbook (Master) > Periodic Motion: Oscillations

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)
Extra Practice
The period T for a simple pendulum is modelled using the equation T=2πL9T=2\pi\sqrt{\frac{L}{9}}
where LL is the length of the pendulum and is the acceleration due to gravity, a constant. Estimate the percent change in the period if the pendulum is brought up 9000 m, where gg decreases by 0.4%.
Kinematics and Simple Harmonic Motion
A 12 g bullet moving horizontally at 30 m/s is fired into a hanging mass of 2.5 kg. The bullet becomes embedded in the hanging mass, which then acts like a pendulum. The cord connecting the mass to the ceiling is 75.0 centimeters.
Practice: Simple pendulum
You pull a simple pendulum to the side and release the mass at an angle of 2.23°2.23\degree.
a) If the length of the pendulum is 25cm, how much time does it take the mass to reach its highest speed? (in s)
b) What is the maximum speed? (in meters per second)
You have landed on an unknown planet. You have a simple pendulum with a length of 25.0 cm25.0\ cm. You measure the frequency of oscillation of the pendulum to be 0.5 Hz0.5\ Hz. What is the gravitational acceleration on this planet (in m/s^2)? What would be the period of the oscillation if you used this pendulum on Earth (in s)?
(Don't include units in your answers)
Simple Harmonic Motion: Pendulum
In two separate experiments a pendulum and a mass-spring are moved from the Earth to the moon surface where gravitational acceleration is 1/6 of the Earth.
a) How does the amplitude of oscillations change?
b) How does the frequency of oscillations change?