Dynamics II: Motion in a Plane – Newton’s Laws in Two Dimensions

Chapter 8: Dynamics II – Motion in a Plane

Introduction

This chapter explores the application of Newton's laws to motion in two dimensions, focusing on projectile and circular motion. Students will learn to analyze forces and accelerations that act in a plane, building on concepts from earlier chapters on kinematics and dynamics.

Newton’s Laws in Two Dimensions

Vector Nature of Newton’s Laws

• Newton’s laws are vector equations and apply equally in one, two, or three dimensions.

• For motion in a plane, forces can be decomposed into components tangent (affecting speed) and perpendicular (affecting direction) to a particle’s trajectory.

• Key Point: A force tangent to the path changes speed; a force perpendicular to the path changes direction.

Example: In circular motion, the centripetal force is always perpendicular to the velocity, changing only the direction, not the speed.

Projectile Motion in Two Dimensions

Review of Projectile Motion

• Projectile motion involves an object moving under the influence of gravity alone (neglecting air resistance).

• Acceleration is always downward, $a_y = -g$.

• The motion can be analyzed by separating it into independent x- and y-components.

Equation for Range:

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

• The maximum range occurs at a launch angle of $45^\circ$.

• These results assume no air resistance.

Example: A projectile launched at $v_0 = 99$ m/s at various angles will have different ranges and trajectories, as shown in the provided graph.

Projectile Motion with Air Resistance

• When air resistance is present, the acceleration components are no longer independent.

• The equations of motion become more complex and often require numerical solutions.

Additional info: Air resistance generally acts opposite to the velocity and reduces both range and maximum height.

Circular Motion

Kinematics of Uniform Circular Motion

• In uniform circular motion, the speed is constant but the direction of velocity changes continuously.

• The acceleration is always directed toward the center of the circle (centripetal acceleration).

Formulas:

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

• Where $a_c$ is centripetal acceleration, $v$ is speed, $r$ is radius, and $\omega$ is angular speed.

Coordinate Systems for Circular Motion

• It is convenient to use a moving coordinate system with radial (r), tangential (t), and perpendicular (z) axes.

• The radial axis points from the particle to the center; the tangential axis is tangent to the path.

Dynamics of Uniform Circular Motion

• A net force directed toward the center is required to maintain circular motion.

• This force is called the centripetal force but is not a new type of force; it is provided by familiar forces (e.g., tension, friction, gravity).

Newton’s Second Law (Radial Direction):

$\sum F_r = m a_c = m \frac{v^2}{r}$

• Without this force, the object would move in a straight line tangent to the circle.

Examples and Applications

• Car Turning on a Level Road: The friction between tires and road provides the centripetal force.

• Banked Curves: The normal force from the road has a component that provides centripetal acceleration, reducing reliance on friction.

• Orbiting Satellites: Gravity provides the necessary centripetal force for circular orbits.

Friction and Circular Motion

Maximum Speed on a Flat Curve

• The maximum speed before sliding occurs is determined by the maximum static friction force.

Formula:

$f_{s,\text{max}} = \mu_s N$

$\mu_s m g = m \frac{v_{\text{max}}^2}{r}$

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

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

Banked Curves

• Banking the road allows the normal force to contribute to the required centripetal force.

• At a specific speed, no friction is needed; at lower or higher speeds, friction acts up or down the slope to prevent slipping.

Formula for Ideal Banking (no friction):

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

• Where $\theta$ is the banking angle, $v$ is the speed, $r$ is the radius, and $g$ is gravity.

Conceptual Checkpoints

• Forces in Circular Motion: The centripetal force is always provided by identifiable agents (e.g., tension in a string, friction, gravity).

• Misconceptions: There is no outward 'centrifugal force' acting on the object in the inertial frame; the net force is always inward.

Summary Table: Forces in Circular Motion

Situation	Agent Providing Centripetal Force
Car on flat curve	Static friction
Car on banked curve (ideal speed)	Normal force
Object on string	Tension
Satellite in orbit	Gravity

Applications and Importance

• Understanding planar motion is essential for analyzing real-world systems such as vehicles, satellites, and rotating machinery.

• This chapter lays the groundwork for more advanced topics like rotational dynamics.