6.1 Combined Translational and Rotational Motion

• Rotational motion involves the movement of an object around an axis
• Match the kinetic energy type with its formula:

  Translational ↔️ \frac{1}{2}mv^2</latex>

  Rotational ↔️ $\frac{1}{2}I\omega^{2}$
• What is translational motion characterized by?

  Movement without orientation change
• What does the variable $I$ represent in rotational kinetic energy?


  Moment of inertia
• The total kinetic energy of an object in combined motion is the sum of its translational and rotational kinetic energies.

  True
• Match the variable with its meaning in the combined motion equation:

  m ↔️ Mass of the object

  v ↔️ Linear velocity

  \omega ↔️ Angular velocity
• Combined motion occurs when an object moves both linearly and rotates around an axis
• In rolling without slipping, the relationship between linear velocity $v$ and angular velocity $\omega$ is $v =$$r\omega$.


  True
• Linear momentum is a vector quantity.


  True
• Order the following types of motion from simplest to most complex:

  1️⃣ Translational

  2️⃣ Rotational

  3️⃣ Combined
• Translational motion focuses on the displacement of the object's center of mass
• What is the formula for the total kinetic energy of an object undergoing combined motion?

  $K =$$\frac{1}{2}mv^{2} +$$\frac{1}{2}I\omega^{2}$
• Match the type of motion with its characteristic:

  Translational ↔️ Linear movement

  Rotational ↔️ Pivoting around an axis
• What is the relationship between linear velocity <m>v</m> and angular velocity <m>\omega</m> in rolling without slipping?

  $v =$$r\omega$
• In rolling without slipping, the relationship between linear and angular velocity is $v =$$r\omega$.


  True
• What is the formula for angular momentum?

  $L =$$I\omega$
• What is the formula for linear momentum?

  $p =$$mv$
• The formula for linear momentum is p = mv
• What does angular momentum measure?

  Rotational inertia
• What distinguishes rotational motion from translational motion?

  Pivoting around an axis
• Match the variable in angular momentum with its unit:

  L ↔️ kg m²/s

  I ↔️ kg m²

  \omega ↔️ rad/s
• Arrange the variables in the linear momentum formula in the correct order:

  1️⃣ $p$ ||| <step_start>$m$ ||| <step_start>$v$
• Linear and angular velocities are directly proportional in rolling motion without slipping.

  True
• What is linear momentum a measure of?

  Inertia in motion
• Linear acceleration is the rate of change of the object's center of mass velocity
• Momentum is a vector quantity.


  True
• What is the relationship between angular acceleration and linear acceleration in combined motion?

  Directly proportional
• The formula for angular momentum is L = Iω
• What is the relationship between linear velocity and angular velocity for an object in rolling motion without slipping?

  $v =$$r\omega$
• What equation relates linear acceleration and angular acceleration for an object in rolling motion without slipping?

  $a =$$r\alpha$
• What is the relationship between linear acceleration and angular acceleration for an object in rolling motion without slipping?

  $a =$$r\alpha$
• Linear and angular accelerations are inversely proportional in rolling motion without slipping.

  False
• The larger the angular acceleration, the larger the linear acceleration for the same radius.


  True
• The linear acceleration of a rolling wheel with radius <m>r</m> and angular acceleration <m>\alpha</m> is given by a = r\alpha
• What is the mathematical representation of the conservation of energy in combined motion?

  $K_{translational} +$$K_{rotational} +$$U =$$constant$
• Translational motion involves movement of an object without changing its orientation.


  True
• In combined motion, the conservation of energy states that the total energy, including translational kinetic, rotational kinetic, and potential energy, remains constant
• In translational motion, the focus is on the displacement of the object's center of mass
• Match the energy type with its formula:

  Translational Kinetic ↔️ $\frac{1}{2}mv^{2}$

  Rotational Kinetic ↔️ $\frac{1}{2}I\omega^{2}$

  Potential ↔️ $mgh$
• What is combined motion?

  Linear and rotational motion together