Glossary

Circular motion is a fascinating aspect of physics, describing objects moving in curved paths. It's all around us, from planets orbiting the sun to riders on a merry-go-round. Understanding the forces and accelerations involved helps explain why objects stay in circular paths.

occurs at constant speed, while nonuniform motion involves changing speeds. Both types involve towards the center, but nonuniform motion adds . These concepts are key to grasping the dynamics of rotating objects and systems.

Uniform Circular Motion

Calculation of centripetal acceleration

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  • Centripetal acceleration (aca_c) points towards the center of the circular path, always perpendicular to the velocity vector
  • Magnitude of centripetal acceleration calculated using the formula ac=v2ra_c = \frac{v^2}{r}
    • vv represents the speed of the object moving in a circular path
    • rr represents the of the circular path (distance from the center to the object)
  • Centripetal acceleration can also be expressed using (ω\omega): ac=ω2ra_c = \omega^2r
    • relates to speed through the equation v=ωrv = \omega r
    • Angular velocity measures the rate of change of angular position ( per second)

Equations for circular motion dynamics

  • In uniform circular motion, speed (vv) remains constant while acceleration points towards the center
  • (TT) represents the time for one complete revolution, calculated using T=2πrvT = \frac{2\pi r}{v}
    • 2πr2\pi r represents the circumference of the circular path
  • (ff) measures the number of revolutions per unit time, calculated using f=1Tf = \frac{1}{T}
    • Frequency is the reciprocal of the period
  • Angular velocity (ω\omega) relates to frequency through the equation ω=2πf\omega = 2\pi f
    • 2π2\pi represents the number of radians in one complete revolution
  • (FcF_c) causes the centripetal acceleration, calculated using Fc=mac=mv2rF_c = ma_c = m\frac{v^2}{r}
    • mm represents the mass of the object
    • always points towards the center of the circular path
    • Centripetal force overcomes the object's to maintain circular motion

Forces in Circular Motion

  • can provide the centripetal force in vertical circular motion (e.g., loop-the-loop)
  • in a string or rope can act as the centripetal force for objects in horizontal circular motion
  • Friction on helps vehicles maintain circular motion without slipping

Nonuniform Circular Motion

Centripetal vs tangential acceleration

  • In , speed changes with time, resulting in
  • Tangential acceleration (ata_t) is parallel to the velocity vector and changes the speed
    • Tangential acceleration calculated using at=dvdta_t = \frac{dv}{dt}, the rate of change of speed
  • Centripetal acceleration (aca_c) remains perpendicular to the velocity vector, changing the direction of motion
    • Magnitude of centripetal acceleration in nonuniform circular motion still calculated using ac=v2ra_c = \frac{v^2}{r}
    • Centripetal acceleration points towards the center of the circular path

Total acceleration in nonuniform motion

  • (aa) in nonuniform circular motion is the vector sum of centripetal and tangential accelerations
  • Magnitude of total acceleration calculated using a=ac2+at2a = \sqrt{a_c^2 + a_t^2}
    • Pythagorean theorem used to find the resultant of the two perpendicular components
  • Direction of total acceleration depends on the relative magnitudes of centripetal and tangential components
    1. If ac>ata_c > a_t, total acceleration is more directed towards the center of the circular path
    2. If at>aca_t > a_c, total acceleration is more aligned with the tangential direction (along the velocity vector)
  • Angle (θ\theta) between total acceleration and centripetal acceleration calculated using tan⁡θ=atac\tan \theta = \frac{a_t}{a_c}
    • Tangent function used to find the angle between the two components
    • Angle helps determine the orientation of the total acceleration vector

Key Terms to Review (21)

