5.2 Connecting Linear and Rotational Motion

• Match the linear motion concept with its definition:

  Displacement ↔️ The change in position of an object

  Velocity ↔️ The rate of change of position

  Acceleration ↔️ The rate of change of velocity

  Momentum ↔️ The product of mass and velocity
• Newton's second law for rotation is $I\alpha =$$\sum \tau$.


  True
• Arrange the following linear motion concepts from simplest to most complex:

  1️⃣ Displacement

  2️⃣ Velocity

  3️⃣ Acceleration

  4️⃣ Momentum
• What is the formula for linear distance in terms of angular displacement and radius?

  s = r\theta</latex>
• What is the rotational counterpart of displacement in linear motion?

  Angle
• What does angular momentum measure?

  Tendency to rotate
• Linear velocity is equal to the product of radius and angular velocity.

  True
• What equation relates linear velocity and angular velocity?

  $v =$$r\omega$
• Displacement measures the change in an object's position.


  True
• Match the linear motion concept with its definition:

  Displacement ↔️ Change in position

  Velocity ↔️ Rate of change of position

  Acceleration ↔️ Rate of change of velocity

  Momentum ↔️ Product of mass and velocity
• The rate of change of angular velocity over time is called angular acceleration
• What is the variable used to denote the radius of rotation in linear-angular relationships?

  r
• How far does a rider travel on a carousel with a radius of 5 meters after one full revolution?

  31.4 meters
• Steps to convert between linear and angular velocity using the formula $v =$$r\omega$


  1️⃣ Identify the given values for $r$ and $\omega$

  2️⃣ Multiply the radius by the angular velocity

  3️⃣ The result is the linear velocity $v$
• The formula relating linear acceleration and angular acceleration is $a =$$r\alpha$, where $r$ is the radius and $\alpha$ is the angular acceleration
• What does the moment of inertia measure?

  Resistance to rotational acceleration
• Match the rotational concept with its example:

  Angle ↔️ A wheel making one complete revolution

  Angular Velocity ↔️ A spinning top completing revolutions per second

  Angular Acceleration ↔️ A fan increasing its speed

  Angular Momentum ↔️ A spinning disk with constant velocity
• What is the formula for linear velocity (v) in terms of radius (r) and angular velocity (ω)?

  $v =$$r\omega$
• Match the linear quantity with its corresponding angular quantity:

  Linear Distance (s) ↔️ Angle (θ)

  Linear Velocity (v) ↔️ Angular Velocity (ω)

  Linear Acceleration (a) ↔️ Angular Acceleration (α)
• If a carousel with a radius of 5 meters takes 2 seconds to reach full speed, the linear acceleration is approximately 1.57 m/s².

  True
• The formula for angular velocity is ω = v / r.

  True
• Linear acceleration is defined as the rate of change of linear velocity.
• The moment of inertia of a uniform rod rotating about its center is $\frac{1}{12}ML^{2}$.


  True
• What does Newton's second law for rotation state?

  $I\alpha =$$\sum \tau$
• Torque is defined as $\tau =$$Fr\sin\theta$


  True
• What is the angular momentum of a disk with a moment of inertia of 0.5 kg·m² rotating at an angular velocity of 10 rad/s?

  $5 \, \text{kg} \cdot \text{m}^{2} / \text{s}$
• What is the angular momentum of a wheel with a moment of inertia of 0.5 kg·m² and an angular velocity of 10 rad/s?

  $5 \, \text{kg} \cdot \text{m}^{2} / \text{s}$
• Match the rotational concept with its analogous linear concept:

  Angle (θ) ↔️ Displacement (x)

  Angular Velocity (ω) ↔️ Velocity (v)

  Angular Acceleration (α) ↔️ Acceleration (a)

  Angular Momentum (L) ↔️ Momentum (p)
• What is the formula for linear velocity in terms of displacement and time?

  $v =$$\frac{\Delta x}{\Delta t}$
• The rate of change of angle over time is called angular velocity.
• Linear and angular quantities are related through the radius of rotation.
• Linear acceleration is the product of radius and angular acceleration.
• The units of linear velocity are meters per second.
• What is the formula for velocity in terms of displacement and time?

  $v =$$\Delta x / \Delta t$
• The measure of rotation around an axis is called the angle
• Angular momentum is a measure of an object's tendency to rotate.

  True
• The formula relating linear distance and angle is $s =$$r\theta$, where $s$ is the linear distance, $r$ is the radius, and $\theta$ is the angle
• If a carousel with a radius of 5 meters completes one revolution in 10 seconds, its linear velocity is approximately 3.14 m/s.
• Match the formula with the quantity and units:

  $v =$$r\omega$ ↔️ Linear Velocity, m/s

  $\omega =$$v / r$ ↔️ Angular Velocity, rad/s
• A carousel with a radius of 5 meters and an angular acceleration of 0.5 rad/s² has a linear acceleration of 2.5 m/s² at its edge.

  True