Compression or Normal Force

The Main Idea

The compression force, most commonly known as the normal force, [math]\displaystyle{ F_N }[/math], is a simple fundamental concept that must be understood before attempting any contact-force problems. First, it is important to understand that the normal force is NOT a kind of fundamental force, such as the electric or gravitational force. It is a force used to describe the interactions between atoms in contact. As hinted by the name, this force simply points in the perpendicular/normal/orthogonal direction to the surface(s) that the object is in contact with. The magnitude of the normal force is often equal to the weight of the object ([math]\displaystyle{ F_g }[/math]), however, objects can move along surfaces with various angles. This results in the normal force being present at an angle to the horizontal. Understanding information about the normal force on an object can also help to finding the magnitude of other forces present in a system, such as the force of friction.

A Mathematical Model

Since the normal force is completely dependent upon on how much force the object is exerting on the surface, there is no single equation to describe it. However, there a 3 main cases that will likely come up:

1) An object lays on a flat surface with no forces pushing up or down on the object, besides gravity:

The normal force ([math]\displaystyle{ F_N }[/math]) will be equal in magnitude and opposite in direction of the weight or gravitational force ([math]\displaystyle{ F_g }[/math]) on the object:

[math]\displaystyle{ F_N = F_g = Mg }[/math]

[math]\displaystyle{ M }[/math] is the mass of the object and [math]\displaystyle{ g }[/math] is the acceleration on the object due to gravity.

2) An object lies on an angled surface with no forces pushing up or down on it, besides gravity:

The normal force [math]\displaystyle{ F_N }[/math] will be equal to the magnitude of the vertical component of the gravitational force ([math]\displaystyle{ F_{g_y} }[/math]) on the object:

[math]\displaystyle{ F_N = F_g \ \text{cos}(\theta) = Mg \times \text{cos}(\theta) }[/math]

[math]\displaystyle{ M }[/math] is the mass of the object, [math]\displaystyle{ g }[/math] is the acceleration on the object due to gravity, and [math]\displaystyle{ \theta }[/math] is the angle between the horizontal and the angled surface.

3) An object lies on a surface, which may be angled (a) or not (b), with a force pushing down on the object:

a) An object lays on an angled surface with a force, such as an applied force, pushing down on the object. The normal force [math]\displaystyle{ F_N }[/math] will be equal to the vertical component of the gravitational force on the object plus the component of the extra force that pushes straight down (if the extra force is angled, so that it is not perfectly vertical, only the vertical component will increase the normal force):

[math]\displaystyle{ F_N = F_g \ \text{cos}(\theta) + F = Mg \times \text{cos}(\theta) + F }[/math]

[math]\displaystyle{ M }[/math] is the mass of the object, [math]\displaystyle{ g }[/math] is the acceleration on the object due to gravity, [math]\displaystyle{ \theta }[/math] is the angle between the horizontal and the angled surface, and [math]\displaystyle{ F }[/math] is the extra force pushing down on the object.

---or---

[math]\displaystyle{ F_N = F_g \ \text{cos}(\theta) + F \ \text{sin}(\phi) = Mg \times \text{cos}(\theta) + F \times \text{sin}(\phi) }[/math]

[math]\displaystyle{ M }[/math] is the mass of the object, [math]\displaystyle{ g }[/math] is the acceleration on the object due to gravity, [math]\displaystyle{ \theta }[/math] is the angle between the horizontal and the angled surface, [math]\displaystyle{ F }[/math] is the extra force pushing down on the object, and [math]\displaystyle{ \phi }[/math] is the angle between the horizontal and the extra force [math]\displaystyle{ F }[/math].

b) An object lays on a flat surface with a force, such as an applied force, pushing down on the object. The normal force [math]\displaystyle{ F_N }[/math] will be equal to the magnitude of the gravitational force on the object plus the component of the extra force that pushes straight down (if the extra force is angled, so that it is not perfectly vertical, only the vertical component will increase the normal force):

[math]\displaystyle{ F_N = F_g + F = Mg + F }[/math]

[math]\displaystyle{ M }[/math] is the mass of the object, [math]\displaystyle{ g }[/math] is the acceleration on the object due to gravity, and [math]\displaystyle{ F }[/math] is the extra force pushing down on the object.

