4.1 Force

Definition

A force is a push or pull exerted on an object resulting from its interaction with another object. A force can cause an object to accelerate by changing its speed, altering its direction of velocity, or causing both. Force is a vector quantity, possessing both magnitude and direction, and follows the rules of vector addition.

SI Unit

The SI unit of force is the newton $(\text{N})$. One newton is defined as the amount of net force required to accelerate a mass of one kilogram at a rate of one meter per second squared: $1 \text{ N} = 1 \text{ kg}\cdot\text{m/s}^2$

Contact and Field Forces

Forces in classical mechanics are categorized into two fundamental classes based on how objects interact:

1. Contact Forces

Contact forces arise from direct physical contact between the interacting surfaces or bodies. Common examples include:

• Normal force exerted perpendicular to a supporting surface
• Tension transmitted through a taut string, cord, or cable
• Frictional force acting parallel to contacting surfaces to oppose sliding motion
• Applied forces exerted by direct pushing, pulling, or muscle action

2. Field Forces

Field forces act across empty space without requiring physical contact between bodies. An object creates a physical field throughout surrounding space that exerts a force on any other responsive body placed within that region. Common examples include:

• Gravitational force between masses (such as Earth pulling downward on an object)
• Electrostatic force between stationary electric charges
• Magnetic force between magnetic poles or moving electric charges

Force as a Vector Quantity

Because force is a vector, a complete description of any force requires specifying its magnitude, direction, and point of application. A force $\vec{F}$ acting in a two-dimensional Cartesian plane can be resolved into perpendicular components:

$\vec{F} = F_x \hat{i} + F_y \hat{j}$

Where:

• $F_x$ is the component of the force along the x-axis
• $F_y$ is the component of the force along the y-axis
• $\hat{i}$ and $\hat{j}$ are standard unit vectors pointing along the positive x- and y-axes, respectively

The magnitude F of the force is given by:

$F = \sqrt{F_x^2 + F_y^2}$

The angle \theta that the force vector makes relative to the positive x-axis is determined by:

$\tan\theta = \frac{F_y}{F_x}$

Net Force and the Principle of Superposition

When multiple forces act simultaneously on a single body, their combined physical effect is identical to that of a single equivalent force called the net force (or resultant force) $\vec{F}{\text{net}}$:

$\vec{F}{\text{net}} = \sum \vec{F} = \vec{F}_1 + \vec{F}_2 + \dots + \vec{F}_n$

The components of the net force along the coordinate axes are calculated by summing the corresponding components of all applied forces:

$F_{\text{net},x} = \sum F_x = F_{1x} + F_{2x} + \dots + F_{nx}$

$F_{\text{net},y} = \sum F_y = F_{1y} + F_{2y} + \dots + F_{ny}$

The overall magnitude of the net force is:

$F_{\text{net}} = \sqrt{\left(\sum F_x\right)^2 + \left(\sum F_y\right)^2}$

The direction \theta of the resultant force vector relative to the positive x-axis is:

$\tan\theta = \frac{\sum F_y}{\sum F_x}$

If the vector sum of all forces equals zero $(\sum \vec{F} = 0)$, the forces are balanced, and the object maintains a constant state of motion. If $\sum \vec{F} \neq 0$, the forces are unbalanced, producing an acceleration in the direction of the net force vector.

Example 1: Net Force Along a Straight Line

Two horizontal forces act concurrently on a crate resting on a smooth floor. Force $\vec{F}_1$ has a magnitude of$45 \text{ N}$ and acts toward the right (+x). Force $\vec{F}_2$ has a magnitude of $20 \text{ N}$ and acts toward the left (-x). Find the magnitude and direction of the net force acting on the crate.

Calculation:

Assigning positive to the right (+x) and negative to the left (-x):

$F_1 = +45 \text{ N}$

$F_2 = -20 \text{ N}$

Summing the collinear components yields:

$F_{\text{net},x} = F_1 + F_2$

$F_{\text{net},x} = 45 + (-20) = +25 \text{ N}$

The net force acting on the crate has a magnitude of $25 \text{ N}$ and is directed to the right.

Example 2: Net Force in Two Dimensions

Two ropes pull simultaneously on an eye-bolt anchored to a wall. Rope A exerts a force $\vec{F}_A$ of magnitude $60.0 \text{ N}$ directed at an angle of $30.0^\circ$ above the horizontal (+x). Rope B exerts a force $\vec{F}_B$ of magnitude $80.0 \text{ N}$ directed horizontally along the positive x-axis. Determine the magnitude and direction of the resultant force acting on the eye-bolt.

Vector Components:

For Rope A:

$F_{Ax} = F_A \cos(30.0^\circ) = (60.0)(0.8660) \approx 51.96 \text{ N}$

$F_{Ay} = F_A \sin(30.0^\circ) = (60.0)(0.5000) = 30.00 \text{ N}$

For Rope B:

$F_{Bx} = 80.0 \text{ N}$

$F_{By} = 0 \text{ N}$

Net Force Components:

Summing along the x-axis:

$F_{\text{net},x} = F_{Ax} + F_{Bx} = 51.96 + 80.0 = 131.96 \text{ N}$

Summing along the y-axis:

$F_{\text{net},y} = F_{Ay} + F_{By} = 30.00 + 0 = 30.00 \text{ N}$

Magnitude of the Resultant Force:

$F_{\text{net}} = \sqrt{(F_{\text{net},x})^2 + (F_{\text{net},y})^2}$

$F_{\text{net}} = \sqrt{(131.96)^2 + (30.00)^2} = \sqrt{17413.4 + 900} = \sqrt{18313.4} \approx 135.3 \text{ N}$

Direction of the Resultant Force:

$\tan\theta = \frac{F_{\text{net},y}}{F_{\text{net},x}} = \frac{30.00}{131.96} \approx 0.2273$

$\theta = \arctan(0.2273) \approx 12.8^\circ$

The net force acting on the eye-bolt has a magnitude of $135.3 \text{ N}$ and acts at an angle of $12.8^\circ$ above the horizontal.

Key Points

• A force is a vector push or pull resulting from an interaction between two bodies.
• The SI unit of force is the newton $(\text{N})$, equivalent to $1 \text{ kg}\cdot\text{m/s}^2$.
• Contact forces require physical contact between objects; field forces act over distance through gravitational, electric, or magnetic fields.
• Forces obey the principle of superposition, combining by vector addition into a single net force $\vec{F}_{\text{net}} = \sum \vec{F}$.
• Balanced forces produce a net force of zero $(\sum \vec{F} = 0)$, whereas unbalanced forces cause acceleration.