Newton's Second Law in Rotational Form - AP Physics C: Mechanics
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What happens to $\alpha$ if net external torque is zero?
What happens to $\alpha$ if net external torque is zero?
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$\alpha = 0$ (no angular acceleration). Newton's first law: no net torque means constant motion.
$\alpha = 0$ (no angular acceleration). Newton's first law: no net torque means constant motion.
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State the relationship between torque and angular momentum.
State the relationship between torque and angular momentum.
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Torque is the rate of change of angular momentum. Fundamental relationship: $\tau = \frac{dL}{dt}$.
Torque is the rate of change of angular momentum. Fundamental relationship: $\tau = \frac{dL}{dt}$.
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State the formula for torque ($\tau$) in rotational dynamics.
State the formula for torque ($\tau$) in rotational dynamics.
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$\tau = rF \sin \theta$. Torque equals lever arm times force times sine of angle.
$\tau = rF \sin \theta$. Torque equals lever arm times force times sine of angle.
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What is the moment of inertia for a solid sphere?
What is the moment of inertia for a solid sphere?
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$I = \frac{2}{5}MR^2$. Moment of inertia for uniform solid sphere.
$I = \frac{2}{5}MR^2$. Moment of inertia for uniform solid sphere.
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What does $I$ represent in the formula $\tau = I\alpha$?
What does $I$ represent in the formula $\tau = I\alpha$?
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Moment of inertia. Rotational inertia, resistance to angular acceleration.
Moment of inertia. Rotational inertia, resistance to angular acceleration.
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What effect does increasing $r$ have on torque for a given force?
What effect does increasing $r$ have on torque for a given force?
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Increases torque. Longer lever arm increases rotational effect.
Increases torque. Longer lever arm increases rotational effect.
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What is the formula for the moment of inertia of a hoop?
What is the formula for the moment of inertia of a hoop?
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$I = MR^2$. All mass concentrated at radius for thin ring.
$I = MR^2$. All mass concentrated at radius for thin ring.
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Identify the reference point for calculating torque in lever systems.
Identify the reference point for calculating torque in lever systems.
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The pivot or axis of rotation. Fixed point about which rotation occurs.
The pivot or axis of rotation. Fixed point about which rotation occurs.
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Which variable represents angular acceleration?
Which variable represents angular acceleration?
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$\alpha$. Greek letter alpha represents angular acceleration.
$\alpha$. Greek letter alpha represents angular acceleration.
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What is the unit of torque in the SI system?
What is the unit of torque in the SI system?
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Newton-meter (N·m). Torque has units of force times distance.
Newton-meter (N·m). Torque has units of force times distance.
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State the principle of superposition for torques.
State the principle of superposition for torques.
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Net torque is the vector sum of individual torques. Torques add vectorially like forces.
Net torque is the vector sum of individual torques. Torques add vectorially like forces.
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What is the moment of inertia of a solid cylinder rotating about its axis?
What is the moment of inertia of a solid cylinder rotating about its axis?
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$I = \frac{1}{2}MR^2$. Formula for solid cylinder about central axis.
$I = \frac{1}{2}MR^2$. Formula for solid cylinder about central axis.
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Find the torque with $r = 0.5 \text{ m}$, $F = 10 \text{ N}$ at $\theta = 90^\circ$.
Find the torque with $r = 0.5 \text{ m}$, $F = 10 \text{ N}$ at $\theta = 90^\circ$.
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$\tau = 5 \text{ N·m}$. Use $\tau = rF\sin\theta$ with $\sin 90^\circ = 1$.
$\tau = 5 \text{ N·m}$. Use $\tau = rF\sin\theta$ with $\sin 90^\circ = 1$.
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What is the unit of angular acceleration in the SI system?
What is the unit of angular acceleration in the SI system?
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radians per second squared (rad/s²). Units of angular velocity per time.
radians per second squared (rad/s²). Units of angular velocity per time.
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What is the effect of increasing $I$ on angular acceleration for a given torque?
What is the effect of increasing $I$ on angular acceleration for a given torque?
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Decreases angular acceleration. Higher inertia reduces acceleration for same torque.
Decreases angular acceleration. Higher inertia reduces acceleration for same torque.
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What does $\alpha$ represent in the formula $\tau = I\alpha$?
What does $\alpha$ represent in the formula $\tau = I\alpha$?
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Angular acceleration. Rate of change of angular velocity.
Angular acceleration. Rate of change of angular velocity.
