Dynamics II: Motion in a Plane – Study Notes

Dynamics II: Motion in a Plane

Introduction to Planar Motion

Motion in a plane involves analyzing objects that move in two dimensions, such as projectiles, satellites, and vehicles on curved paths. Newton's laws apply equally in two or three dimensions, but the analysis often requires careful selection of coordinate systems and consideration of forces acting in multiple directions.

• Newton's Laws: Remain valid as vector equations in any number of dimensions.

• Coordinate Systems: Choice of axes (e.g., x-y, r-t-z) is crucial for simplifying the analysis.

• Applications: Includes projectile motion, circular motion, orbital motion, and motion on curved surfaces.

Projectile Motion

Projectile Motion Without Air Resistance

Projectiles move under the influence of gravity alone when air resistance is neglected. The motion can be decomposed into independent horizontal (x) and vertical (y) components.

• Acceleration: Only the vertical component is affected by gravity; horizontal acceleration is zero.

• Equations of Motion:

$y(t) = v_{0y} t - \frac{1}{2} g t^2$ $x(t) = v_{0x} t$

• Range (R): For a projectile launched at speed $v_0$ and angle $\theta$:

$R = \frac{v_0^2 \sin 2\theta}{g}$

• Maximum Range: Occurs at $\theta = 45^\circ$ (without air resistance).

Projectile Motion With Air Resistance (Drag)

When drag is included, the x- and y-components of acceleration are no longer independent, and the trajectory is not a parabola. The launch angle for maximum range is less than $45^\circ$.

• Equations: Must be solved numerically due to coupling between components.

• Effect: Maximum range is achieved at an angle less than $45^\circ$.

Example: Rocketing in the Wind

• Scenario: A rocket is launched vertically but experiences a constant horizontal wind force.

• Analysis: Both vertical and horizontal accelerations are constant; the trajectory is a straight line.

• Key Result: The sideways deflection at a given height can be calculated using kinematics.

• Formula for Deflection: If $F_{wind}$ is the horizontal force, $m$ is mass, and $y$ is height:

$x = \frac{F_{wind}}{2 F_{thrust} - 2mg} y$

• Example Calculation: For a 30 kg rocket, 1500 N thrust, 20 N wind, at 1 km height, deflection is 17 m.

Circular Motion

Uniform Circular Motion

Uniform circular motion occurs when an object moves in a circle at constant speed. The velocity is always tangent to the circle, and the acceleration (centripetal) points toward the center.

• Centripetal Acceleration:

$a_c = \frac{v^2}{r} = \omega^2 r$

• Coordinate System: r-t-z (radial-tangential-perpendicular) coordinates are used:

  • r-axis: Points toward the center

  • t-axis: Tangent to the circle (counterclockwise)

  • z-axis: Perpendicular to the plane

• Net Force: Must point toward the center (radial direction).

Dynamics of Uniform Circular Motion

• Newton's Second Law (Radial):

$F_{net, r} = m a_c = m \frac{v^2}{r}$

• Source of Centripetal Force: Provided by tension, friction, gravity, or normal force, depending on the context.

• Centrifugal Force: A fictitious force observed only in non-inertial (rotating) frames; not a real force in inertial frames.

Example: Car Turning a Corner

• Maximum Speed on a Flat Curve: Determined by static friction.

• Formula:

$v_{max} = \sqrt{\mu_s g r}$

• Where: $\mu_s$ is the coefficient of static friction, $g$ is acceleration due to gravity, $r$ is the curve radius.

• Key Point: Mass cancels out; only friction, gravity, and radius matter.

Banked Curves

Banked curves are tilted to help provide the required centripetal force through the normal force, reducing reliance on friction.

• Banking Angle for No Friction:

$\tan \theta = \frac{v^2}{r g}$

• At Lower Speeds: Static friction prevents slipping down the bank.

• At Higher Speeds: Static friction prevents slipping up the bank.

Nonuniform Circular Motion

When speed changes along the circular path, the object experiences both radial (centripetal) and tangential acceleration.

• Radial Acceleration: Changes direction of velocity.

• Tangential Acceleration: Changes magnitude (speed) of velocity.

• Newton's Second Law (General):

$\vec{F}_{net} = m \vec{a} = m (a_r \hat{r} + a_t \hat{t})$

Orbital Motion

Circular Orbits

Satellites and planets in circular orbits are in continuous free fall, with gravity providing the required centripetal force.

• Gravitational Force:

$F_g = \frac{G M m}{r^2}$

• Orbital Speed:

$v_{orbit} = \sqrt{\frac{G M}{r}}$

• Period of Low-Earth Orbit: For $r \approx R_E = 6400$ km, $T \approx 90$ minutes.

• Weightlessness: Astronauts feel weightless because they are in free fall, not because gravity is absent.

Loop-the-Loop and Vertical Circles

Roller Coaster Loops

Objects moving in vertical circles, such as roller coasters, experience varying normal forces at different points in the loop.

• At the Bottom: Normal force is greater than weight; you feel heavier.

• At the Top: Normal force and gravity both point downward; normal force can be zero at the critical speed.

• Critical Speed at Top:

$v_{min} = \sqrt{g r}$

• If speed drops below this, the object will lose contact with the track.

Problem-Solving Strategy for Circular Motion

1. Model: Simplify the object as a particle; make reasonable assumptions.

2. Visualize: Draw diagrams, choose r-t-z coordinates, and identify forces.

3. Solve: Apply Newton's second law in component form; solve for acceleration, velocity, and position.

4. Review: Check units, significant figures, and physical reasonableness.

Key Concepts Table

Concept	Description	Key Equation
Projectile Motion (no drag)	Parabolic trajectory; independent x and y acceleration	$R = \frac{v_0^2 \sin 2\theta}{g}$
Projectile Motion (with drag)	Trajectory not parabolic; coupled acceleration	Numerical solution required
Uniform Circular Motion	Constant speed; acceleration toward center	$a_c = \frac{v^2}{r}$
Banked Curve (no friction)	Normal force provides centripetal force	$\tan \theta = \frac{v^2}{r g}$
Critical Speed (loop)	Minimum speed to stay on track at top	$v_{min} = \sqrt{g r}$
Orbital Speed	Speed for circular orbit under gravity	$v_{orbit} = \sqrt{\frac{G M}{r}}$

Summary of Important Points

• Newton's laws apply in all dimensions; vector analysis is essential.

• Projectile motion is parabolic without drag; with drag, the path is more complex.

• Circular motion requires a net inward (centripetal) force, provided by tension, friction, gravity, or normal force.

• Banked curves and orbits are special cases of circular motion with unique force considerations.

• Fictitious forces (like centrifugal force) appear only in non-inertial frames.

• Problem-solving involves modeling, visualizing, applying Newton's laws, and reviewing results for consistency.

Additional info: For more advanced analysis, consider energy methods and nonuniform circular motion where tangential acceleration is present. Always check the direction and source of forces in free-body diagrams.