Simple and Physical Pendulums - AP Physics C: Mechanics
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Calculate the period of a simple pendulum with $L = 1 \text{ m}$ on Earth.
Calculate the period of a simple pendulum with $L = 1 \text{ m}$ on Earth.
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$T \approx 2\pi \sqrt{\frac{1}{9.8}} \approx 2.01 \text{ s}$. Substituting $L = 1$ m and $g = 9.8$ m/s² into period formula.
$T \approx 2\pi \sqrt{\frac{1}{9.8}} \approx 2.01 \text{ s}$. Substituting $L = 1$ m and $g = 9.8$ m/s² into period formula.
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State the expression for the angular frequency of a simple pendulum.
State the expression for the angular frequency of a simple pendulum.
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$\omega = \sqrt{\frac{g}{L}}$. Angular frequency is $\omega = \frac{2\pi}{T}$.
$\omega = \sqrt{\frac{g}{L}}$. Angular frequency is $\omega = \frac{2\pi}{T}$.
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Calculate the period of a pendulum with $L = 2 \text{ m}$ on the Moon ($g=1.6 \text{ m/s}^2$).
Calculate the period of a pendulum with $L = 2 \text{ m}$ on the Moon ($g=1.6 \text{ m/s}^2$).
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$T \approx 2\pi \sqrt{\frac{2}{1.6}} \approx 7.02 \text{ s}$. Lower gravity increases period significantly.
$T \approx 2\pi \sqrt{\frac{2}{1.6}} \approx 7.02 \text{ s}$. Lower gravity increases period significantly.
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What does the period of a simple pendulum NOT depend on?
What does the period of a simple pendulum NOT depend on?
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Mass of the bob. Period formula shows no mass dependence.
Mass of the bob. Period formula shows no mass dependence.
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State the restoring torque in a physical pendulum.
State the restoring torque in a physical pendulum.
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$\tau = -mgd \sin(\theta)$. Negative sign indicates restoring nature of torque.
$\tau = -mgd \sin(\theta)$. Negative sign indicates restoring nature of torque.
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Identify the type of energy at the highest point of a pendulum swing.
Identify the type of energy at the highest point of a pendulum swing.
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Maximum potential energy. All kinetic energy converts to potential at extremes.
Maximum potential energy. All kinetic energy converts to potential at extremes.
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State the expression for the angular frequency of a simple pendulum.
State the expression for the angular frequency of a simple pendulum.
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$\omega = \sqrt{\frac{g}{L}}$. Angular frequency is $\omega = \frac{2\pi}{T}$.
$\omega = \sqrt{\frac{g}{L}}$. Angular frequency is $\omega = \frac{2\pi}{T}$.
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State the formula for the period of a physical pendulum.
State the formula for the period of a physical pendulum.
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$T = 2\pi \sqrt{\frac{I}{mgd}}$. Accounts for distributed mass via moment of inertia.
$T = 2\pi \sqrt{\frac{I}{mgd}}$. Accounts for distributed mass via moment of inertia.
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Determine the change in gravitational potential energy at the bottom of the swing.
Determine the change in gravitational potential energy at the bottom of the swing.
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Zero; all energy is kinetic. Reference level chosen at lowest swing point.
Zero; all energy is kinetic. Reference level chosen at lowest swing point.
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Calculate the angular frequency for a $2 \text{ m}$ pendulum on Earth.
Calculate the angular frequency for a $2 \text{ m}$ pendulum on Earth.
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$\omega = \sqrt{\frac{9.8}{2}} \approx 2.21 \text{ rad/s}$. Using $\omega = \sqrt{g/L}$ with given values.
$\omega = \sqrt{\frac{9.8}{2}} \approx 2.21 \text{ rad/s}$. Using $\omega = \sqrt{g/L}$ with given values.
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What is the effect of increasing the mass on the period of a simple pendulum?
What is the effect of increasing the mass on the period of a simple pendulum?
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No effect. Mass cancels out in period equation derivation.
No effect. Mass cancels out in period equation derivation.
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Identify the linear displacement in a pendulum swing.
Identify the linear displacement in a pendulum swing.
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Arc length from equilibrium. Arc length equals $L\theta$ for small angles.
Arc length from equilibrium. Arc length equals $L\theta$ for small angles.
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Calculate the torque for a $5 \text{ kg}$ pendulum with $d = 0.5 \text{ m}$ and $\theta = 30^{\circ}$.
Calculate the torque for a $5 \text{ kg}$ pendulum with $d = 0.5 \text{ m}$ and $\theta = 30^{\circ}$.
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$\tau = -5 \times 9.8 \times 0.5 \times \sin(30^{\circ}) \approx -12.25 \text{ Nm}$. Using torque formula with $\sin(30°) = 0.5$.
$\tau = -5 \times 9.8 \times 0.5 \times \sin(30^{\circ}) \approx -12.25 \text{ Nm}$. Using torque formula with $\sin(30°) = 0.5$.