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Determine the direction of torque with $F$ applied clockwise.
Determine the direction of torque with $F$ applied clockwise.
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Clockwise torque is negative. Convention: clockwise rotation is negative direction.
Clockwise torque is negative. Convention: clockwise rotation is negative direction.
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What is the formula for moment of inertia for a point mass?
What is the formula for moment of inertia for a point mass?
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$I = mr^2$. Mass times radius squared for point masses.
$I = mr^2$. Mass times radius squared for point masses.
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Explain the significance of the perpendicular distance in torque calculation.
Explain the significance of the perpendicular distance in torque calculation.
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Determines effective lever arm length. Only perpendicular force component creates torque.
Determines effective lever arm length. Only perpendicular force component creates torque.
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State Newton's Second Law in rotational form.
State Newton's Second Law in rotational form.
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$\tau = I\alpha$. Rotational analog of $F = ma$.
$\tau = I\alpha$. Rotational analog of $F = ma$.
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Calculate $I$ with $r = 1 \text{ m}$, $m = 4 \text{ kg}$ for a point mass.
Calculate $I$ with $r = 1 \text{ m}$, $m = 4 \text{ kg}$ for a point mass.
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$I = 4 \text{ kg·m}^2$. Use $I = mr^2$ formula: $4 \times 1^2$.
$I = 4 \text{ kg·m}^2$. Use $I = mr^2$ formula: $4 \times 1^2$.
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Find $\alpha$ given $\tau = 20 \text{ N·m}$ and $I = 4 \text{ kg·m}^2$.
Find $\alpha$ given $\tau = 20 \text{ N·m}$ and $I = 4 \text{ kg·m}^2$.
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$\alpha = 5 \text{ rad/s}^2$. Apply $\tau = I\alpha$: $\alpha = 20/4$.
$\alpha = 5 \text{ rad/s}^2$. Apply $\tau = I\alpha$: $\alpha = 20/4$.
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What is the rotational analog of force in linear dynamics?
What is the rotational analog of force in linear dynamics?
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Torque ($\tau$). Rotational equivalent of linear force.
Torque ($\tau$). Rotational equivalent of linear force.
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Identify the relationship between net torque and angular acceleration.
Identify the relationship between net torque and angular acceleration.
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$\tau_{\text{net}} = I\alpha$. Direct statement of Newton's second law rotation.
$\tau_{\text{net}} = I\alpha$. Direct statement of Newton's second law rotation.
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What is the role of the angle $\theta$ in the torque formula $\tau = rF \sin \theta$?
What is the role of the angle $\theta$ in the torque formula $\tau = rF \sin \theta$?
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$\theta$ is the angle between force and lever arm. Sine function gives effective force component.
$\theta$ is the angle between force and lever arm. Sine function gives effective force component.
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Calculate the net torque: $\tau_1 = 5 \text{ N·m}$, $\tau_2 = -3 \text{ N·m}$.
Calculate the net torque: $\tau_1 = 5 \text{ N·m}$, $\tau_2 = -3 \text{ N·m}$.
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$\tau_{\text{net}} = 2 \text{ N·m}$. Add torques algebraically: $5 + (-3) = 2$.
$\tau_{\text{net}} = 2 \text{ N·m}$. Add torques algebraically: $5 + (-3) = 2$.
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What is the rotational equivalent of linear momentum?
What is the rotational equivalent of linear momentum?
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Angular momentum. Rotational equivalent of $p = mv$.
Angular momentum. Rotational equivalent of $p = mv$.
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What is the angular acceleration if $I = 3 \text{ kg·m}^2$ and $\tau = 9 \text{ N·m}$?
What is the angular acceleration if $I = 3 \text{ kg·m}^2$ and $\tau = 9 \text{ N·m}$?
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$\alpha = 3 \text{ rad/s}^2$. Apply $\alpha = \tau/I = 9/3$.
$\alpha = 3 \text{ rad/s}^2$. Apply $\alpha = \tau/I = 9/3$.
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Which variable represents torque in rotational dynamics equations?
Which variable represents torque in rotational dynamics equations?
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$\tau$. Greek letter tau denotes torque.
$\tau$. Greek letter tau denotes torque.
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Determine the torque direction if $F$ is applied counterclockwise.
Determine the torque direction if $F$ is applied counterclockwise.
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Counterclockwise torque is positive. Convention: counterclockwise rotation is positive direction.
Counterclockwise torque is positive. Convention: counterclockwise rotation is positive direction.
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