Physics 201: Vectors, Forces, and Newton's Laws in Two Dimensions

Vectors and Forces in Two Dimensions

Introduction to Vectors

In physics, a vector is a quantity that has both magnitude (size) and direction. Many physical quantities, such as force, velocity, and acceleration, are vectors. Understanding vectors is essential for analyzing motion and forces in two or more dimensions.

• Vector Notation: Vectors are often denoted with an arrow above the letter (e.g., $\vec{A}$).

• Magnitude: The magnitude of a vector $\vec{A}$ is written as $|\vec{A}|$ or simply $A$.

• Components: Any vector in two dimensions can be broken into x and y components: $A_x$ and $A_y$.

• Direction: The direction of a vector is often specified by the angle it makes with a coordinate axis.

Trigonometry Review for Vectors

Trigonometric functions are used to relate the components of a vector to its magnitude and direction.

• Sine: $\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}$

• Cosine: $\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}$

• Tangent: $\tan \theta = \frac{\text{opposite}}{\text{adjacent}}$

• Pythagorean Theorem: $\text{hypotenuse} = \sqrt{(\text{adjacent})^2 + (\text{opposite})^2}$

Relating Magnitude to Components

The components of a vector $\vec{A}$ with magnitude $A$ and angle $\theta$ (measured from the x-axis) are:

• $A_x = A \cos \theta$

• $A_y = A \sin \theta$

• The magnitude can be found from the components: $A = \sqrt{A_x^2 + A_y^2}$

Example: A vector of magnitude 5 units at 30° above the x-axis has components $A_x = 5 \cos 30° = 4.33$, $A_y = 5 \sin 30° = 2.5$.

Multiplication by a Scalar

Multiplying a vector by a scalar changes its magnitude but not its direction (unless the scalar is negative, which reverses the direction).

• If $\vec{A}$ is a vector and $k$ is a scalar, then $k\vec{A}$ has magnitude $|k|A$ and the same (or opposite) direction as $\vec{A}$.

Vector Addition: Graphical and Numerical Methods

Vectors can be added graphically or numerically:

• Tip-to-Tail (Graphical) Method: Place the tail of the second vector at the tip of the first. The resultant vector is drawn from the tail of the first to the tip of the last.

• Parallelogram Method: Both vectors are drawn from the same point; the diagonal of the parallelogram represents the sum.

• Numerical (Component) Method: Add the x-components and y-components separately:

$R_x = A_x + B_x + C_x + \ldots$ $R_y = A_y + B_y + C_y + \ldots$ $R = \sqrt{R_x^2 + R_y^2}$

Example: If $\vec{A} = (3, 4)$ and $\vec{B} = (1, -2)$, then $\vec{A} + \vec{B} = (3+1, 4+(-2)) = (4, 2)$.

Forces and Newton's Laws in Two Dimensions

Labeling Forces in Diagrams

Careful labeling of forces is crucial, especially when multiple objects or forces are involved. The notation typically specifies:

• Type of force (e.g., tension $T$, gravitational $F_g$)

• Object exerting the force

• Object experiencing the force

For example, $\vec{F}_{G, E\text{ on }B}$ denotes the gravitational force exerted by Earth on object B.

When writing Newton's Second Law for an object, if there is no ambiguity, subscripts may be omitted for simplicity.

Newton's Second Law in Two Dimensions

Newton's Second Law relates the net force on an object to its acceleration:

$\vec{F}_{\text{net}} = m\vec{a}$

In component form:

$F_{\text{net},x} = m a_x$ $F_{\text{net},y} = m a_y$

Newton's Third Law (Action-Reaction Law)

Newton's Third Law states that forces always come in pairs:

• If object 1 exerts a force on object 2, then object 2 exerts an equal and opposite force on object 1.

$\vec{F}_{1\text{ on }2} = -\vec{F}_{2\text{ on }1}$

• These forces act on different objects.

• They are called interaction pairs or action-reaction pairs.

Example: When a mosquito collides with a truck, the force the mosquito exerts on the truck is equal in magnitude and opposite in direction to the force the truck exerts on the mosquito. However, their accelerations differ due to their different masses (see Newton's Second Law).

Scales and Apparent Weight

Scales measure the normal force exerted by the surface, not the gravitational force directly. In an elevator accelerating upward, the normal force (and thus the scale reading) increases; if accelerating downward, it decreases.

• At rest: $N = mg$

• Accelerating upward: $N = m(g + a)$

• Accelerating downward: $N = m(g - a)$

Example: A 70 kg person in an elevator accelerating upward at $2\,\text{m/s}^2$ experiences a normal force $N = 70 \times (9.8 + 2) = 826$ N.

General Problem-Solving Approach in Physics

Using Physics Models

Physicists use models and general principles to analyze and predict the behavior of physical systems. The process involves:

• Translating the problem into physical terms

• Drawing relevant diagrams and graphs

• Identifying the appropriate physics models (e.g., Newton's laws)

• Applying conceptual, qualitative, and quantitative reasoning

• Writing out general concepts, equations, and all intermediate steps

Tip: Always show your reasoning and steps in calculations to receive full credit in exams and quizzes.

Table: Vector Addition by Components

The following table summarizes how to add vectors by their components:

Vector	x-component	y-component
$\vec{A}$	$A_x$	$A_y$
$\vec{B}$	$B_x$	$B_y$
Resultant	$A_x + B_x$	$A_y + B_y$

Additional info: If more vectors are present, continue summing their respective components in each column.