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 y-components
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 x-direction
• 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 y-components 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:
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 x- and 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 = m g 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