Topic 4

The Laws of Motion

The Laws of Motion

Two main factors need to be addressed to answer questions about why the
motion of an object will change.
§ Forces acting on the object
§ The mass of the object

Dynamics studies the causes of motion.

§Will start with three basic laws of motion
§ Formulated by Sir Isaac Newton
Force

Forces in everyday experience

§ Push on an object to move it
§ Throw or kick a ball
§ May push on an object and not be able to move it

Forces are what cause any change in the velocity of an object.

§ A force is that which causes an acceleration
Classes of Forces

Contact forces involve physical contact between two objects

§ Examples a, b, c
Field forces act through empty space
§ No physical contact is required
§ Examples d, e, f
More About Forces

A spring can be used to calibrate the magnitude of a force.

Doubling the force causes double the reading on the spring.
When both forces are applied, the reading is three times the initial reading.
Vector Nature of Forces

The forces are applied perpendicularly

to each other.
The resultant (or net) force is the
hypotenuse.
Forces are vectors, so you must use
the rules for vector addition to find the
resultant (net) force acting on an object.
Newton’s First Law

If an object does not interact with other objects, it is possible to identify a

reference frame in which the object has zero acceleration.
§ This is also called the law of inertia.
§ It defines a special set of reference frames called inertial frames.
§ We call this an inertial frame of reference.
Inertial Frames

Any reference frame that moves with constant velocity relative to an inertial frame
is itself an inertial frame.
If you accelerate relative to an object in an inertial frame, you are observing the
object from a non-inertial reference frame.

We can consider the Earth to be an inertial frame, although it has a small

centripetal acceleration associated with its motion.
Newton’s First Law – Alternative Statement

In the absence of external forces, when viewed from an inertial reference frame,
an object at rest remains at rest and an object in motion continues in motion with
a constant velocity.
§ Newton’s First Law describes what happens in the absence of a force.
§ either at rest or moving at a constant velocity
§ Also tells us that when no force acts on an object, the acceleration of the
object is zero
The First Law also allows the definition of force as that which causes a change
in the motion of an object.
Inertia and Mass
The tendency of an object to resist any attempt to change its velocity is
called inertia.

Mass is that property of an object that specifies how much resistance an

object exhibits to changes in its velocity.
§ Mass is a scalar quantity.
§ The SI unit of mass is kg.

Masses can be defined in terms of the accelerations produced by a given

force acting on them:

m1 a2
º
m2 a1
§ The magnitude of the acceleration acting on an object is inversely
proportional to its mass.
Mass vs. Weight

Mass and weight are two different quantities.

Weight is equal to the magnitude of the gravitational force exerted on the object.
§ Weight will vary with location (due to different g value).
Mass is the same everywhere.
Newton’s Second Law
When viewed from an inertial reference frame, the acceleration of an object is
directly proportional to the net force acting on it and inversely proportional to its
mass.
§ Force is the cause of changes in motion, as measured by the acceleration.
§ Remember, an object can have motion (with constant velocity) in the absence of
forces.
§ Do not interpret force as the cause of motion (force is the cause of changes in
motion).

Algebraically,

∑ 𝐅⃗ = 𝑚𝐚 (Eq. 1)

§ This equation is valid only when the speed of the object is much less than
the speed of light.
More About Newton’s Second Law
!
å F is the resultant force (net force)
§ This is the vector sum of all the forces acting on the object.

Newton’s Second Law can be expressed in terms of components:

§ SFx = m ax
§ SFy = m ay
§ SFz = m az

The SI unit of force is the newton (N).

§ 1 N = 1 kg·m / s2
Example 1
Example 1
Gravitational Force
!
The gravitational force, Fg , is the force that the earth exerts on an object.
This force is directed toward the center of the earth.
From!Newton’s Second Law:
!
§ Fg = mg (Eq. 2)
Its magnitude is called the weight of the object.
§ Weight = Fg= mg (Eq. 3)

Because the weight is dependent on g, the weight varies with location.

§ g, and therefore the weight, is less at higher altitudes.
§ This can be extended to other planets, but the value of g varies from planet
to planet, so the object’s weight will vary from planet to planet.
 Newton’s Third Law
!
If two objects interact, the force F12 exerted by object 1 on object 2 is equal in
!
magnitude and opposite in direction to the force F exerted by object 2 on object 1.
21
! !
§ F12 = -F21 (Eq. 4)
!
§ Note on notation: FAB is the force exerted by A on B.
Newton’s Third Law, Alternative Statement
The action force is equal in magnitude to the reaction force and opposite in
direction.
§ One of the forces is the action force, the other is the reaction force.
§ It doesn’t matter which is considered the action and which the reaction.
§ The action and reaction forces must act on different objects and be of the
same type.
Action-Reaction Example
The normal force (table on monitor) is
the reaction of the force the monitor
exerts on the table.
§ Normal means perpendicular, in
this case
The action (Earth on monitor) force is
equal in magnitude and opposite in
direction to the reaction force, the force
the monitor exerts on the Earth.

