Rotational Motion: Concepts, Formulas, and Applications

Rotational Motion

Types of Circular Motion

Rotational motion involves objects moving in a circular path. There are two main types of circular motion:

• Rotation: An object spins around an internal axis (an axis located within the object itself).

• Revolution: An object moves around an external axis (an axis located outside the object).

Examples:

• Rotation: The Earth spinning on its axis.

• Revolution: The Earth orbiting around the Sun.

Describing Circular Speeds

Circular motion can be described using two types of speeds:

• Rotational (Angular) Speed: The rate at which something spins, measured in revolutions per minute (RPM) or radians per second (rad/s).

• Linear (Tangential) Speed: The distance traveled around the circle divided by the time taken, measured in meters per second (m/s).

Example: If a merry-go-round makes 1 complete rotation in 1.25 seconds, its rotational speed is:

• Number of rotations per second: $\frac{1}{1.25} = 0.8$ rotations/second

• To convert to RPM: $0.8 \times 60 = 48$ RPM

Period (T): The time it takes to complete one full revolution.

• For a wheel with diameter 39 m making one revolution in 8 seconds, the linear speed $v$ is:

$v = \frac{2\pi r}{T}$

where $r$ is the radius and $T$ is the period.

Centripetal Acceleration

Centripetal acceleration is the rate at which the direction of velocity changes for an object moving in a circle. It always points toward the center of the circle.

• Formula: $a_c = \frac{v^2}{r}$

• Units: m/s2

Example: For a rider on a wheel with linear speed $v$ and radius $r$, plug values into the formula above to find $a_c$.

Accelerations are often described in terms of "g's":

• 1 g = 9.8 m/s2

• The more "g's" you experience, the heavier you feel.

• At 3g's, you feel three times as heavy as normal.

• Many people pass out at around 5g's.

• Astronauts on the ISS feel weightless (0g).

Centripetal Force

Centripetal force is the net force required to keep an object moving in a circular path, always directed toward the center of the circle. It is not a new type of force; rather, it can be provided by tension, gravity, friction, or other forces.

• Formula: $F_c = m \frac{v^2}{r}$

• Depends on the object's mass ($m$), speed ($v$), and radius of the circle ($r$).

Example: For a 920 kg car traveling at 68 m/s on a curve of radius 75 m:

$F_c = 920 \times \frac{68^2}{75} = 57,088 \text{ N}$

To find how many g's the rider feels:

$\text{g's} = \frac{a_c}{9.8}$

Moving objects tend to go in a straight line (inertia). If centripetal force is removed, the object moves tangentially to the circle.

Centrifugal Force

Centrifugal force is a perceived force that seems to push objects away from the center of a circle. In reality, it is not a true force but the result of inertia—the tendency of an object to move in a straight line.

• What feels like a force is actually your body wanting to continue in a straight line while the path curves.

Sample Problems: The Rotor Ride

A carnival ride called the Rotor has a radius of 3.00 m and takes 2.20 seconds to complete one revolution. For a person of mass 75 kg:

1. Linear speed: $v = \frac{2\pi r}{T} = \frac{2\pi \times 3.00}{2.20} = 8.57 \text{ m/s}$

2. Centripetal acceleration: $a_c = \frac{v^2}{r} = \frac{(8.57)^2}{3.00} = 24.5 \text{ m/s}^2$

3. Centripetal force: $F_c = m \frac{v^2}{r} = 75 \times 24.5 = 1,837.5 \text{ N}$

Additional info: These calculations illustrate the relationships between speed, acceleration, and force in rotational motion, and how they apply to real-world amusement park rides.