---or---

[math]\displaystyle{ F_N = F_g + F \ \text{sin}(\phi) = Mg + F \times \text{sin}(\phi) }[/math]

[math]\displaystyle{ M }[/math] is the mass of the object, [math]\displaystyle{ g }[/math] is the acceleration on the object due to gravity, [math]\displaystyle{ F }[/math] is the extra force pushing down on the object, and [math]\displaystyle{ \phi }[/math] is the angle between the horizontal and the extra force [math]\displaystyle{ F }[/math].

It is useful to note the frictional force ([math]\displaystyle{ F_f }[/math]) is defined in terms of the normal force ([math]\displaystyle{ F_N }[/math]) by:

[math]\displaystyle{ F_f = \mu F_N }[/math]

[math]\displaystyle{ \mu }[/math] is the characteristic coefficient of friction between the two materials in contact.

A Computational Model

from __future__ import division

from visual import *

from visual.graph import *

scene.title = "Normal Force"

scene.background = color.black

scene.center = (0.7, 1, 0)

inclinedplane = box(pos = vector(0.6, 0, 0), size = (1.2, 0.02, 0.2), color = color.green, opacity = 0.5)

cart = box(size = (0.2, 0.06, 0.06), color = color.red)

trail = curve(color = color.yellow, radius = 0.01)

cart.m = 0.5

cart.pos = vector(0, 0.04, 0.08)

cart.v = vector(0,0,0)

theta = 25 * (pi / 180.0)

cart.v = norm(inclinedplane.axis)

cart.v.mag = 3

inclinedplane.rotate(angle = theta, origin = (0,0,0), axis = (0,0,1))

cart.rotate(angle = theta, origin = (0,0,0), axis = (0,0,1))

g = 9.8

M = cart.m

F = 100

t = 0

dt = 0.005

while cart.pos.y > 0.02:

   rate(1000)
   F_N = M * g * cos(theta)
   a = norm(inclinedplane.axis)
   a.mag = g * sin(theta)
   cart.v = cart.v + a*dt
   cart.pos = cart.pos + cart.v*dt
   trail.append(pos = cart.pos)

   t = t+dt
   print("Normal Force:", F_N)

Examples

The simplest normal force problem involves an object resting on a flat surface with only gravity and the normal force acting on the object. To raise the difficulty a little, one may position the object on an inclined plane, so that the normal force is now at an angle. Finally, other considerations such as incorporating a tension force or applied force could demonstrate other difficulties.

Simple

A [math]\displaystyle{ 7 \ \text{kg} }[/math] block rests on a flat table.

a) Find the normal force ([math]\displaystyle{ F_N }[/math]) that is being exerted on the block:

Since the block is at rest on a flat surface, we know the net force ([math]\displaystyle{ F_{net} }[/math]) must be [math]\displaystyle{ 0 }[/math]:

[math]\displaystyle{ F_{net} = \sum F = Ma = 0 }[/math]

We also know that only the normal force ([math]\displaystyle{ F_N }[/math]) and the gravitational force are acting on the block ([math]\displaystyle{ F_g }[/math]). Therefore, they must be equal in magnitude and opposite in direction. Since we know gravity is acting straight downwards, the normal force must act straight upwards.

