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Circular Motion & Rotational Motion Summary

SECTION 3.7 — CIRCULAR MOTION

• Key Concept:
  • Constant speed does not imply constant velocity due to the change in direction of the velocity vector.
• Centripetal Acceleration:
  • Formula: $a_c = \frac{v^2}{r}$
  • This acceleration always points toward the center of the circular path.
• Velocity and Acceleration Relationship:
  • The velocity vector is tangent to the circular path, while the centripetal acceleration vector is directed radially inward.
• Centripetal Force:
  • Formula: $F_c = \frac{mv^2}{r}$
  • Represents the net inward force necessary to maintain circular motion.

SECTION 6.1 — ROTATION ANGLE & ANGULAR VELOCITY

• Rotation Angle:
  • Formula: $\theta = \frac{s}{r}$
  • Where:
  • $\theta$ is the rotation angle,
  • $s$ is the arc length,
  • $r$ is the radius of the circular path.
• Angular Velocity:
  • Formula: $\omega = \frac{d\theta}{dt}$
• Relationships:
  • Arc length and radius: $s = r\theta$
  • Linear velocity and angular velocity: $v = r\omega$
  • Note: Angular velocity $\omega$ is the same for all points on the circular path, while linear velocity $v$ is dependent on radius $r$.

SECTION 6.2 — ANGULAR ACCELERATION

• Angular Acceleration:
  • Formula: $\alpha = \frac{d\omega}{dt}$
• Tangential and Radial Accelerations:
  • Tangential acceleration relation: $a_t = r\alpha$
  • Radial acceleration relation: $a_r = \frac{v^2}{r} = r\omega^2$
• There are two types of acceleration to consider during rotational motion:
  • Tangential Acceleration: Responsible for change in the speed along the circular path.
  • Radial Acceleration: Responsible for change in direction of the velocity vector.

SECTION 6.3 — ROTATIONAL KINEMATICS

• If angular acceleration $\alpha$ is constant:
  • Angular velocity: $\omega = \omega_0 + \alpha t$
  • Rotation angle: $\theta = \omega_0 t + \frac{1}{2}\alpha t^2$
  • Relation between angular velocities: $\omega^2 = \omega_0^2 + 2\alpha \theta$

SECTION 7.1 — DYNAMICS OF CIRCULAR MOTION

• Net Radial Force:
  • Formula: $F_{net, r} = \frac{mv^2}{r}$
• Key Point:
  • The forces acting on a body in circular motion must produce an inward radial component that equals the required centripetal force for maintaining the circular path.

SECTION 7.2 — REAL SITUATIONS

7.2a — Friction as Centripetal Force

• Friction provides the necessary centripetal force to keep an object moving in a circular path.
  • Formula: $f_s = \frac{mv^2}{r}$
• Maximum Safe Speed:
  • Formula: $v<em>{max} = \sqrt{\mu</em>s r g}$
  • Where:
  • $\mu_s$ is the coefficient of static friction,
  • $g$ is the acceleration due to gravity.

7.2b — Banked Curves

• Banked Curve Analysis:
  • Formula: $\tan\theta = \frac{v^2}{rg}$
  • The banking angle $\theta$ allows normal force to provide the necessary inward radial component of the force.

7.2c — Vertical Circles

• Minimum Speed at Top:
  • Formula: $v_{min} = \sqrt{rg}$
• Tension in Vertical Circles:
  • At the bottom of the circle: tension is maximum.
  • At the top of the circle: tension is minimum.

EXAM QUICK REVIEW

• Key Equations:
  • Centripetal acceleration: $a_c = \frac{v^2}{r} = r\omega^2$
  • Linear velocity: $v = r\omega$
  • Tangential acceleration: $a_t = r\alpha$
  • Centripetal force: $F_c = \frac{mv^2}{r}$
  • Maximum speed: $v<em>{max} = \sqrt{\mu</em>s r g}$
  • Minimum speed at the top of a loop: $v_{min} = \sqrt{rg}$
• Conceptual Understandings:
  • Differentiate between tangential versus radial acceleration.
  • Recognize that all forms of circular motion require an inward net force to maintain circular motion.