2026-08-16 08:45:40: Circular Motion
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[[Force and Translational Dynamics]]
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Circular motion happens when an object moves along a curved path.
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Centripetal acceleration ($a_c=v^2/r$) always points towards the center to keep it turning.
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Circular motion means an object has an inward center-seeking component of acceleration called centripetal acceleration. The magnitude is $a_c=v^2/r$ and the inward net force is supplied by real forces like gravity and friction and normal force.
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Centripetal acceleration points towards the center and changes direction, not speed.
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Tangential acceleration changes an objects speed and points along the path. The net acceleration is the vector sum of centripetal and tangential forces.
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Period and frequency are reciprocals. $T=1/f$. For constant speed circular motion $T=2\pi r/v$.
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At the top of a vertical loop with minimum speed, gravity supplied the centripetal force being equal to $v=\sqrt{gr}$.
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For a circular orbit, gravity provides the centripetal accel, leading to keplers third law: $T^2=\frac{4\pi^2}{GM}R^3$
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![[Pasted image 20260816081041.png]]
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## Centripetal Acceleration
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Centripetal accel keeps an object moving in a circle by changing the direction of velocity towards the center.
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Magnitude is the tangential speed squared divided by the radius: $a_c=\frac{v^2}{r}$
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It points perpendicularly to the objects instantaneous velocity (no shit).
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It can come from a single force, several forces, or several components acting on it.
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There is no "centripetal force" when drawing a FBD, you only draw the forces affecting the object like gravity or friction, which are what give it centripetal force.
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Forces that supply centripetal accel:
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* An object needs a minimum speed to stay on a circular path, at minimum speed gravity is the only thing acting on the object, giving the formula $v=\sqrt{gr}$
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* Banked curves: Components of the normal force and static friction can point towards the center.
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* Conical pendulum, a horizontal component of the string tension supplies the centripetal force while vertical component balances gravity.
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## Tangential Accel and Net Accel
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Tangential accel is the accel that causes an object to change velocity magnitudes, and it points tangent to the objects circular path. When an object speeds up or down as it goes along a circular path, there is tangential accel. This produces nonuniform circular motion.
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Net accel of an object moving in a circular motion is the vector sum of the centripetal and tangential components.
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## Period and Frequency
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Period ($T$) is the amount of time for one revolution, frequency ($f$) is the amount of revolution in a second. $$T=1/f$$
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For an object moving at a constant speed around a circle, the formula is $T=\frac{2\pi r}{v}$
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## Kepler's Third Law for Circular Orbits
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For a satellite in circular orbit, the only relevant acting force is the gravity of the planet is surrounds. Because of this, orbital period depends on the orbit radius and mass of the planet: $$T^2=\frac{4\pi r^2}{GM}R^3$$
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$T$ is orbital period
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$R$ is orbital radius
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$M$ is the mass
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$G$ is universal gravitational constant
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throwback to $F/m_2 = \frac{Gm_1}{r^2}$
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| Term | Definition |
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| ------------------------ | --------------------------------------------------------------------------------------------------------------------------------------------------------------------------- |
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| banked surface | An inclined surface on which an object travels in a circular path, where normal force and friction components contribute to centripetal acceleration. |
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| centripetal acceleration | The component of an object's acceleration directed toward the center of its circular path. |
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| circular orbit | The path of a satellite moving around a central body at a constant distance, where gravitational force provides the centripetal force needed to maintain the circular path. |
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| circular path | The trajectory followed by an object moving in a circle around a fixed center point. |
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| conical pendulum | A pendulum that moves in a horizontal circle, with tension providing a component of the centripetal force. |
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| frequency | The number of complete oscillations or cycles of simple harmonic motion that occur per unit time, typically measured in hertz (Hz). |
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| gravitational attraction | The force of gravity exerted by a central body on a satellite, which provides the centripetal force necessary for circular orbital motion. |
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| gravitational force | The attractive force due to mass, which can serve as the sole source of centripetal acceleration at the top of a vertical circular loop. |
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| Kepler's third law | A principle stating that the square of a satellite's orbital period is proportional to the cube of its orbital radius, expressed as T² = (4π²/GM)R³. |
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| net acceleration | The vector sum of centripetal acceleration and tangential acceleration for an object moving in a circle. |
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| normal force | The perpendicular component of the force exerted on an object by a surface, directed away from the surface. |
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| orbital period | The time it takes for a satellite to complete one full orbit around a central body. |
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| orbital radius | The distance from the center of a central body to a satellite in circular orbit. |
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| period | The time required for an object to complete one full circular path, rotation, or cycle. |
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| radius | The distance from the center of a circular path to the object traveling on that path. |
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| static friction | A friction force that acts between two surfaces in contact that are not moving relative to each other, preventing an object from slipping or sliding. |
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| tangential acceleration | The component of linear acceleration directed along the tangent to the circular path of a rotating point, related to angular acceleration by a_T = rα. |
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| tangential speed | The instantaneous speed of an object moving along a circular path, directed tangent to the circle. |
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| tension | The macroscopic net force that segments of a string, cable, chain, or similar system exert on each other in response to an external force. |
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| uniform circular motion | Motion of an object traveling in a circular path at constant speed. |
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| vertical circular loop | A circular path oriented vertically, where an object must maintain a minimum speed at the top to continue circular motion. |
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