Post on 03-Jan-2016
Types of physical quantities
• In physics, quantities can be divided into such general categories as scalars, vectors, matrices, etc.
• Scalars – physical quantities that can be described by their value (magnitude) only
• Vectors – physical quantities that can be described by their value and direction
Vectors
• Vectors are labeled either a or
• Vector magnitude is labeled either |a| or a
• Two (or more) vectors having the same magnitude and direction are identical
a
Vector sum (resultant vector)
• Not the same as algebraic sum
• Triangle method of finding the resultant:a) Draw the vectors “head-to-tail”b) The resultant is drawn from the tail of A to the head of B
A
B
R = A + B
Addition of more than two vectors
• When you have many vectors, just keep repeating the process until all are included
• The resultant is still drawn from the tail of the first vector to the head of the last vector
Multiplying a vector by a scalar
• The result of the multiplication is a vector
c A = B
• Vector magnitude of the product is multiplied by the scalar
|c| |A| = |B|
• If the scalar is positive (negative), the direction of the result is the same as (opposite to that) of the original vector
Vector components
• Component of a vector is the projection of the vector on an axis
• To find the projection – drop perpendicular lines to the axis from both ends of the vector – resolving the vector
Unit vectors
• Unit vector:A) Has a magnitude of 1 (unity)B) Lacks both dimension and unitC) Specifies a direction
• Unit vectors in a right-handed coordinate system
Chapter 3Problem 42
Vector A has magnitude 1.0 m and points 35° clockwise from the x-axis. Vector B has magnitude 1.8 m. Find the direction of B such that A + B is in the y-direction.
Displacement
The displacement vector is defined as the change in its position,
if rrr
)ˆˆˆ()ˆˆˆ( kzjyixkzjyixr iiifff
kzzjyyixx ifififˆ)(ˆ)(ˆ)(
kzjyixr ˆˆˆ
r
Velocity
• Average velocity
• Instantaneous velocity
t
kzjyix
t
rv
ˆˆˆ
kdt
dzj
dt
dyi
dt
dx
dt
rdv ˆˆˆ
kvjvivv zyxˆˆˆ
Instantaneous velocity
• Vector of instantaneous velocity is always tangential to the object’s path at the object’s position
Acceleration
• Acceleration – the rate of change of velocity (vector)
• The magnitude of the velocity (the speed) can change – tangential acceleration
• The direction of the velocity can change – radial acceleration
• Both the magnitude and the direction can change
Chapter 3Problem 23
What are (a) the average velocity and (b) the average acceleration of the tip of the 2.4-cm-long hour hand of a clock in the interval from noon to 6 PM? Use unit vector notation, with the x-axis pointing toward 3 and the y-axis toward noon.
Projectile motion
• A special case of 2D motion
• An object moves in the presence of Earth’s gravity
• We neglect the air friction and the rotation of the Earth
• As a result, the object moves in a vertical plane and follows a parabolic path
• The x and y directions of motion are treated independently
Projectile motion – X direction
• A uniform motion: ax = 0
• Initial velocity is
• Displacement in the x direction is described as
iixi vv cos
tvxx iii )cos(
Projectile motion – Y direction
• Motion with a constant acceleration: ay = – g
• Initial velocity is
• Therefore
• Displacement in the y direction is described as
iiyi vv sin
2
2
1)sin( gttvyy iii
gtvv iiy sin
Projectile motion: putting X and Y together
constvv iix cos
2
2
1)sin( gttvyy iii
gtvv iiy sin
tvxx iii )cos(
Chapter 3Problem 33
A carpenter tosses a shingle horizontally off an 8.8-m-high roof at 11 m/s. (a) How long does it take the shingle to reach the ground? (b) How far does it move horizontally?
Uniform circular motion
• A special case of 2D motion
• An object moves around a circle at a constant speed
• Period – time to make one full revolution
• The x and y directions of motion are treated independently
v
rT
2
Uniform circular motion
• Velocity vector is tangential to the path
• From the diagram
• Using
• We obtain
jvivjvivv yxˆ)cos(ˆ)sin(ˆˆ
r
xpcosr
y psin
jr
vxi
r
vyv pp ˆˆ
Centripetal acceleration
dt
vda
j
dt
dx
r
vi
dt
dy
r
v pp ˆˆ
jvr
viv
r
v ˆ)sin(ˆ)cos(
)sinˆcosˆ(2
jir
v
22yx aaa
r
v2
x
y
a
atan
cos)/(
sin)/(2
2
rv
rv
tan
222
sincos r
v
jdt
dvi
dt
dv yx ˆˆ
jr
vxi
r
vyv pp ˆˆ
jvivjvivv yxˆ)cos(ˆ)sin(ˆˆ
Centripetal acceleration
During a uniform circular motion:
• the speed is constant
• the velocity is changing due to centripetal (“center seeking”) acceleration
• centripetal acceleration is constant in magnitude (v2/r), is normal to the velocity vector, and points radially inward
Chapter 3Problem 38
How fast would a car have to round a 75-m-radius turn for its accelerationto be numerically equal to that of gravity?
Relative motion
• Reference frame: physical object and a coordinate system attached to it
• Reference frames can move relative to each other
• We can measure displacements, velocities, accelerations, etc. separately in different reference frames
Relative motion
• If reference frames A and B move relative to each other with a constant velocity
• Then
• Acceleration measured in both reference frames will be the same
BAPBPA vvv
BAPBPA rrr
PBPA aa
BAv