Forces and Motion in Two and Three Dimensions

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