Chapter 8 Dynamics II: Motion in a Plane

Dynamics in Two Dimensions

• Newton's second law applies in 2D and 3D motion: $\vec{a} = \frac{\vec{F}_{net}}{m}$
• Forces can be decomposed along orthogonal axes.

Uniform Circular Motion

• Uses the rtz-coordinate system.
  • $t$-axis: Tangential to the circle, counterclockwise.
  • $r$-axis: Radial, from particle to circle's center.
  • $z$-axis: Perpendicular to the motion plane.
• Velocity vector: Only a tangential component ($v_t = \omega r$).
• Acceleration vector: Only a radial component ($a_r = \frac{v^2}{r} = \omega^2 r$).
• Net force (central force) is directed towards the circle's center: $F_{net} = m\frac{v^2}{r} = m\omega^2 r$

Key Concepts and Clicker Questions

• Inertia: Without a central force, an object moves in a straight path.
• Static Friction: Provides the necessary centripetal force for a car to round a curve. $\mu_s = \frac{v^2}{Rg}$
  • $\mu<em>s > 1$ for tires on dry asphalt, \mus < 0.1 for tires on icy roads.
• Banked Track: On an icy banked road, the net force is horizontal (towards the center) with $F_{net} = \frac{mv^2}{R}$.
• Banked Curve with Friction: Car 1 (no friction) vs. Car 2 (moving slower) - Static friction on Car 2 is directed up the banked curve.

Circular Orbits

• Occur when the trajectory curvature is parallel to the planet's curvature.
• Objects are perpetually in free fall.
• Gravitational force: $\vec{F}_G = m\vec{g}$, toward the center of the planet.

Reasoning About Circular Motion

• Centrifugal Force: A fictitious force in a non-inertial reference frame.
• On the rotating Earth, $F_G = mg$ where $g$ accounts for the rotation.

Nonuniform Circular Motion

• Tangential force component leads to tangential acceleration.
• Net force is the sum of radial and tangential forces and does not point towards the center.

Loop-the-Loop

• Top: Normal force and weight are in the same direction.
  • Minimum speed to stay on track: $v<em>{min} = \sqrt{gR}$, where $F</em>{cent} = \frac{mv^2}{R} = mg$
• Bottom: $F_{cent} = \frac{mv^2}{R} = n - mg$
• Side: $F_{cent} = \frac{mv^2}{R} = n$
• Tangential Acceleration: Occurs when the net force has a tangential component.

Well of Death Stunt

• Normal force provides the centripetal force: $F<em>{cent} = n</em>{wc} = \frac{mv^2}{r}$
• Minimum speed to maintain height: $v = \sqrt{\frac{gr}{\mu_s}}$