Nicholas J. Giordano
www.cengage.com/physics/giordano
Forces and Motion in Two and Three
Dimensions
Newton’s Laws and Motion
• We will extend Newton’s Laws to multiple dimensions as the
foundation for explaining motion
• We can extend ideas of motion to two- and three-dimensional
cases
• Still interested in displacement, velocity and acceleration
• Must allow for the vector addition of many quantities
• Force, acceleration, velocity, and displacement
• Express directions in terms of a chosen coordinate system
• Apply the same principles and problem-solving techniques as
used with one-dimensional motion
Introduction
Using Newton’s Second Law
• Newton’s Second Law in vector form states
• Generally start by determining all the individual forces
•
•
•
•
acting on the object
Construct a free body diagram
Add the individual forces as vectors
Use Newton’s Second Law to find the acceleration
Once the acceleration is found, it can be used to determine
velocity and displacement
Section 4.1
Statics
• Statics
• Deals with objects at rest
• We will look at the conditions for translational equilibrium
• Equilibrium examples use the same approach as the one-
dimensional problems previously studied
• Friction may need to be included in equilibrium problems
Section 4.1
Statics and Equilibrium
• Statics is an area of mechanics dealing with problems in
which both the velocity and acceleration are zero
• The object is also said to be in translational equilibrium
• Often the “translational” is dropped
• If the acceleration is zero, then
• This is the condition for translational equilibrium
Section 4.1
Equilibrium Example: Refrigerator
• Four forces act on the
refrigerator:
• Gravity and normal force
in y-direction
• Force exerted by person
(push) and static friction
in x-direction
• Draw a free body diagram
(b) and include a
coordinate system
Section 4.1
Refrigerator Example, cont.
• Express the forces in terms of their x- and y-components
• In this case, the forces are aligned along the axes
• Apply the condition for equilibrium:
• ΣFx = 0 and ΣFy = 0
• For y-direction: N - m g = 0
• For x-direction: Fpush - Ffriction = 0
Section 4.1
Equilibrium Example: Sled
• All the forces do not all align with the x- or y-axes
• Find the x- and y-components of all forces that are not on an axis
• Applying Newton’s Second Law:
• ΣFx = Tx – Ffriction = T cos θ – Ffriction = 0
• ΣFy = N – mg + Ty = N – mg + T sin θ = 0
Equilibrium Example:
Tightrope Walker
• Both sections of the rope
exert a tension force at the
center where the walker is
standing
• The walker and the rope
are at rest
• The forces acting at the
center are shown in the
free body diagram
• Tension forces on the
right and on the left
• Weight of the walker
Section 4.1
Tightrope Walker Example, cont.
• Choose the usual x-y coordinate axes along the horizontal
and vertical directions
• Express all the forces in terms of the x- and y-components
• The tensions in both sides of the rope are equal
• Solve for the unknown quantities
Section 4.1
Problem Solving Strategy for Statics Problems
• Recognize the principle
• For static equilibrium, the sum of the forces must be zero
• Use
• Sketch the problem
• Show the given information in the picture
• Include a coordinate system
• Identify the relationships
• Use all the forces to construct a free body diagram
• Express all the forces on the object in terms of their x- and
y-components
• Apply ΣFx = 0 and ΣFy = 0
• May also include ΣFz = 0
Section 4.1
Problem Solving Strategy for Statics Problems,
cont.
• Solve
• Solve all the equations
• The number of equations must equal the number of
unknown quantities
• Check
• Consider what your answer means
• Check that your answer makes sense
Section 4.1
Inclines (Hills)
• The normal force (N) acts
perpendicular to the incline
(plane)
• The friction force acts up the
incline
• The motion would tend to
be down the incline
• Friction opposes the motion
• The force due to gravity acts
straight down
• These forces compose your
free body diagram
Section 4.1
Inclines, cont.
• Choose a coordinate system
• Choose axes parallel and perpendicular to the incline
• Less components
• If acceleration is present, the acceleration would be along
the incline
• Find the components of the gravitational force
• The rest of the forces are along the axes
• The normal force is not equal to mg
• The value of N depends on the angle of the hill
Section 4.1
Angle of Incline To Not Slip
• Analysis of the problem indicates the minimum frictional
force to keep the object from slipping is
• Ffriction = m g sin θ
• Since this is static friction, Ffriction ≤ μstatic N
• Assuming it is just in equilibrium (so Ffriction =
μstatic N), the angle of the incline at which the object is on
the verge of slipping is tan θ = μs
Section 4.1
Equilibrium Example: Flag
• Determine the tension and
angle
• Two unknowns, so need to
look at two dimensions
• Draw the free body diagram
• Choose horizontal and
vertical directions for your
coordinate system
• Tension has x- and ycomponents
Section 4.1
Flag Example, cont.
