Chapter 10: Motion in a Plane – Physics Study Guide

Motion in a Plane

Overview

This chapter introduces the fundamental concepts and mathematical tools required to analyze motion in two dimensions. It covers vector operations, decomposition of forces, friction, work, and momentum, providing a comprehensive foundation for understanding complex physical systems.

Straight is a Relative Term

Reference Frames and Trajectories

The trajectory of a free-falling object depends on the observer's reference frame. In a moving frame, the path may appear straight, while in a stationary frame, the object exhibits both horizontal and vertical motion.

• Reference Frame: The perspective from which motion is observed (e.g., cart vs. Earth).

• Trajectory: The path followed by an object, which can differ based on the observer's motion.

• Decomposition: Motion can be separated into horizontal (constant velocity) and vertical (free fall) components.

Vectors in a Plane

Vector Addition and Subtraction

Vectors are fundamental in describing motion in two dimensions. Their addition is commutative, but subtraction is not. Vectors can be decomposed into components along perpendicular axes.

• Vector Addition: Place the tail of the second vector at the head of the first; the sum is the vector from the tail of the first to the head of the second.

• Vector Subtraction: Reverse the direction of the vector to be subtracted and add it to the other vector.

• Commutativity: Addition is commutative ($\vec{a} + \vec{b} = \vec{b} + \vec{a}$), subtraction is not ($\vec{a} - \vec{b} \neq \vec{b} - \vec{a}$).

Vector Components and Decomposition

Any vector can be decomposed into x and y components using a rectangular coordinate system. This is essential for analyzing motion and forces in two dimensions.

• Component Vectors: $\vec{A} = A_x \hat{i} + A_y \hat{j}$

• Magnitude: $A = \sqrt{A_x^2 + A_y^2}$

• Angle: $\theta = \tan^{-1}(A_y/A_x)$

Acceleration Components

In two-dimensional motion, acceleration can be decomposed into components parallel and perpendicular to the instantaneous velocity.

• Parallel Component: Changes the speed of the object.

• Perpendicular Component: Changes the direction of velocity but not its magnitude.

Decomposition of Forces

Forces on Inclined Planes

When analyzing motion on an inclined surface, it is useful to decompose forces into components parallel (tangential) and perpendicular (normal) to the surface.

• Tangential Component: Drives motion along the surface.

• Normal Component: Acts perpendicular to the surface, often balancing gravity.

• Coordinate Choice: Choose axes aligned with the direction of acceleration for easier analysis.

Friction

Static and Kinetic Friction

Friction arises from microscopic interactions between surfaces. It opposes relative motion and can be classified as static (no relative motion) or kinetic (surfaces sliding).

• Static Friction: Prevents motion up to a maximum value.

• Kinetic Friction: Acts when surfaces are sliding, generally less than static friction.

• Normal Force: Perpendicular component of contact force, balances gravity.

Work and Friction

Energy Dissipation

Kinetic friction causes energy dissipation and irreversible changes, while static friction does not dissipate energy.

• Static Friction: Elastic, no energy loss.

• Kinetic Friction: Non-elastic, causes energy dissipation.

Vector Algebra

Coordinate Systems

Vectors can be represented in rectangular (Cartesian) or polar coordinates. Conversion between these systems uses trigonometric relationships.

• Rectangular Coordinates: $(x, y)$

• Polar Coordinates: $(r, \theta)$

• Conversion: $x = r \cos \theta$, $y = r \sin \theta$

Vector Decomposition and Addition

• Decomposition: $\vec{A} = A_x \hat{i} + A_y \hat{j}$

• Addition: $R_x = A_x + B_x$, $R_y = A_y + B_y$

Example: Speeding Ball

A ball thrown at 30° to the horizontal with a speed of 30 m/s:

• $v_x = 30 \cos 30^\circ = 26$ m/s

• $v_y = 30 \sin 30^\circ = 15$ m/s

Projectile Motion in Two Dimensions

Kinematics of Projectiles

Projectile motion can be analyzed by decomposing it into horizontal and vertical components. The horizontal velocity remains constant, while the vertical velocity changes due to gravity.

• Position Vector: $\vec{r} = x \hat{i} + y \hat{j}$

• Velocity Components: $v_x = v_{x,i}$ (constant), $v_y = v_{y,i} - g \Delta t$

• Position Components: $x_f = x_i + v_{x,i} \Delta t$, $y_f = y_i + v_{y,i} \Delta t - \frac{1}{2}g \Delta t^2$

Collisions and Momentum in Two Dimensions

Conservation of Momentum

Momentum is conserved in both the x and y directions for isolated systems. The equations for conservation are applied separately to each component.

• Momentum Conservation: $\Delta p_x = m_1(v_{1x,f} - v_{1x,i}) + m_2(v_{2x,f} - v_{2x,i}) = 0$

• Momentum Conservation: $\Delta p_y = m_1(v_{1y,f} - v_{1y,i}) + m_2(v_{2y,f} - v_{2y,i}) = 0$

Work as the Product of Two Vectors

Work and Scalar (Dot) Product

Work is defined as the scalar product of force and displacement. The work done by gravity is independent of the path taken.

• Work: $W = \vec{F} \cdot \vec{d} = Fd \cos \phi$

• Path Independence: For gravity, $W = mgh$ regardless of the path.

• Dot Product Properties: Commutative, zero if vectors are perpendicular.

Coefficients of Friction

Quantitative Description of Friction

The maximum force of static friction is proportional to the normal force, with the proportionality constant called the coefficient of static friction ($\mu_s$). Kinetic friction is also proportional to the normal force, but with a smaller coefficient ($\mu_k$).

• Static Friction: $f_s \leq \mu_s F_N$

• Kinetic Friction: $f_k = \mu_k F_N$

• Procedure: Draw a free-body diagram, decompose forces, solve for normal and frictional forces.

Summary Table: Types of Friction

Type	Formula	Coefficient	Motion
Static Friction	$f_s \leq \mu_s F_N$	$\mu_s$	No relative motion
Kinetic Friction	$f_k = \mu_k F_N$	$\mu_k$	Surfaces sliding

Summary Table: Vector Operations

Operation	Formula	Commutativity
Addition	$\vec{a} + \vec{b}$	Yes
Subtraction	$\vec{a} - \vec{b}$	No
Dot Product	$\vec{a} \cdot \vec{b} = ab \cos \phi$	Yes

Summary Table: Projectile Motion Equations

Component	Equation
Horizontal Position	$x_f = x_i + v_{x,i} \Delta t$
Vertical Position	$y_f = y_i + v_{y,i} \Delta t - \frac{1}{2}g \Delta t^2$
Horizontal Velocity	$v_x = v_{x,i}$
Vertical Velocity	$v_y = v_{y,i} - g \Delta t$

Summary Table: Conservation of Momentum in Two Dimensions

Component	Equation
x	$\Delta p_x = m_1(v_{1x,f} - v_{1x,i}) + m_2(v_{2x,f} - v_{2x,i}) = 0$
y	$\Delta p_y = m_1(v_{1y,f} - v_{1y,i}) + m_2(v_{2y,f} - v_{2y,i}) = 0$

Summary Table: Work and Energy

Type	Equation
Constant Force	$W = Fd \cos \phi$
Variable Force	$W = \int \vec{F} \cdot d\vec{r}$
Gravity	$W = mgh$