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Motion in a Circle - Key Concepts and Formulas

컨트롤 버튼이 '내비게이션' 모드로 변경되었습니다.
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  • What is circular motion?
    Circular motion is the movement of an object along a circular path centered on an axis of rotation.
  • Define rotational coordinate (θ).
    The rotational coordinate θ is a unitless quantity defined as the arc length s divided by the radius r: \(\theta = \frac{s}{r}\).
  • What is rotational velocity (ω)?
    Rotational velocity ω is the rate of change of the rotational coordinate θ with respect to time: \(\omega = \frac{d\theta}{dt}\).
  • Relationship between tangential velocity and rotational velocity
    Tangential velocity vt is related to rotational velocity by \(v_t = r \omega\), where r is the radius.
  • What is centripetal acceleration?
    Centripetal acceleration is the acceleration directed toward the center of the circle, given by \(a_r = -\frac{v^2}{r}\) or \(a_r = -r \omega^2\).
  • What is tangential acceleration (at)?
    Tangential acceleration is the rate of change of tangential velocity, related to rotational acceleration α by \(a_t = r \alpha\).
  • Formula for total acceleration in circular motion
    Total acceleration magnitude is \(a = \sqrt{a_r^2 + a_t^2}\), combining radial and tangential components.
  • What is rotational acceleration (α)?
    Rotational acceleration α is the rate of change of rotational velocity: \(\alpha = \frac{d\omega}{dt} = \frac{d^2 \theta}{dt^2}\).
  • State the parallel-axis theorem.
    The rotational inertia about an axis parallel to one through the center of mass is \(I = I_{cm} + md^2\), where d is the distance between axes.
  • Define rotational inertia (I).
    Rotational inertia I measures an object's resistance to changes in rotational velocity and depends on mass distribution relative to the axis.
  • Rotational kinetic energy formula
    Rotational kinetic energy is \(K_{rot} = \frac{1}{2} I \omega^2\), where I is rotational inertia and ω is rotational speed.
  • Angular momentum (L) of a rotating object
    Angular momentum is \(L = I \omega\), where I is rotational inertia and ω is rotational velocity.
  • Angular momentum of a particle moving in a straight line
    Angular momentum relative to an axis is \(L = r_\perp m u\), where r⊥ is the lever arm perpendicular distance.
  • Law of conservation of angular momentum
    Angular momentum can be transferred but not created or destroyed; it remains constant if no tangential forces act.
  • What force causes centripetal acceleration?
    A net force directed toward the center of the circular path, called the centripetal force, causes centripetal acceleration.
  • How does speed vary on a rotating disk?
    All points have the same rotational velocity ω, but tangential speed increases with radius from the center.
  • What is the period (T) in circular motion?
    The period T is the time taken for one complete revolution around the circle.
  • Rotational kinematics equations for constant acceleration

    \(\theta_f = \theta_i + \omega_i \Delta t + \frac{1}{2} \alpha (\Delta t)^2\)

    \(\omega_f = \omega_i + \alpha \Delta t\)

  • What happens to rotational kinetic energy if both I and ω double?
    Rotational kinetic energy quadruples because \(K_{rot} = \frac{1}{2} I \omega^2\) depends on ω squared.
  • How to find lever arm distance r⊥?
    Draw the line of action of velocity, find the perpendicular distance from the rotation axis to this line; this distance is r⊥.