7.5 Simple and Physical Pendulums

• A simple pendulum consists of a mass (called the bob) suspended from a massless, rigid string
• The period of a simple pendulum depends on the length of the string/rod and the acceleration due to gravity
• The period of a simple pendulum is given by $T =$$2\pi\sqrt{\frac{l}{g}}$, where $l$ is the length and $g$ is the acceleration due to gravity
• Steps to derive the period equation for a simple pendulum:

  1️⃣ Identify the restoring force (gravity): $F =$$mg\sin\theta$

  2️⃣ Apply Newton's 2nd law: $ma =$$mg\sin\theta$

  3️⃣ Simplify using small angle approximation: $a =$$- \frac{g}{l}\theta$

  4️⃣ Recognize the equation of a simple harmonic oscillator: $\omega =$$\sqrt{\frac{g}{l}}$

  5️⃣ Calculate the period: $T =$$2\pi\sqrt{\frac{l}{g}}$
• The mass of the bob has no effect on the period of a simple pendulum.
• Match the features with the correct type of pendulum:

  Mass Distribution ↔️ Concentrated at bob (Simple) ||| Distributed throughout body (Physical)

  Moment of Inertia ↔️ $I =$$ml^{2}$ (Simple) ||| $I$ varies (Physical)

  Period Equation ↔️ $T =$$2\pi\sqrt{\frac{l}{g}}$ (Simple) ||| $T =$$2\pi\sqrt{\frac{I}{mgl}}$ (Physical)
• The period of a simple pendulum is $2\pi\sqrt{\frac{l}{g}}$


  True
• Longer pendulums have longer periods.

  True
• What property must be considered when deriving the period equation for a physical pendulum?

  Moment of inertia
• What is the key feature of the bob in a simple pendulum?

  Suspended mass
• The period of a simple pendulum depends on the mass of the bob.

  False
• An increase in the acceleration due to gravity decreases the period of a simple pendulum.


  True
• The angular frequency of a physical pendulum is $\omega =$$\sqrt{\frac{mgl}{I}}$
• Steps to derive the period equation for a physical pendulum

  1️⃣ Consider the torque acting on the pendulum: $\tau =$$- mgl\sin\theta$

  2️⃣ Use Newton's second law for rotation: $\tau =$$I\alpha$

  3️⃣ For small angles, approximate $\sin\theta \approx \theta$

  4️⃣ Substitute angular acceleration $\alpha =$$\frac{d^{2}\theta}{dt^{2}}$

  5️⃣ Solve the resulting differential equation to find angular frequency $\omega$

  6️⃣ Calculate the period $T =$$\frac{2\pi}{\omega}$
• What is the angular frequency $\omega$ of a physical pendulum in terms of $m, g, l, I$?


  $\omega =$$\sqrt{\frac{mgl}{I}}$
• What does the physical pendulum equation account for that the simple pendulum equation does not?

  Distribution of mass
• The simple pendulum assumes all mass is concentrated at a single point
• Steps to solve problems involving simple and physical pendulums

  1️⃣ Identify the type of pendulum

  2️⃣ Apply the appropriate period equation

  3️⃣ Plug in the given values

  4️⃣ Calculate the period
• What is the bob of a simple pendulum?

  The suspended mass
• The period of a simple pendulum depends on its mass

  False
• What is the natural frequency of a simple pendulum?

  $\omega =$$\sqrt{\frac{g}{l}}$
• What is the pivot point of a simple pendulum?

  The fixed point
• How does increasing gravity affect the period of a simple pendulum?

  Decreases the period
• What is a physical pendulum?

  A pendulum with non-negligible mass
• What is the natural frequency of a simple pendulum?

  $\sqrt{\frac{g}{l}}$
• Which factor has no effect on the period of a simple pendulum?

  Mass of the bob
• The moment of inertia for a simple pendulum is $I =$$ml^{2}$


  True
• What is the mass distribution assumption for a simple pendulum?

  All mass at the bob
• A simple pendulum consists of a mass called the bob
• The natural frequency of a simple pendulum is given by $\omega =$$\sqrt{\frac{g}{l}}$
• The moment of inertia for a physical pendulum is always $I =$$ml^{2}$.


  False
• The period $T$ of a physical pendulum is related to its angular frequency $\omega$ by $T =$$\frac{2\pi}{\omega} =$$2\pi\sqrt{\frac{I}{mgl}}$, where $I$ is the moment of inertia
• For small angles, $\sin\theta \approx \theta$ is a valid approximation in pendulum motion


  True
• The moment of inertia $I$ of a simple pendulum is $ml^{2}$


  True
• In a physical pendulum, the mass is distributed throughout the body


  True
• Why is the physical pendulum equation more general than the simple pendulum equation?

  Accounts for mass distribution
• The mass of the bob affects the period of a simple pendulum.

  False
• The moment of inertia $I$ of a physical pendulum depends on its shape and mass distribution.


  True
• The period of a physical pendulum is given by $T =$$2\pi\sqrt{\frac{I}{mgl}}$, where $I$ is the moment of inertia of the pendulum body.
• Match the components of a simple pendulum with their descriptions:

  Mass (bob) ↔️ The suspended mass

  String/Rod ↔️ A massless, rigid support

  Pivot Point ↔️ The fixed point of swing