Chapter 04 : Newton's Laws

Force vs Acceleration

Units:

[acceleration units] = [force units]/[mass]
[force units] = [mass]*[acceleration units]
MKS : kg·m/s² = Newtons (N)
English units : 1 pound (lb) = 4.44822 N or 1 N = 0.2248 lb
CGS : g·cm/s² = dyne (N)

Multiple Forces

\[ \sum \vec{F}_i = m\vec{a} \]

Useful to determine an unknown force.
A single force can be replaced with other forces (aligned with your coordinate system) that add up to that original force.

Newton's Laws

First Law
If there is no net force on an object, it will continue to move with a constant velocity (which could be zero but doesn't have to be).
\[ \sum \vec{F}_i = 0 \hspace{1em} implies \hspace{1em} \vec{a}=0 \]

\[ \vec{a}=0 \hspace{1em} implies \hspace{1em} \sum \vec{F}_i = 0 \]

Second Law
If there is a net force on an object that it will cause that object to accelerate according to \( \sum \vec{F}_i = m\vec{a} \)
Note the picky wording there: The forces acting on an object are what produce the acceleration of that object.
Other forces may be present elsewhere, and may indirectly work their way onto the object, but ultimately it's the forces acting directly on the object that are all it `knows' about.
(Will become important when multiple objects are involved.)

Third Law
All forces are interaction forces. If object A exerts a force on B, then B exerts an equal and opposite force on A: same magnitude, opposite direction.

Gravity

Objects with mass attract one another. We'll see the full version of this in chapter 6 but near the surface of the Earth (or any planet/moon/etc) this can be approximated as:
\[ \vec{F}_g = m\vec{g} \]

Apply Newton's Laws to an object of mass m:
\[ \sum \vec{F}_i = m\vec{a} \]

\[ m\vec{g} = m\vec{a} \]

\[ \vec{g} = \vec{a} \]

The object will accelerate downward with |a|=|g| (independent of its mass)

Normal Force

Consider object sitting on table: gravity is still present, but the object isn't accelerating so there must be another force present.
This contact force or normal force prevents objects from passing through each other. (Ultimately it's an electric force down at the atomic scale.)

The name normal force refers to the fact that it's direction is normal (i.e. perpendicular) to the contact surface between the two objects. It just prevents things from moving through each other.

Object on Scale: Apparent Weight

ELEVATOR AT REST

A 60 kg person is standing on a scale, in an elevator, at rest.

(a) What will the scale read?

ELEVATOR ACCELERATING UPWARD

The same person is in an elevator that is accelerating upward at 2 m/s².

(a) What will the scale read?

Remember: \( \vec{a} = d\vec{v}/dt \) so the sign of the acceleration is related to changes in the velocity, not the velocity itself. This particular elevator might be:

Initially at rest, just starting to move upward.
Already moving upward and now moving up faster and faster.
Presently moving downward but slowing down, coming to the next lower floor perhaps.

Applying Newton's Laws to an Object

Scenario Forces Present Free-body Appropriate Coordinates

• Problem Statement : visualize (sketch) what's going on.
• Select the OBJECT to which we'll apply Newton's Laws.
• Note ALL forces acting on THAT object. (Gravity? Normal force? Others?)
• Optional free-body diagram version
• Select appropriate coordinate system
• Finally, apply Newton's Laws in each coordinate direction:
\( \sum F_x = ma_x \) \( \sum F_y = ma_y \) etc

Example: Pushing Crate

A 100 kg crate is being pushed across the floor with a force of 200 N directed 30^o below the horizontal.
Determine:
(a) the acceleration of the crate
(b) the magnitude of the normal force

Example: Object Sliding Down Incline

An object of unknown mass M is placed (at rest) on a 30^o incline.
It slides down the incline until it reaches the floor, travelling a distance of 2 m along the incline.

(a) How fast is the object moving when it reaches the bottom of the ramp?

Example: Object on Incline Held in Place

An object of mass M=100 kg is placed (at rest) on a 40^o ramp.

If we want to hold the object in place by pushing on it horizontally as shown in the figure, how much force do we need to apply?
What will the normal force be in this situation?

Force vs Acceleration
Units: [acceleration units] = [force units]/[mass] *[force units] = [mass][acceleration units] MKS : kg·m/s² = Newtons (N) English units : 1 pound (lb) = 4.44822 N or 1 N = 0.2248 lb CGS : g·cm/s² = dyne (N)**
Multiple Forces
\[ \sum \vec{F}_i = m\vec{a} \] Useful to determine an unknown force. A single force can be replaced with other forces (aligned with your coordinate system) that add up to that original force.
Newton's Laws
First Law If there is no net force on an object, it will continue to move with a constant velocity (which could be zero but doesn't have to be). \[ \sum \vec{F}_i = 0 \hspace{1em} implies \hspace{1em} \vec{a}=0 \] \[ \vec{a}=0 \hspace{1em} implies \hspace{1em} \sum \vec{F}_i = 0 \]
Second Law If there is a net force on an object that it will cause that object to accelerate according to \( \sum \vec{F}_i = m\vec{a} \) Note the picky wording there: The forces acting on an object are what produce the acceleration of that object. Other forces may be present elsewhere, and may indirectly work their way onto the object, but ultimately it's the forces acting directly on the object that are all it `knows' about. (Will become important when multiple objects are involved.)
Third Law All forces are interaction forces. If object A exerts a force on B, then B exerts an equal and opposite force on A: same magnitude, opposite direction.