Circular Motion and Centripetal Forces: From Skateboard Loops to Galaxy Rotation

The Loop of Death Case Study: Skateboarder Lizzie Armanto became the first female skater to successfully complete a full $360$ -degree loop (the "loop of death").
- The Physics Challenge: Success requires achieving a specific velocity—enough speed to maintain contact with the ramp at the top of the loop, but not so much that the skater is slammed against the wall by excessive force.
- Universal Application: Circular motion (or rotational motion) is found at all scales, from spinning coins and records to children on toys, planets orbiting stars, stars orbiting the centers of galaxies, and galaxies orbiting each other.

Newton's First Law and Inertia: According to Newton’s First Law, an object in motion will maintain a constant velocity in a straight line unless acted upon by a force.
- The Perception of Turning: When a car turns left, passengers feel pushed to the right. This is not a force throwing them rightward, but rather the passenger's inertia attempting to maintain a straight path while the car (via the seat, seatbelt, or door) pushes them to the left.
Direction of Force: For an object to move in a circle, a force must pull or push it constantly toward the center.
- The Egg and String Demonstration: An egg on a string orbits due to the tension in the string. If the string is released, the egg immediately travels in a straight line (tangent to the circle), proving that the inward force was what maintained the circular path.
- Key Identification: The force in circular motion is always directed inwards toward the center of the circle.

The General Formula: the magnitude of the force required to keep an object of mass $m$ moving at a linear velocity $v$ in a circle of radius $r$ is defined by: $F_c = \frac{m \times v^2}{r}$
Dimensional Analysis and Units:
- Mass ( $m$ ) is in $kg$ .
- Velocity squared ( $v^2$ ) is in $m^2/s^2$ .
- Radius ( $r$ ) is in $m$ .
- Combining these: $\frac{kg \times m^2/s^2}{m} = \frac{kg \times m}{s^2}$ , which is the definition of a Newton ( $N$ ).
Centripetal Acceleration: Using Newton's Second Law ( $F = m \times a$ ), the centripetal acceleration ( $a_c$ ) can be derived by setting $m \times a = \frac{m \times v^2}{r}$ .
- The masses cancel out: $a = \frac{v^2}{r}$ .
- Mass Independence: Centripetal acceleration depends only on velocity and radius; it is the same for a lightweight object as it is for a massive one (e.g., a rock or a person).

Fritz Zwicky (1930s): Observed that galaxies rotated much faster than they should, based on the amount of visible mass available to provide gravitational centripetal force.
Vera Rubin (1970s): Confirmed these observations, showing that galaxies were spinning so fast they should fly apart if only visible matter were holding them together.
Dark Matter: These findings led to the conclusion that there must be extra, invisible mass providing additional gravity. Dark matter is now believed to account for $85\%$ of all matter in the universe, though its exact nature remains unknown.

Centripetal Force: A real, physical force (such as tension, gravity, or friction) that acts on an object to keep it moving toward the center of a circle.
Centrifugal Force: Often called a "fictitious force," it is the outward push felt by an observer within a rotating reference frame.
- Example (The Gravitron): In this carnival ride, riders feel pinned against the wall as the floor drops. From the perspective of the rider, they feel an outward centrifugal force.
- Perspective Difference: In physics problems described from an outside (inertial) frame, one only considers the inward-acting centripetal force.

Scenario: A $1\,kg$ ball is on a $2\,m$ string, completing one revolution every $2\,s$ .
Variables:
- $m = 1\,kg$
- $r = 2\,m$
- $T (\text{period}) = 2\,s$
Find Velocity ( $v$ ):
- v = \frac{\text{circumference}}{\text{time}} = \frac{2 \times \text{\pi} \times r}{T}
- v = \frac{2 \times \text{\pi} \times 2\,m}{2\,s} = 2\text{\pi}\,m/s
Calculate Force ( $F_c$ ):
- F_c = \frac{1\,kg \times (2\text{\pi}\,m/s)^2}{2\,m}
- $F_c \text{ is approximately } \frac{1 \times 36}{2} = 18\,N$ .

The PVC Pipe Experiment: If a ball is spun on a string passed through a pipe with weights hanging at the other end, the weights provide a constant centripetal force through gravity.
The Paradox: If the radius is shortened by pulling the string down, the ball appears to speed up.
Linear Velocity ( $v$ ): The distance per second traveled around the edge of the circle (meters per second). To keep force constant ( $F = \frac{mv^2}{r}$ ), if $r$ decreases, $v$ must actually decrease.
Angular Velocity ( $\omega$ ): The number of rotations per second. While the linear velocity decreases when the radius is tightened, the angular velocity increases because the circumference has shrunk significantly, creating the illusion of speeding up.

Friction and Turning: On a flat road, friction between the tires and the pavement provides the centripetal force required to turn a car. On icy roads, the lack of friction causes cars to continue in a straight line (crashing).
Banked Roads: Engineers tilt roads (at circular racetracks or freeway overpasses) so that the Normal Force ( $F_n$ ) helps provide the inward centripetal force.
The "Perfect Angle" Calculation:
- Goal: Find the angle ( $\theta$ ) where a car can turn at velocity ( $v$ ) without needing any friction.
- Force Balance: The component of gravity pulling the car down the ramp (mg \times \text{\sin}(\theta)) must be balanced by the component of the centripetal force pushing it up/along the ramp (F_c \times \text{\cos}(\theta)).
- m \times g \times \text{\sin}(\theta) = \frac{m \times v^2}{r} \times \text{\cos}(\theta)
- \frac{\text{\sin}(\theta)}{\text{\cos}(\theta)} = \frac{v^2}{g \times r}
- \text{\tan}(\theta) = \frac{v^2}{g \times r}
Observations:
- The angle is independent of mass; the same bank works for a scooter as it does for a giant SUV.
- The units of the right side ( $v^2/gr$ ) are dimensionless, matching the dimensionless nature of the tangent function.

Sanity Checks for Banked Curves:
- Maximum Angle ( $90$ Degrees): As $\theta \rightarrow 90^{\circ}$ , \text{\tan}(\theta) \rightarrow \text{\infty}. This implies that to drive on a vertical wall with no friction, one would need infinite velocity.
- Zero Gravity ( $g = 0$ ): If gravity is zero, the formula implies an angle of $90$ degrees. In space, turning requires a bank that is essentially a cylinder.
Spinning Space Habitats: Giant spinning cylinders can simulate Earth's gravity ( $1\,g$ ) via the centrifugal effect felt by inhabitants inside the rotating frame.
- Media Examples: 2001: A Space Odyssey, The Martian.
- Ringworld: Larry Niven’s 1970 sci-fi novel features a world shaped like a thin spinning cylinder with a radius the size of Earth's orbit around the sun, creating an artificial environment with an atmosphere and oceans.

Rule 1: The force required for circular motion is the centripetal force ( $F_c$ ), directed toward the center of the circle.
Rule 2: For any circular motion problem, always apply the equation $F_c = \frac{m \times v^2}{r}$ .