Kinematics of Particles

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Kinematics of Particles

• ENGR 221
• April 2, 2003

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Lecture Goals

• Newtons Laws
• Position, Velocity and Acceleration
• