Download Chapter 19 Outline The First Law of Thermodynamics

Chapter 3 Outline
Motion in Two or Three Dimensions
• Position and velocity vectors
• Acceleration vectors
• Parallel and perpendicular components
• Projectile motion
• Uniform circular motion
• Relative velocity
Position Vector
• The position vector points
from the origin to point 𝑃,
the position of the object.
• We can express the vector in
terms of its 𝑥, 𝑦, and 𝑧
components.
𝒓 = 𝑥 𝒊 + 𝑦𝒋 + 𝑧 𝒌
• The position unit vector
gives the direction from the
origin to the object.
𝒓
𝒓=
𝑟
Velocity Vector
• The velocity vector is found from the
time derivative of the position vector.
𝑑𝒓
𝒗=
𝑑𝑡
• The velocity is tangent to the path at
each point.
• In component form:
𝑑𝒓 𝑑
𝒗=
=
𝑥 𝒊 + 𝑦𝒋 + 𝑧 𝒌
𝑑𝑡 𝑑𝑡
𝑑𝑥
𝑑𝑦
𝑑𝑧
𝒗=
𝒊+
𝒋+ 𝒌
𝑑𝑡
𝑑𝑡
𝑑𝑡
Velocity Vector
• The velocity in each direction is just
the time derivative of the coordinate
of that direction.
𝑑𝑥
𝑣𝑥 =
𝑑𝑡
𝑑𝑦
𝑣𝑦 =
𝑑𝑡
𝑑𝑧
𝑣𝑧 =
𝑑𝑡
• The magnitude of the velocity
(speed) is given by:
𝑣=
𝑣𝑥2 + 𝑣𝑦2 + 𝑣𝑧2
Acceleration Vector
• The acceleration vector is found from
the time derivative of the velocity
vector.
𝑑𝒗
𝒂=
𝑑𝑡
• While we might typically think of
acceleration as a change in speed, it
is very important that we understand
that it is a change in velocity.
• As we will discuss later in this chapter,
in uniform circular motion, the speed is
not changing, but the direction, and
therefore velocity is constantly changing.
Acceleration Vector
• In component form:
𝑑𝑣𝑦
𝑑𝑣𝑥
𝑑𝑣𝑧
𝒂=
𝒊+
𝒋+
𝒌
𝑑𝑡
𝑑𝑡
𝑑𝑡
• Or,
𝑑2 𝑥
𝑑2𝑦
𝑑2 𝑧
𝒂= 2𝒊+ 2𝒋+ 2𝒌
𝑑𝑡
𝑑𝑡
𝑑𝑡
Parallel and Perpendicular
Components of Acceleration
• We can resolve the acceleration into its
components parallel to the velocity
(along the path) and perpendicular to
the velocity.
• The parallel component, 𝑎∥ only changes
the magnitude of the velocity, its speed.
• The perpendicular component, 𝑎⊥ only
changes the direction of the velocity, so its
speed remains constant.
Projectile Motion
• Any body that is given an initial velocity and follows a
path determined solely by the effects of gravity and air
resistance is a projectile.
• The path the projectile follows is its trajectory.
• Initially, we will consider the simplest model in which we neglect
the effects of air resistance, and the curvature of the earth.
Projectile Motion
• While a projectile moves in
three-dimensional space, we
can always reduce the problem
to two dimensions by choosing
to work in the vertical 𝑥-𝑦
plane that contains the initial
velocity.
• We can simplify this further
by treating the 𝑥 and 𝑦
components separately.
• The vertical and horizontal
motions are independent
Projectile Motion
• In the ideal model, we only consider the force due to
gravity, so there is no acceleration in the 𝑥 direction.
𝑣𝑥 = 𝑣0𝑥
𝑥 = 𝑥0 + 𝑣0𝑥 𝑡
• In the 𝑦 direction, we have an acceleration due to gravity
of 𝑔 = 9.8 m/s 2 downward. For the following equations,
we will use a coordinate system in which up is positive.
𝑣𝑦 = 𝑣0𝑦 − 𝑔𝑡
𝑦 = 𝑦0 + 𝑣0𝑦 𝑡 − 12𝑔𝑡 2
• While not required, it is often simplest to set the origin as the
initial position of the projectile, so that 𝑥0 = 𝑦0 = 0 at 𝑡 = 0.