Figure (b) shows two forces that act on

the monitor: the normal force and the
force of gravity.
 Free Body Diagram

The most important step in solving problems involving

Newton’s Laws is to draw the free body diagram.
§Be sure to include only the forces acting on the object of
interest.
§The particle model is used by representing the object as a
dot in the free body diagram.
§The forces that act on the object are shown as being
applied to the dot.
The free body helps isolate only those forces acting on the
object and eliminate the other forces from the analysis.
Two Analysis Models Using Newton’s Second Law
Assumptions
§ Objects can be modeled as particles.
§ Interested only in the external forces acting on the object
§ Can neglect reaction forces
§ Initially dealing with frictionless surfaces
§ Masses of strings or ropes are negligible.
§ The force the rope exerts is away from the object and parallel to the rope.
§ When a rope attached to an object is pulling it, the magnitude of that force is
the tension in the rope.
1st Analysis Model: The Particle in Equilibrium

If the acceleration of an object that can be modeled as a particle is zero, the

object is said to be in equilibrium.
§ The model is the particle in equilibrium model.
Mathematically, the net force acting on the object is zero.

!
åF = 0
åF x = 0 and åF
y =0
Example 2
Example 2
2nd Analysis Model: The Particle Under a Net Force

If an object that can be modeled as a particle experiences an acceleration, there

must be a nonzero net force acting on it.
§ Model is particle under a net force model .
Draw a free-body diagram.
Apply Newton’s Second Law in component form.

! 𝐅⃗ = 𝑚𝐚

! 𝐹! = 𝑚𝑎!

! 𝐹" = 𝑚𝑎"
Newton’s Second Law, Example

Forces acting on the crate:

§ A tension, acting through the rope,
is the magnitude of force
§ The gravitational force,
§ The normal force exerted by the
floor

Apply Newton’s Second Law in component

form:
åF = T = ma
x x

åF y = n - Fg = 0 ® n = Fg
Solve for the unknown(s)
If the tension is constant, then a in x
direction is a constant and the kinematic
equations can be used to more fully
describe the motion of the crate.
Note About the Normal Force

The normal force is not always equal to

the gravitational force of the object.
For example, in this case
åF y = n - Fg - F = 0
and n = mg + F
! !
n may also be less than Fg
Problem-Solving Hints – Applying Newton’s Laws
Conceptualize
§ Draw a diagram
§ Choose a convenient coordinate system for each object
Categorize
§ Is the model a particle in equilibrium?
§ If so, SF = 0
§ Is the model a particle under a net force?
§ If so, SF = m a
Problem-Solving Hints – Applying Newton’s Laws, cont.
Analyze
§ Draw free-body diagrams for each object
§ Include only forces acting on the object
§ Find components along the coordinate axes
§ Be sure units are consistent
§ Apply the appropriate equation(s) in component form
§ Solve for the unknown(s)
Finalize
§ Check your results for consistency with your free-body diagram
§ Check extreme values
Inclined Planes
Categorize as a particle under a net
force since it accelerates.
Forces acting on the object:
§ The normal force acts
perpendicular to the plane.
§ The gravitational force acts straight
down.
Choose the coordinate system with x
along the incline and y perpendicular to
the incline.
Replace the force of gravity with its
components.
Apply the model of a particle under a
net force to the x-direction and a
particle in equilibrium to the y-direction.
 Example 3

A car of mass m is on an icy driveway

inclined at an angle 𝜃 as shown in this
figure.
(a) Find the acceleration of the car,
assuming the driveway is frictionless
(b) Suppose the car is released from rest
at the top of the incline and the
distance from the car’s front bumper
to the bottom of the incline is d. How
long does it take the front bumper to
reach the bottom of the hill, and what
is car’s speed as it arrives there?
Example 3
(a)
∑ 𝐹! = 𝑚𝑔 sin 𝜃 = 𝑚𝑎! (eq. 1)
∑ 𝐹" = 𝑛 − 𝑚𝑔 cos 𝜃 = 0 (eq. 2)
Solving eq. 1 for ax, 𝑎! = 𝑔 sin 𝜃

(b)
Defining the initial position of the front bumper as 𝑥# = 0 and its final position as
𝑥$ = 𝑑, and recognizing that 𝑣!# = 0, choose the particle under constant
%
acceleration model, 𝑥$ = 𝑥# + 𝑣!# 𝑡 + 𝑎! 𝑡 & , then we will get
&
% &' &'
𝑑 = 𝑎! 𝑡 & , solving for t, we get 𝑡 = =
& (! ) *+, -