Angular velocity: Angular velocity is the rate at which an object rotates around a fixed axis. It is measured in radians per second (rad/s).
Angular Velocity: Angular velocity is a measure of the rate of change of the angular position of an object. It describes the speed of rotation or the change in the orientation of an object around a fixed axis or point. This concept is fundamental in understanding the motion of objects undergoing circular or rotational motion.
Banked Curves: Banked curves refer to curved sections of a road or track that are designed with a tilted or angled surface. This angled surface, known as the banking, helps vehicles navigate the curve more efficiently and safely by providing an additional centripetal force that counteracts the outward force experienced during the turn.
Centripetal Acceleration: Centripetal acceleration is the acceleration experienced by an object moving in a circular path, directed towards the center of the circular motion. It is the rate of change in the direction of the velocity vector, causing the object to continuously change direction and move in a curved trajectory.
Centripetal force: Centripetal force is the force that keeps an object moving in a circular path, directed towards the center of the circle. It is necessary for maintaining circular motion and depends on mass, velocity, and radius of the path.
Centripetal Force: Centripetal force is the force that causes an object to move in a circular path, constantly changing the direction of the object's motion. It is the force that acts perpendicular to the object's velocity and points towards the center of the circular path.
Frequency: Frequency is a fundamental concept in physics that describes the number of occurrences of a repeating event or phenomenon per unit of time. It is a crucial parameter in various areas of physics, including wave behavior, oscillations, and sound propagation.
Inertia: Inertia is the property of an object that resists changes to its state of motion. It depends solely on the mass of the object.
Inertia: Inertia is the property of an object that resists changes to its state of motion. It is the tendency of an object to remain at rest or in motion unless acted upon by an unbalanced force.
Nonuniform Circular Motion: Nonuniform circular motion refers to the motion of an object traveling in a circular path where the speed or direction of the object is constantly changing. This type of motion is characterized by a varying centripetal acceleration, which is the acceleration directed toward the center of the circular path.
Normal Force: Normal force is the support force exerted by a surface perpendicular to the object resting on it, preventing the object from falling through the surface. It plays a crucial role in balancing other forces acting on an object, particularly in scenarios involving gravity and acceleration.
Orbital period: The orbital period is the time taken for a satellite or celestial body to complete one full orbit around another object. It is typically measured in seconds, minutes, hours, or years.
Period: The period of a periodic phenomenon is the time taken for one complete cycle or repetition of the event. This concept is fundamental in understanding various physics topics, including uniform circular motion, simple harmonic motion, and wave phenomena.
Radians: Radians are a unit of angular measurement used in mathematics and physics, defined as the angle subtended at the center of a circle by an arc whose length is equal to the radius of that circle. This unit connects linear and angular dimensions, making it essential for understanding circular motion, rotation, and oscillatory motion.
Radius: The radius is a fundamental geometric concept that represents the distance from the center of a circle or sphere to its circumference or surface. It is a crucial parameter in various physics topics related to circular and rotational motion.
Schwarzschild radius: The Schwarzschild radius is the radius of a sphere such that, if all the mass of an object were to be compressed within that sphere, the escape velocity from its surface would equal the speed of light. It is a key concept in understanding black holes.
Tangential acceleration: Tangential acceleration is the rate of change of the tangential velocity of an object moving along a circular path. It is directed along the tangent to the path of motion.
Tangential Acceleration: Tangential acceleration is the acceleration component that is perpendicular to the radius of a curved path, causing an object to change its speed along the curve. It is a crucial concept in understanding the motion of objects undergoing uniform circular motion, rotation with constant angular acceleration, and the relationship between angular and translational quantities.
Tension: Tension is a force that acts to pull or stretch an object, often along the length of a string, rope, or cable. It is a vector quantity, meaning it has both magnitude and direction, and it plays a crucial role in various physics concepts related to forces, motion, and equilibrium.
Total acceleration: Total acceleration in uniform circular motion is the vector sum of the radial (centripetal) and tangential accelerations. It describes how both the speed and direction of an object change as it moves along a circular path.
Uniform Circular Motion: Uniform circular motion is the motion of an object moving at a constant speed along a circular path. Although the speed remains constant, the direction of the object's velocity changes continuously, resulting in an acceleration that is directed toward the center of the circle, which is essential for maintaining this circular path.
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