Physical Pendulum - notes PDF

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SHM: Physical Pendulum

Learning Outcomes By the end of this lesson, you should be able to: • Show how a physical pendulum will result in SHM. • Apply the SHM equation for a physical pendulum to determine the period, frequency, length or moment of inertia of a pendulum.

The physical pendulum • Simple pendulum: all mass concentrated in single point. long arm massless string

• Physical pendulum: different mass distributions, suspended from a pivot and made to oscillate about that pivot point.

Motion of a physical pendulum • Suppose an object of mass 𝑀 pivots about a point. • 𝜃 is the angle from the vertical. • ℓ is the distance from the pivot point to the center of mass. • The object’s oscillation will now be a rotation around the pivot point, so let’s set up the equation of motion for this rotation.

gravity acts at the center of the mass

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Motion of a physical pendulum The torque acting on the object due to its own weight is

𝜏 = −ℓ𝐹 sin 𝜃 = −ℓ𝑀𝑔 sin 𝜃 and acts in the direction to return the object to its equilibrium position – it is a restoring torque (hence the minus sign).

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Motion of a physical pendulum Newton’s 2nd law:

𝜏 = 𝐼𝛼 where 𝐼 is the moment of inertia and 𝛼 is the angular acceleration:

𝑑!𝜃 𝛼= ! 𝑑𝑡 𝑑!𝜃 ⟹𝜏=𝐼 ! 𝑑𝑡

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Motion of a physical pendulum 𝑑!𝜃 𝜏 = −ℓ𝑀𝑔 sin 𝜃 𝜏=𝐼 ! 𝑑𝑡 Equating both expressions for the torque yields: 𝑑!𝜃 −ℓ𝑀𝑔 sin 𝜃 = 𝐼 ! 𝑑𝑡 Once again, we make the small-angle approximation, so sin 𝜃 ≈ 𝜃 and we get:

𝑑!𝜃 𝑑!𝜃 𝑀𝑔ℓ 𝜃 −ℓ𝑀𝑔𝜃 = 𝐼 ! ⟹ ! = − 𝐼 𝑑𝑡 𝑑𝑡

displacement from equilirbium positions are going to be small.

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Motion of a physical pendulum The equation of motion for the physical pendulum is thus 𝑀𝑔ℓ 𝑑!𝜃 = − 𝜃 𝐼 𝑑𝑡 ! This is again the same form of equation we’ve seen before. The solution is SHM:

𝜃 𝑡 = 𝐴 cos(𝜔𝑡 + 𝜑)

with

𝜔! =

𝑀𝑔ℓ 𝐼

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Physical vs simple pendulum For the physical pendulum:

𝜃 𝑡 = 𝐴 cos(𝜔𝑡 + 𝜑) 𝜔! =

𝑀𝑔ℓ 𝐼

A simple pendulum can be seen as a physical pendulum with

𝐼 = 𝑀ℓ! In that case, the frequency reduces to

𝜔! =

𝑀𝑔ℓ 𝑀𝑔ℓ 𝑔 = = 𝐼 𝑀ℓ! ℓ

which of course is the result we got before.

Simple & physical pendula: summary • A simple pendulum undergoes SHM if the angle of oscillation is small. • So does a physical pendulum. • So does almost every other oscillating system: as long as the system has a linear restoring force, the resulting motion is simple harmonic. • The mathematical description of the motion is the same in each case, only the physical variables change.

𝑑!𝑥 = −𝜔! 𝑥 ! 𝑑𝑡

Example: Leg Oscillations • When you walk, your leg swings forward like a pendulum. Modeling your leg as a uniform rod with a total length of 90 cm, calculate the time it will take you to take one step....