Download Two Dimensional Motion 2

Two Dimensional Motion
Two Dimensional Motion
One Dimensional Motion –
Describes Motion in a straight line
Two Dimensional Motion – Describes,


Motion in a Curved Path
(Circular Motion and Projectile Motion)
Two Dimensional Force Vectors
(Equilibrant Vector and Inclined Planes)
Equilibrium
If the Net Force acting on the
object is zero
FNET = 0 a = 0
The object is either stationary
(v = 0) or traveling with a constant
velocity (v = constant)
Equilibrant
The Equilibrant (-R) is the single force
that when added to a group of forces,
produces Equilibrium.
The Equilibrant (-R) is equal and
opposite to the Resultant Force (R).
To Find the Equilibrant
1. Find the Resultant (R) of the group of
force vectors. R = A + B + C
2. Draw the Equilibrant (-R) with the
same magnitude in the opposite
direction as the Resultant (R)
Inclined Planes
An Inclined Plane is a flat surface with
one end elevated higher than the other
The Weight is a force that always acts
downward
Since the Weight Vector (mg) acts
straight downward, the Weight Vector
splits into 2 perpendicular components
Weight Components on an
Inclined Plane
One component of the weight acts
parallel to the surface of the incline
and down the incline (mg||)
The other component of the weight acts
perpendicular to the surface of the
incline and into the incline (mg|)
Weight Components on an
Inclined Plane
mg|| - component of the weight acting
parallel to the incline
mg|– component of the weight acting
perpendicular to the incline
Weight Components on an
Inclined Plane
Since the components are perpendicular
to each other they form a right triangle
and we can find the components with
the sine and cosine functions
mg|| = mg sin(q)
mg| = mg cos(q)
Weight Components on an
Inclined Plane
Note:
That as q increases mgsin(q) increases
and mgcos(q) decreases
These components of the weight are
forces and can be used on a FBD with
Newton’s Laws (Net Force Equation)
Projectile Motion
A Projectile is anything thrown, shot
or dropped into the air
Projectile Motion is a combination of
two independent motions
1.
2.
Horizontal Motion (x-motion)
Vertical Motion (y-motion)
Perpendicular Vectors
Vectors that are Perpendicular to
each other are INDEPENDENT of
each other.
 Ex. Boat traveling perpendicular
to the current.
 Ex. Projectile thrown in the air.
Perpendicular Vectors and
Projectile Motion
Since Projectile Motion is a combination
of two perpendicular motions (vectors),
each motion is independent of each
other!
Gravity (y-direction) does not affect the
Horizontal Speed (vX)
vX has no affect on vY
Projectile Motion
There are no forces affecting the
Horizontal Motion (aX = 0). We use
only vX = dX/t for the x-motion
The Vertical Motion is accelerated by
gravity (aY = g). We use the Kinematic
Equations for y-motion
Projectile Motion Problems
There are three types of Projectile
Motion Problems
1. Objects shot straight upwards (vX = 0)
2. Objects shot horizontally
(half of a parabola)
3. Objects shot at an angle
(whole parabola)
Projectile Motion Terminology
dX – Range – horizontal distance traveled
dY – How High the projectile traveled
vertically
tH – half time – time the projectile travels
upward or downward (Bottom to Top or
Top to Bottom)
tT – Total time – the total time the
projectile is in the air
Uniform Circular Motion
Uniform Circular Motion (UCM)
describes an object traveling in a
circular path at a constant speed
Does an object in U.C.M. accelerate?
YES!!

Because there is a change in Direction!!
Finding the Acceleration for
Uniform Circular Motion
We cannot use a = (vf – vi)/t for
uniform circular motion because the
change in velocity is a change in
direction (not a change in speed)!
We need to derive another acceleration
equation that uses vector diagrams to
quantify this change in direction
Centripetal Acceleration
ac = centripetal acceleration
ac = (circular speed)2
radius of the circle
ac = vc2/r
S.I. Unit = m/s2
Centripetal Acceleration
Acceleration that causes a change in
direction.
An object in circular motion has a
centripetal acceleration.
Centripetal (means “center-seeking”)
Acceleration always acts towards the
center of the circle.
Circular Speed
Circular speed (vc) is the speed an
object travels in a circle with.
vc = distance = circumference
time
time
vc = 2pr/t
Centripetal Force
Centripetal (Fc) Force is the Force
that causes a change in direction
We can use Newton’s 2nd Law to
calculate the Centripetal Force (Fc)
Fc = mac
The Centripetal Force (Fc) is a
NET FORCE
Vertical Circle
In a Horizontal Circle, gravity acts the
same at every point in the circle
In a Vertical Circle, gravity speeds up
the object on the way down and slows
down an object on the way up
We need to use the Net Force Equation
at different points in the circle to
describe the motion
Vertical Circle
The Centripetal Force is a NET FORCE!
An object in a Vertical Circle usually
has a Maximum Velocity at the Bottom
of the circle and a Minimum at the Top
A Vertical Circle has a minimum speed
that must be maintained in order to
stay in a circular path (Critical Velocity)
Rotary Motion
Rotation or Rotary Motion is motion
around a central axis (not a central
point).
Axis is the line that goes through
the object around which the object
rotates
Torque and Rotary Motion
Translational (Linear) Motion is
changed by a FORCE!
Rotary Motion is changed by a
TORQUE!
Torque
Torque (t) causes rotation (analogous
to Force in Linear Motion)
Torque = Force x Lever Arm = t = Fl
Lever Arm (l) is the perpendicular
distance from the force to the axis
line
Torque
Directions for Torque (Vector)
 Clockwise (-)
 Counter Clockwise (+)
Unit = Newton-meter (Nm)
Torques can balance just like
Forces (Clockwise and CounterClockwise)
Frames of Reference
Frame of Reference or Reference Point
are the objects used to determine
what motion has taken place
There are two types of frames
1.
2.
Inertial Frames (Non-accelerated)
Non-inertial Frames (Accelerated)
Frames of Reference
Inertial Reference Points are not being
accelerated (v=0 or v=constant).
Newton’s laws are easily observable
Non-inertial Reference Points are
accelerated frames of reference and
Newton’s Laws appear to break down in
these reference frames
Centrifugal Force and
Non-inertial Reference Frames
Centrifugal Force – is the force that
seems to push an object to the outside
of the circular or curved path
The Centrifugal Force is fictitious
because the object is in an accelerated
frame of reference and can be
explained with Newton’s First Law