Summing these forces gives:

[math]\displaystyle{ F_{net} = F_g - F_N = 0 }[/math]

Therefore:

[math]\displaystyle{ F_N = F_g = Mg = 7 \times 9.81 = 68.67 \ \text{Newtons} }[/math]

Middling

A [math]\displaystyle{ 14 \ \text{kg} }[/math] block is lying on an inclined plane that makes an angle [math]\displaystyle{ \theta \ \text{of} \ 48^\text{o} }[/math] with the horizontal.

a) Find the normal force ([math]\displaystyle{ F_N }[/math]) acting on the block:

First, we know the normal force ([math]\displaystyle{ F_N }[/math]) will be perpendicular to the inclined plane. Hence, it will be convenient to use a coordinate plane where the x-axis is parallel to the inclined plane, and the y-axis is perpendicular to the inclined plane. With this coordinate system, the normal force should be along the y-axis, which is convenient.

Second, the only force acting along the y-axis other than the normal force is the gravitational force ([math]\displaystyle{ F_g }[/math]). Therefore, if we can find the y-component of the gravitational force, we will be able to find the normal force.

Finally, since the block is at rest with respect to the y-axis, we know its acceleration along the y-axis ([math]\displaystyle{ a_y }[/math])must be [math]\displaystyle{ 0 }[/math].

We start with Newton's Second Law along the y-axis:

[math]\displaystyle{ F_{net} = \sum F_y = F_{g_y} - F_N = F_g \text{cos}(\theta) - F_N = Ma_y = 0 }[/math]

Therefore:

[math]\displaystyle{ F_N = F_g \times \text{cos}(\theta) = Mg \times \text{cos}(48^\text{o}) = 91.90 \ \text{Newtons} }[/math]

Difficult

In a pulley system, a block of mass [math]\displaystyle{ M }[/math] is hanging over the edge of an inclined plane that makes an angle [math]\displaystyle{ \alpha }[/math] with the horizontal. It is attached to another block of mass [math]\displaystyle{ m }[/math]. The block of mass [math]\displaystyle{ m }[/math] is sliding upwards on the ramp as the block of mass [math]\displaystyle{ M }[/math] is slowly moving downwards.

a) Draw a free body diagram for the block of mass [math]\displaystyle{ M }[/math] and the block of mass [math]\displaystyle{ m }[/math]:

b) Find the normal force ([math]\displaystyle{ F_N }[/math]) acting on the block of mass [math]\displaystyle{ m }[/math]:

We know the normal force ([math]\displaystyle{ F_N }[/math]) must be normal to the inclined plane. We can see from our free body diagram for the block of mass [math]\displaystyle{ m }[/math], the only forces working perpendicular to the inclined plane are a component of gravity and the normal force. Using this realization, Newton's Second Law, and the fact that [math]\displaystyle{ a_{\perp} }[/math] must equal [math]\displaystyle{ 0 }[/math], we can sum the forces perpendicular to the surface to find the normal force acting on the block of mass [math]\displaystyle{ m }[/math]:

[math]\displaystyle{ F_{\perp} = F_{g_{\perp}} - F_N = ma_{\perp} = 0 }[/math]

Therefore:

[math]\displaystyle{ F_N = F_{g_{\perp}} = mg \times \text{cos}(\alpha) }[/math]

Connectedness

The normal force is very crucial to our understanding of everyday life because of its endless necessity, application, and usefulness. It is the reason why gravity doesn't keep pulling us down and explains this not-so-apparent force that is acting on an object. Why can you place your textbooks on your desk and not have to worry about them falling through? The normal force. The application of the normal force can be seen in many industrial situations, where the weights of large objects must be watched. Also, it must be taken into account when drawing free body diagrams, such as for construction and architectural purposes.

History

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Sir Isaac Newton

The normal force is a direct application of Newton's Third Law of Motion. Sir Isaac Newton was a famous scientist from England who roamed the Earth from 1643 to 1727. His work in math and physics set the stage for many of the principles and theories that we have today. His main work, the Philosophiae Naturalis Principia Mathematic, discussed many of his theories about physics and stated his three laws of motion. The application of Newton's Third Law of Motion extends strongly into the modern era due to the high prevalence of the normal force in real world situations, such as landing a rocket on the moon.