Using 𝑣!$ & = 𝑣!# & + 2𝑎! (𝑥$ − 𝑥# ) with 𝑣!# = 0 to find the final velocity of the car
𝑣!$ & = 2𝑎! 𝑑

𝑣!$ = 2𝑎! 𝑑 = 2𝑔𝑑 sin 𝜃

Multiple Objects

When two or more objects are connected or in contact, Newton’s laws may be
applied to the system as a whole and/or to each individual object.
Whichever you use to solve the problem, the other approach can be used as a
check.
Multiple Objects – Atwood’s Machine
When two object of unequal mass are
hang vertically over a frictionless pulley
of negligible mass as in this figure, it is
called an Atwood machine.

Forces acting on the objects:

§ Tension (same for both objects,
one string)
§ Gravitational force
Each object has the same acceleration
since they are connected.
Draw the free-body diagrams
Apply Newton’s Laws
Solve for the unknown(s)
 Example 4

Determine the magnitude of the

acceleration of the two objects and
the tension in the lightweight string.

• Note the acceleration is the same for both

objects
• The tension is the same on both sides of
the pulley as long as you assume a
massless, frictionless pulley.
Example 4
Forces of Friction

When an object is in motion on a surface or through a viscous medium, there will

be a resistance to the motion.
§ This is due to the interactions between the object and its environment.
This resistance is called the force of friction.
Static Friction

Static friction acts to keep the

object from moving.
As long as the object is not
moving, ƒs = F
! !
If F increases, so does ƒs
! !
If F decreases, so does ƒs
ƒs £ µs n (Eq. 5)
§ Remember, the equality holds
when the surfaces are on the
verge of slipping.
Kinetic Friction

The force of kinetic friction acts when

the object is in motion.
Although µk can vary with speed, we
shall neglect any such variations.
ƒk = µk n (Eq. 6)
Explore Forces of Friction

Vary the applied force

§The net force F - fk in the x direction
produces an acceleration to the right,
according to Newton’s second law.
§If F = fk, the acceleration is zero and
the trash can moves to the right with
constant speed.

§If the applied force 𝐅⃗ is removed from

the moving can, the friction force 𝐟𝐤
acting to the left provides an
acceleration of the trash can in the –x
direction and eventually brings it to
rest.
Forces of Friction, summary.

Friction is proportional to the normal force.

§ ƒs £ µsn and ƒk= µk n
§ μ is the coefficient of friction; n is the magnitude of the normal force
§ These equations relate the magnitudes of the forces; they are not vector
equations.
§ For static friction, the equals sign is valid only at impeding motion, the
surfaces are on the verge of slipping.

The direction of the frictional force is opposite the direction of motion and parallel
to the surfaces in contact.
!
Friction is a force, so it simply is included in the å F in Newton’s Laws.
The rules of friction allow you to determine the direction and magnitude of the
force of friction.
Friction Example to Determine the Coefficient of Friction
The block is sliding down the plane, so
friction acts up the plane.
! 𝐹! = 𝑚𝑔 sin 𝜃 − 𝑓/ = 0

! 𝐹" = 𝑛 − 𝑚𝑔 cos 𝜃 = 0
𝑛
𝑓/ = 𝑚𝑔 sin 𝜃 = sin 𝜃 = 𝑛 tan 𝜃
cos 𝜃

ƒs £ µsn and ƒk= µkn

𝜇𝑛 = 𝑛 tan 𝜃
𝜇 = tan 𝜃

For µs, use the angle where the block just slips.
For µk, use the angle where the block slides down at a constant speed.
 Example 5

A hockey puck on a frozen pond is given an

initial speed of 20.0 m/s. if the puck always
remains on the ice and slides 115 m before
coming to rest, determine the coefficient of
kinetic friction between the puck and ice.

• Draw the free-body diagram, including the

force of kinetic friction.
§ Opposes the motion
§ Is parallel to the surfaces in contact
• Continue with the solution as with any
Newton’s Law problem.
• This example gives information about the
motion which can be used to find the
acceleration to use in Newton’s Laws.
Example 5
Analysis Model Summary

Particle under a net force

§ If a particle experiences a non-zero net force, its acceleration is related to
the force by Newton’s Second Law.
§ May also include using a particle under constant acceleration model to relate
force and kinematic information.
Particle in equilibrium
§ If a particle maintains a constant velocity (including a value of zero), the
forces on the particle balance and Newton’s Second Law becomes.
!
åF = 0