Projectile Motion
• We can represent the initial
velocity in terms of its
components to rewrite the
equations for position and
velocity.
• We have also taken the initial
position to be at the origin.
𝑥 = 𝑣0 cos 𝛼0 𝑡
𝑦 = 𝑣0 sin 𝛼0 𝑡 − 12𝑔𝑡 2
𝑣𝑥 = 𝑣0 cos 𝛼0
𝑣𝑦 = 𝑣0 sin 𝛼0 − 𝑔𝑡
Trajectory Shape
• The previous equations tell us the
position and velocity at each
time, but to see the shape of the
trajectory, we need to look at the
vertical position as a function of
horizontal position. (𝑦 𝑥 =?)
𝑡 = 𝑥/ 𝑣0 cos 𝛼0
𝑣0 sin 𝛼0 𝑥 1
𝑦=
− 2𝑔
𝑣0 cos 𝛼0
𝑥
𝑣0 cos 𝛼0
𝑔
2
𝑦 = tan 𝛼0 𝑥 − 2
𝑥
2𝑣0 cos 2 𝛼0
• Note that 𝑦 is a function of 𝑥 2 .
This gives rise to a parabola.
2
Projectile Motion Example #1
Projectile Motion Example #2
Projectile Motion Exam Question Example
• Some nice pirates realize that their friends are carelessly
sailing away without any cannonballs, so they decide to
launch one towards the ship. The forgetful sailors are
traveling directly away from the sailors at 15 m/s. The
pirates’ cannon has a muzzle velocity of 100 m/s, and it is
unfortunately stuck at an angle 70° above the horizontal.
If the pirates want to deliver the cannonball, they should
fire the cannon when the ship is how far away?
Circular Motion
• Changes in direction mean changes in velocity.
• Special case: Uniform circular motion
• Constant speed
• No tangential acceleration, only perpendicular
Uniform Circular Motion
• Centripetal acceleration, 𝑎rad
• Constant magnitude
• Pointing towards center
𝑎rad
𝑣2
=
𝑅
• Period, 𝑇
• Time for one full circle
• Circumference (2𝜋𝑅) divided by speed
2𝜋𝑅
𝑇=
𝑣
• Centripetal acceleration in terms of period
𝑎rad
4𝜋 2 𝑅
=
𝑇2
Nonuniform Circular Motion
• What if the speed is changing?
• Still have centripetal acceleration, 𝑎rad ,
towards the center.
• Also have tangential acceleration, 𝑎tan , along
the path.
𝑎tan
𝑑𝒗
𝑑𝑣
=
=
𝑑𝑡
𝑑𝑡
• Note that 𝑣 is the magnitude of the velocity.
•
𝑑𝒗
𝑑𝑡
is not the same as
𝑑𝒗
= 𝒂 =
𝑑𝑡
𝑑𝒗
𝑑𝑡
!
2
2
𝑎rad
+ 𝑎tan
Circular Motion Example
Relative Velocity
• When you are driving, and pass a car, it appears to be
moving backwards.
• Relative to the ground, it is still moving forward.
• Relative to you, a moving frame of reference, it is moving
backwards.
• What is the actual velocity?
• Any frame that is moving at a constant velocity is equally valid.
• This, along with the constant speed of light, is the basis of special
relativity.
• Generally, if we do not explicitly state a frame, we
measure relative to the ground.
Relative Position in One Dimension
𝑥𝑃/𝐴 = 𝑥𝑃/𝐵 + 𝑥𝐵/𝐴
• Notation:
• Object at position 𝑃.
• In coordinate system 𝐴, the
object is at 𝑥𝑃/𝐴 .
• In coordinate system 𝐵, the
object is at 𝑥𝑃/𝐵 .
• The origin of coordinate
system 𝐵, with respect to 𝐴,
is at 𝑥𝐵/𝐴 .
