Transcript Physics 218 - Purdue University

Today: (Ch. 3)

Forces and Motion in Two and Three
Dimensions

Equilibrium and Examples
Tomorrow: (Ch. 4)

Projectile Motion
Example
A passenger weighing 598 N rides in an elevator. The
gravitational field strength is 9.8 N/kg. What is the
apparent weight of the passenger in each of the
following situations? In each case the magnitude
of elevator’s acceleration is 0.5 m/s2.
(a)The passenger is on the 1st floor and has pushed
the button for the 15th floor i. e. the elevator is
beginning to move upward.
(b)The elevator is slowing down as it nears the 15th
floor.
Statics and Equilibrium
• Statics
 Deals with objects at rest
• 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
Equilibrium and Newton’s law
According to Newton’s 2nd law


 F  ma
This vector quantity can be written in terms
of two perpendicular components
F
x
 max
F
y
 may
When object in equilibrium the net force acting
on it is zero: For object to be in equilibrium
F
x
 max  0 and
F
y
 may  0
Equilibrium on an inclined plane

N

Fa
h


W  mg
Identify all the forces in the
d system
φ
y
x
Choose the axes, which are
convenient to work with…
Wy
φ

Wx W
φ
Using Newton’s Second Law
• Newton’s Second Law
F  m a
• Determine 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
• Acceleration can be used to determine velocity and
displacement
Equilibrium Example
• Forces acting in A
– Gravity and normal force in ydirection
– Force exerted by person (push)
and static friction in x-direction
• Free body diagram in B
• Forces in x and y components and
apply condition of equilibrium
 ΣFx = 0 and ΣFy = 0
 For y-direction: N - m g = 0
 For x-direction: Fpush - Ffriction = 0
Equilibrium Example 2
• 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 3
• 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
– Tension on the right and on the left & Weight of the
walker
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
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
Inclines (Hills)
• Normal force (N) : perpendicular to the
incline (plane)
• Friction force : up the incline
– Opposite to motion
• The force due to gravity acts straight
down
• Coordinate system
– Axes parallel and perpendicular to
the incline
• (If) Acceleration : along the incline
• Components of the gravitational force
• The normal force is not equal to mg
Angle of Incline To Not Slip
• 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
Equilibrium Example, Flag
• Find out tension and angle (two
unknowns so 2D problem)
• Free body diagram
– Horizontal and vertical
directions : coordinate system
– Tension has x- and ycomponents
• Equations for equilibrium in the
x- and y-directions
Motion in Two Dimensions
• Two dimensions is just like one dimension,
done twice:
• for the x components, then
• for the y components.
• The x and y motions are independent of each
other!
• The x part of the motion occurs exactly as if
the y part did not exist.
• And, the y part of the motion occurs
exactly as if the x part did not exist.
Projectile Motion
The x and y motions are independent of each other!
In general, for the x components:
 assume that there is no air resistance,
so that ax = 0
 then vx = constant = v0x
In general, for the y components:
 ay = 9.8 m/s2 down
 then vy changes
Acceleration due to gravity is always 9.8 m/s2
downward, throughout the path of the projectile!
y
v
θ
x
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
HW Question
Calculate the distance between the window
and the ceiling using kinematics
A roofing tile falls from rest off the roof of a building. An
observer from across the street notices that it takes
0.54 s for the tile to pass between two windowsills that
are 2.5 m apart. How far is the sill of the upper window
from the roof of the building?
The Three Equations
(1)
v  v0  at
(2)
1 2
x  v0 t  at
2
(3)
v  v  2ax
2
2
0
Tomorrow: (ch 4)

Projectile Motion

Reference Frames & Relative Velocity

Example Involving Newton’s Law