• Write the equations for equilibrium in the x- and y-
directions
• Solve for the unknown quantities
• Check to be sure the answers make sense
Section 4.1
Equilibrium in Three Dimensions
• Many situations can be simplified by choosing the x-y
plane to match the geometry of the problem
• If three dimensions are necessary, the same basic approach
is used
• Include a corresponding relation for the z-direction:
• ΣFz = 0
Section 4.1
Projectile Motion
• Consider objects in motion
and the forces acting on
them
• Projectile motion is one
example of this type of
motion
• We will ignore the force
from air drag
• For now
• Components of gravity are
Fgrav, x = 0, Fgrav, y = - m g
Section 4.2
Projectile Motion, cont.
• Accelerations are ax = 0 and ay = - g
• The acceleration in both x- and y-directions is constant
• The motions in the x- and y-directions are independent of each
other
• The motion in the x-direction is constant acceleration with
a=0
• The motion in the y-direction is constant acceleration with
a=-g
•
This is free fall
• The relationships among displacement, velocity, acceleration,
and time for constant acceleration apply directly to projectile
motion
Section 4.2
Projectile Motion Example:
Rolling off a Cliff
• The car rolls off the cliff
• Its initial velocity is
directed along the
horizontal
• Choose the coordinate
system to be the horizontal
and vertical directions
• Apply the relations for
motion with constant
acceleration
Section 4.2
Independence of x- and y- motions
• The time it takes for the
object to reach the ground
is independent of its
motion along the xdirection
• Here, the two balls strike
the ground at the same
time
Section 4.2
Projectile Motion Example:
Shooting at a Target
• If the rifle is fired horizontally, the bullet misses the target
• The bullet falls a distance Δy while traveling between the rifle
and the target
• To compensate, the rifle is aimed above the target
• The value of Δy depends on the distance to the target and the
speed of the bullet
Section 4.2
Projectile Motion Example:
Baseball Throw
• The baseball has an initial
position of x = 0 and y = h
• h is the height of the ball
when it leaves the bat
• The ball is hit with an initial
velocity of vo at an angle of
θ above the horizontal
• vox = vo cos θ
• voy = vo sin θ
Section 4.2
Baseball Example: Trajectory
• To find the trajectory of the
ball, you need the x- and ycomponents of its motion
x = vo (cos θ) t
• y = h + vo (sin θ) t – ½ g t2
• The graphs show x and y as
functions of time
•
Section 4.2
Baseball Example: Velocity
• To describe the motion, the
velocity components as functions
of time are also needed
• vx = vo cos θ
• vy = vo sin θ – g t
• The plots show the velocities as
functions of time
• Note that vx is constant
• vy varies linearly with time
•
The slope is -g
Section 4.2
Baseball Example: Velocity, cont.
• The graph shows the
components of the
velocity at various points
along the trajectory
• The total velocity at any
point is the vector sum of
its components:
v = v x2 + v y2
Section 4.2
Projectile Motion, Final Notes
• For a symmetrical trajectory:
• tlands = 2 tto top
• Speed at landing is equal to its initial speed
• The range of the projectile is the horizontal distance it
travels
• Applies only if air drag is negligible
• Applies only if the motion is symmetrical
• Maximum range will occur at θ = 45°
Section 4.2
Reference Frames
• A reference frame is an
observer’s choice of
coordinate system for
making measurements
• It will include an origin
• Newton’s Laws give a
correct description of the
motion in any reference
frame that moves with a
constant velocity
Section 4.3
Relative Velocity
• The two cars are traveling at constant velocities along
the x-direction
• For an observer on the sidewalk, the cars have
velocities
Relative Velocity, cont.
