5.3 Rotational Dynamics

• What is the study of the relationship between torque and rotational motion called?

  Rotational dynamics
• Rotational inertia is analogous to mass in linear dynamics.


  True
• Newton's second law for rotation states that torque equals rotational inertia multiplied by angular acceleration
• The formula for Newton's second law for rotation is $\tau =$$I\alpha$.


  True
• What is the formula for torque in terms of force, distance, and angle?

  $\tau =$$rF\sin(\theta)$
• Steps for applying Newton's second law to rotational problems:

  1️⃣ Identify the pivot point

  2️⃣ Calculate the torque

  3️⃣ Determine the rotational inertia

  4️⃣ Apply Newton's second law: $\tau =$$I\alpha$

  5️⃣ Solve for angular acceleration
• The moment of inertia for a point mass is $I =$$mr^{2}$.


  True
• What is the moment of inertia for a solid cylinder rotating about its axis?

  $I =$$\frac{1}{2}mr^{2}$
• What is the moment of inertia formula for a point mass?

  $I =$$mr^{2}$
• The moment of inertia formula for a hollow cylinder is $I =$$mr^{2}$.


  True
• What is the moment of inertia formula for a solid sphere?

  $I =$$\frac{2}{5}mr^{2}$
• Moment of inertia determines how much torque is required to produce a given angular acceleration.
• Moment of inertia is analogous to mass in linear motion.


  True
• Match the object shape with its moment of inertia formula:

  Point mass ↔️ $I =$$mr^{2}$

  Solid cylinder ↔️ $I =$$\frac{1}{2}mr^{2}$

  Solid sphere ↔️ $I =$$\frac{2}{5}mr^{2}$
• Arrange the key concepts of rotational dynamics in a logical order:

  1️⃣ Torque (\( \tau \))

  2️⃣ Rotational Inertia (\( I \))

  3️⃣ Angular Acceleration (\( \alpha \))
• Rotational inertia is analogous to mass in linear dynamics.


  True
• The rotational inertia of an object depends on its shape.

  True
• Angular acceleration is calculated as the change in angular velocity divided by the change in time.
• What two factors does the moment of inertia depend on?

  Mass distribution and shape
• In the moment of inertia formula, \( m \) represents the mass and \( r \) represents the radius of the object.


  True
• Newton's second law for rotation relates torque, moment of inertia, and angular acceleration.
• A wheel with a moment of inertia of \(5 \, \text{kg} \cdot \text{m}^2\) is subjected to a torque of \(10 \, \text{N} \cdot \text{m}\). What is its angular acceleration?

  $2 \, \text{rad / s}^{2}$
• What is the rotational analog of force in Newton's second law for rotation?

  Torque
• What does angular momentum measure in rotational motion?

  Resistance to changes
• The formula for angular momentum is \(L = I \omega\)
• What does the principle of conservation of angular momentum state?

  Total angular momentum remains constant
• The formula for conservation of angular momentum is \(L_{i} = L_{f}\)
• What type of energy does an object possess due to its rotational motion?

  Rotational kinetic energy
• Rotational kinetic energy depends on moment of inertia and angular velocity.

  True
• Rotational dynamics explores how forces applied at a distance from a pivot point cause objects to rotate
• What is rotational dynamics the study of?

  Torque and rotational motion
• Rotational dynamics explores how forces applied at a distance from a pivot point cause objects to rotate
• Understanding rotational dynamics is crucial for analyzing the motion of spinning objects.


  True
• What are the key concepts in rotational dynamics?

  Torque, inertia, acceleration
• Match the concept with its description:

  Torque ↔️ The force that causes rotation

  Rotational Inertia ↔️ Resistance to changes in rotational motion

  Angular Acceleration ↔️ Rate of change of angular velocity
• What is the formula for torque?

  $\tau =$$rF\sin(\theta)$
• What is the relationship between torque, rotational inertia, and angular acceleration?

  $\tau =$$I\alpha$
• What does the moment of inertia depend on?

  Mass distribution and shape
• A wheel with a moment of inertia of \(5 \, \text{kg} \cdot \text{m}^2\) is subjected to a torque of \(10 \, \text{N} \cdot \text{m}\). What is its angular acceleration?

  $\alpha =$$2 \, \text{rad / s}^{2}$
• Newton's second law for rotation relates torque, moment of inertia, and angular acceleration