• Subscripts: object/frame
Relative Velocity in One Dimension
• Since velocity is the time
derivative of position,
𝑑𝑥𝑃/𝐴 𝑑𝑥𝑃/𝐵 𝑑𝑥𝐵/𝐴
=
+
𝑑𝑡
𝑑𝑡
𝑑𝑡
𝑣𝑃/𝐴−𝑥 = 𝑣𝑃/𝐵−𝑥 + 𝑣𝐵/𝐴−𝑥
• If 𝐴 and 𝐵 are any two
frames of reference,
𝑣𝐴/𝐵−𝑥 = −𝑣𝐵/𝐴−𝑥
Relative Velocity in Two or Three Dimensions
• We can extend this to two or three dimensions.
𝒓𝑃/𝐴 = 𝒓𝑃/𝐵 + 𝒓𝐵/𝐴
𝒗𝑃/𝐴 = 𝒗𝑃/𝐵 + 𝒗𝐵/𝐴
𝒗𝐴/𝐵 = −𝒗𝐵/𝐴
Relative Motion Example
Chapter 3 Summary
Motion in Two or Three Dimensions
• Position and velocity vectors
• 𝒓 = 𝑥 𝒊 + 𝑦 𝒋 + 𝑧𝒌
• 𝒗=
𝑑𝑥
𝑑𝑦
𝒊+ 𝒋
𝑑𝑡
𝑑𝑡
+
𝑑𝑧
𝒌
𝑑𝑡
• Acceleration vector: 𝒂 =
𝑑𝑣𝑥
𝒊
𝑑𝑡
+
𝑑𝑣𝑦
𝑑𝑡
𝒋+
• Parallel component changes speed.
• Perpendicular component changes direction.
• Treat each component separately.
𝑑𝑣𝑧
𝒌
𝑑𝑡
Chapter 3 Summary
Motion in Two or Three Dimensions
• Projectile motion
• 𝑥 = 𝑣0 cos 𝛼0 𝑡; 𝑣𝑥 = 𝑣0 cos 𝛼0
• 𝑦 = 𝑣0 sin 𝛼0 𝑡 − 12𝑔𝑡 2 ; 𝑣𝑦 = 𝑣0 sin 𝛼0 − 𝑔𝑡
• 𝑦 = tan 𝛼0 𝑥 −
𝑔
𝑥2
2
2
2𝑣0 cos 𝛼0
• Circular motion
• 𝑎rad =
• 𝑇=
•
𝑑𝒗
𝑑𝑡
𝑣2
𝑅
2𝜋𝑅
𝑣
= 𝒂 =
2
2
𝑎rad
+ 𝑎tan
• Relative velocity
• 𝒓𝑃/𝐴 = 𝒓𝑃/𝐵 + 𝒓𝐵/𝐴
• 𝒗𝑃/𝐴 = 𝒗𝑃/𝐵 + 𝒗𝐵/𝐴
• 𝒗𝐴/𝐵 = −𝒗𝐵/𝐴
Chapter 4 Outline
Newton’s Laws of Motion
• Forces
• Contact and long range
• Superposition
• Newton’s first law
• Inertial frames of reference
• Newton’s second law
• Mass vs. weight
• Newton’s third law
• Inertial frames of reference
• Free-body diagrams
Forces
• Forces are interactions between two
bodies or between a body and the
environment.
• Contact force
• Push, pull, friction…
• Long-range
• Gravitational, electric, magnetic…
• Vectors!
Superposition of Forces
• When more than one force is acting on a body, the net
force is the vector sum of the forces.
𝑭net =
𝑭𝑖 = 𝑭1 + 𝑭2 + 𝑭3 + ⋯
𝑖
• Sometimes the net force is called the resultant force, 𝑹.
Newton’s First Law
• Natural state of an object
• Aristotle, Galileo, Descartes…
• From Pricipia, the Latin followed by an English translation.
“Lex I: Corpus omne perseverare in statu suo quiescendi vel
movendi uniformiter in directum, nisi quatenus a viribus
impressis cogitur statum illum mutare.”
“Law I: Every body persists in its state of being at rest or of
moving uniformly straight forward, except insofar as it is
compelled to change its state by force impressed.”
• From out text: A body acted on by no net force moves with
constant velocity (which may be zero) and zero
acceleration.