• Consider a reference frame at rest with respect to car 1
• The reference frame is defined by x’ and y’ in the previous
figure
• The velocity of car 2 with respect to car 1 is
• The velocity of car 2 relative to the observer = velocity of
car 2 relative to car 1 + velocity of car 1 with respect to the
observer
•
This gives the general way velocities in different reference frames
are related
• The same ideas apply to motion in two dimensions
• Apply the ideas to each component
Section 4.3
Further Applications of Newton’s Laws
• There are many forces that can play a role in Newton’s
Laws
• Since we are dealing with two-dimensional problems,
directions can also be found
• In terms of angles
• Examples will show how to apply the general problem-
solving strategy to specific instances
Section 4.4
Newton’s Second Law –
General Problem Solving Strategy
• Recognize the principle
• Consider all the forces acting on the object
• Compute the total force
•
•
If in equilibrium, the total force must be zero
If the object is not in equilibrium, set the total force equal to ma
• Sketch the problem
• Define a coordinate system to include in your sketch
• It also needs to contain all the forces in the problem
• Include all the given information
Section 4.4
Newton’s Second Law –
General Problem Solving Strategy, cont.
• Identify the relationships
• Construct the free body diagram
• Express all the forces in their components along x and y
• Apply Newton’s Second Law in component form
•
ΣFx = m ax and ΣFy = m ay
• If the acceleration is constant along the x- or y-axis or both,
apply the kinematic equations
• Solve
• Solve the equations for all the unknown quantities
• The number of equations must equal the number of
unknown quantities
Section 4.4
Newton’s Second Law –
General Problem Solving Strategy, final
• Check
• Consider what your answer means
• Be sure the answer makes sense
Section 4.4
Traveling Down a Hill
• Assume a frictionless
surface
• Draw the free body
diagram
• The force of gravity has xand y-components
• Looking at the x-direction,
along the hill, gives
• ax = g sin θ
Section 4.4a
Traveling Down a Hill with Friction
• The forces acting on the
sled are
• The normal force exerted
by the road
• The force of gravity
• The force of friction
between the sled and the
hill
•
Use kinetic friction since the
sled is slipping relative to
the hill
Section 4.4
Newton’s Second Law Example:
Pulleys and Cables
• Assume a massless cable
• Therefore the tension is the
same in all points of the
cable
• Choose the x-direction
parallel to the string
• The “positive” direction
will be opposite for the two
crates
• Apply Newton’s Second
Law to each crate
• Solve the equations
Section 4.4
Newton’s Second Law Example:
Cables and Pulleys 2
• Apply general problem-
solving strategy
• Draw free body diagrams
of each object
• The x-direction follows
the string
• Friction could be included
• See example 4.12
Section 4.4
Accelerometer
• An accelerometer is a device
that measures acceleration
• A rock hanging from a string
can act as an accelerometer
• When the airplane moves
with a constant velocity, the
string hangs vertically
• The two forces add to zero
• T=mg
Section 4.5
Accelerometer, cont.
• When the airplane
accelerates, the string will
hang at an angle θ
• Now T cos θ = m g and ax
= g tan θ
• By measuring the angle,
you can determine the
acceleration
Section 4.5
Ear as an Accelerometer
• When your head accelerates, the gelatinous layer lags behind a
small amount
• Very similar to the rock on the string
• This lag causes the hair cells to deflect
• The hair cells send signals to the brain
• The brain interprets the acceleration
Section 4.5
Inertial Reference Frames
• Analyze the motion of the accelerometer in the plane from
inside the plane
• If the acceleration is zero, the string hangs straight down
• Agrees with the observation of the observer outside the plane
• If the acceleration is nonzero to an observer outside the plane,
he does not agree with the observer inside the plane
• The inside observer would observe the angle of the
accelerometer, but would calculate an acceleration of zero
• Newton’s Second Law can only be applied in inertial reference
frames
• Nonaccelerating frames
Section 4.5
Projectile Motion with Air Drag
• With air drag, the maximum range no longer occurs at
45°
• For a baseball, the maximum range occurs when the ball is
projected at an angle of approximately 35°
• Also depends on the initial speed of the ball
• Artillery guns are usually aimed much higher than 45°
• They travel high enough to reach altitudes where the air is
not as dense and so drag is reduced
• A bicycle coasting down a hill reaches a terminal velocity
that depends on the angle of the hill and the density of the
air
Section 4.6
Air Drag and a Bicycle
• Assume a frictionless bicycle
• Forces acting on the bicycle
along the incline are gravity
and air drag
• The terminal (coasting)
velocity of the bicycle will
depend on
• The mass
• The angle of the hill
(incline)
• The density of the air
• The frontal area of the
cyclist
Section 4.6
Newton’s Second Law in
Three Dimensions
• The examples have shown applications of Newton’s
Second Law in two dimensions:
• ∑Fx = m ax and ∑Fy = m ay
• This allows you look at forces and accelerations in both
directions
• To deal with a problem that involves three dimensions,
add a third equation to the two above
• ∑Fz = m az
• The basic approach is the same as in two dimensions