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What is the relationship between period and frequency?
What is the relationship between period and frequency?
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$f = \frac{1}{T}$. Inverse relationship between period and frequency.
$f = \frac{1}{T}$. Inverse relationship between period and frequency.
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What is the phase of a pendulum at its maximum displacement?
What is the phase of a pendulum at its maximum displacement?
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$\phi = \frac{\pi}{2}$ or $-\frac{\pi}{2}$. Phase at amplitude depends on initial conditions.
$\phi = \frac{\pi}{2}$ or $-\frac{\pi}{2}$. Phase at amplitude depends on initial conditions.
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Determine the moment of inertia for a rod pendulum pivoted at the end.
Determine the moment of inertia for a rod pendulum pivoted at the end.
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$I = \frac{1}{3}mL^2$. Standard formula for rod rotating about end.
$I = \frac{1}{3}mL^2$. Standard formula for rod rotating about end.
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What is the simple pendulum approximation for small angles?
What is the simple pendulum approximation for small angles?
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$\sin(\theta) \approx \theta$. Valid for angles less than about 15 degrees.
$\sin(\theta) \approx \theta$. Valid for angles less than about 15 degrees.
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Calculate the period change if $g$ is doubled for a simple pendulum.
Calculate the period change if $g$ is doubled for a simple pendulum.
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$T' = \frac{T}{\sqrt{2}}$; period decreases. Period inversely proportional to square root of $g$.
$T' = \frac{T}{\sqrt{2}}$; period decreases. Period inversely proportional to square root of $g$.
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State the condition for the pendulum to oscillate as a simple harmonic oscillator.
State the condition for the pendulum to oscillate as a simple harmonic oscillator.
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Small angle approximation. Required for linear restoring force relationship.
Small angle approximation. Required for linear restoring force relationship.
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What does the period of a simple pendulum NOT depend on?
What does the period of a simple pendulum NOT depend on?
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Mass of the bob. Period formula shows no mass dependence.
Mass of the bob. Period formula shows no mass dependence.
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What is the kinetic energy at the lowest point of a simple pendulum?
What is the kinetic energy at the lowest point of a simple pendulum?
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Maximum kinetic energy. All potential energy converts to kinetic at bottom.
Maximum kinetic energy. All potential energy converts to kinetic at bottom.
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What is the definition of a physical pendulum?
What is the definition of a physical pendulum?
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A pendulum with an extended mass. Unlike simple pendulum's point mass assumption.
A pendulum with an extended mass. Unlike simple pendulum's point mass assumption.
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Identify the moment of inertia in a physical pendulum.
Identify the moment of inertia in a physical pendulum.
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$I$, the rotational inertia about the pivot. Determines rotational resistance to angular acceleration.
$I$, the rotational inertia about the pivot. Determines rotational resistance to angular acceleration.
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For a simple pendulum, what is $L$?
For a simple pendulum, what is $L$?
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Length of the string or rod. Distance from pivot to center of mass.
Length of the string or rod. Distance from pivot to center of mass.
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Calculate the period of a pendulum with $L = 2 \text{ m}$ on the Moon ($g=1.6 \text{ m/s}^2$).
Calculate the period of a pendulum with $L = 2 \text{ m}$ on the Moon ($g=1.6 \text{ m/s}^2$).
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$T \approx 2\pi \sqrt{\frac{2}{1.6}} \approx 7.02 \text{ s}$. Lower gravity increases period significantly.
$T \approx 2\pi \sqrt{\frac{2}{1.6}} \approx 7.02 \text{ s}$. Lower gravity increases period significantly.
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What is the amplitude in a pendulum?
What is the amplitude in a pendulum?
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Maximum angular displacement. Maximum angular distance from equilibrium position.
Maximum angular displacement. Maximum angular distance from equilibrium position.
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Which type of pendulum can have a non-uniform mass distribution?
Which type of pendulum can have a non-uniform mass distribution?
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Physical pendulum. Simple pendulum assumes point mass only.
Physical pendulum. Simple pendulum assumes point mass only.
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State the formula for the torque in a physical pendulum.
State the formula for the torque in a physical pendulum.
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$\tau = -mgd \sin(\theta)$. Restoring torque proportional to angular displacement.
$\tau = -mgd \sin(\theta)$. Restoring torque proportional to angular displacement.
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What is the equilibrium position of a pendulum?
What is the equilibrium position of a pendulum?
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Vertical position where $\theta = 0$. Lowest potential energy configuration.
Vertical position where $\theta = 0$. Lowest potential energy configuration.
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Determine the frequency of a simple pendulum with $T = 2 \text{ s}$.
Determine the frequency of a simple pendulum with $T = 2 \text{ s}$.
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$f = \frac{1}{T} = 0.5 \text{ Hz}$. Frequency is reciprocal of period.
$f = \frac{1}{T} = 0.5 \text{ Hz}$. Frequency is reciprocal of period.
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