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6.2 Uniform Circular Motion

2 min read•june 24, 2024

is all about objects moving in curves. It's like when you swing a ball on a string or drive around a roundabout. The key is that something's always pulling the object toward the center.

This pulling force is called . It changes the direction of motion but not the speed. The faster you go or the tighter the turn, the stronger this force needs to be. It's what keeps things spinning instead of flying off.

Uniform Circular Motion

Centripetal acceleration and velocity

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( $a_c$ $a_{c}$ ) directed towards center of circular path
- Always perpendicular to velocity vector
- Changes direction of object, not speed
Magnitude of $a_c$ $a_{c}$ depends on object's speed ( $v$ $v$ ) and radius ( $r$ $r$ ) of circular path
- Formula: $a_c = \frac{v^2}{r}$
- Higher speed increases $a_c$
- Smaller radius increases $a_c$
$a_c$ $a_{c}$ changes only direction of velocity, not magnitude
- Velocity vector always tangent to circular path (ball on a string, car turning)
Circular motion occurs when an object moves in a curved path and its velocity changes direction continuously

Forces in uniform circular motion

Net force acting on object in uniform circular motion is centripetal force ( $F_c$ $F_{c}$ )
- Directed towards center of circular path
- Formula: $F_c = ma_c = m\frac{v^2}{r}$ , where $m$ is object's mass
can provide $F_c$ $F_{c}$
- Ball attached to string rotating in horizontal circle (tetherball, yo-yo)
  - Tension in string provides $F_c$
can provide $F_c$ $F_{c}$
- Car making turn on flat road (racetrack, highway ramp)
  - between tires and road provides $F_c$
Gravity can provide $F_c$ $F_{c}$
- Satellite orbiting Earth (GPS, weather satellites)
  - between satellite and Earth provides $F_c$
resists changes in motion, requiring a continuous centripetal force for circular motion

Applications of centripetal equations

Identify force providing $a_c$ $a_{c}$
- Tension, friction, gravity, or combination of forces
Determine relevant variables
- Mass of object ( $m$ )
- Speed of object ( $v$ )
- Radius of circular path ( $r$ )
Use appropriate equations to solve for unknown variable
- Centripetal acceleration: $a_c = \frac{v^2}{r}$
- Centripetal force: $F_c = ma_c = m\frac{v^2}{r}$
Example problem: 1000 kg car makes turn with 50 m radius at 15 m/s. Calculate $F_c$ $F_{c}$ and determine force responsible for circular motion
1. Given: $m = 1000 \text{ kg}$ , $v = 15 \text{ m/s}$ , $r = 50 \text{ m}$
2. Calculate $F_c$ : $F_c = m\frac{v^2}{r} = 1000 \text{ kg} \cdot \frac{(15 \text{ m/s})^2}{50 \text{ m}} = 4500 \text{ N}$
3. Force responsible for circular motion is static friction between tires and road (roller coaster loop, merry-go-round)

Special cases of circular motion

: Angled roads that use both friction and components to provide centripetal force
: A weight suspended by a string that moves in a horizontal circle, with tension providing the centripetal force

Key Terms to Review (17)

Angular Velocity: Angular velocity is a measure of the rate of change of the angular position of an object, typically expressed in radians per second. It describes the speed of rotational motion around a fixed axis or point.

Banked Curves: Banked curves refer to the curved sections of a road or track that are designed with a sloped or inclined surface. This banking, or superelevation, helps vehicles navigate the curve at higher speeds by providing an outward force that counteracts the inward centripetal force acting on the vehicle.

Centripetal Acceleration: Centripetal acceleration is the acceleration experienced by an object moving in a circular path, directed toward the center of the circular motion. It is the acceleration that causes the object to continuously change direction and maintain its circular trajectory.

Centripetal Force: Centripetal force is the force that acts on an object moving in a circular path, causing it to continuously change direction and maintain its curved trajectory. It is the force that is directed toward the center of the circular motion, perpendicular to the object's velocity.

Circular Motion: Circular motion is the movement of an object in a circular path around a fixed point or axis. It is characterized by a constant change in the direction of the object's velocity, resulting in a centripetal acceleration that keeps the object moving in a circular trajectory.

Conical Pendulum: A conical pendulum is a type of pendulum where the weight attached to the end of the string or rod moves in a circular path around a central axis, forming a cone shape. This motion is an example of uniform circular motion.

Friction Force: Friction force is the force that opposes the relative motion between two surfaces in contact. It acts parallel to the interface between the surfaces and serves to resist their sliding motion.

Gravitational Force: Gravitational force is the attractive force that exists between any two objects with mass. It is the force that keeps planets in orbit around the Sun, and it is the force that causes objects to accelerate downward towards the Earth's surface. Gravitational force is a fundamental force in the universe and is described by Newton's law of universal gravitation.

Huygens: Huygens is a Dutch mathematician, astronomer, and physicist who made significant contributions to the understanding of wave theory, optics, and uniform circular motion. His principles and theories have become foundational in various areas of physics.

Inertia: Inertia is the tendency of an object to resist changes in its state of motion. It is the property of matter that causes an object at rest to remain at rest, and an object in motion to remain in motion, unless acted upon by an external force.

Newton: A newton is the standard unit of force in the International System of Units (SI). It is named after the renowned English physicist Sir Isaac Newton, who formulated the laws of motion that describe the relationship between an object's mass, acceleration, and the forces acting upon it.

Normal Force: The normal force is the force exerted perpendicular to the surface of contact between two objects. It is the force that acts to support an object and prevent it from sinking into or falling through the surface it is resting on.

Periodicity: Periodicity refers to the repeating or cyclical nature of a phenomenon, where certain characteristics or behaviors recur at regular intervals. This concept is particularly relevant in the context of uniform circular motion, where the motion of an object follows a predictable, recurring pattern.

Radial Force: Radial force is the force that acts on an object moving in a circular path, directing the object towards the center of the circle. This force is responsible for the object's centripetal acceleration and is a critical component in understanding uniform circular motion.

Static Friction: Static friction is the force that resists the relative motion between two surfaces in contact when they are not moving with respect to each other. It is the maximum force that can be applied before the surfaces start to slide against each other.

Tangential Velocity: Tangential velocity is the rate of change of an object's position along the tangent of its circular path. It represents the speed of an object moving in a circular motion, perpendicular to the radius of the circle.

Tension Force: Tension force is the force exerted on an object by a rope, string, cable, or other connector that is pulled taut. It acts to resist the separation of the object from the connector and is always directed along the length of the connector, pulling the object towards the source of the tension.

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Practice QuizGlossary

Practice Quiz Glossary

6.2 Uniform Circular Motion

Uniform Circular Motion

Centripetal acceleration and velocity

Top images from around the web for Centripetal acceleration and velocity

Top images from around the web for Centripetal acceleration and velocity

Forces in uniform circular motion

Applications of centripetal equations

Special cases of circular motion

Key Terms to Review (17)

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AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.

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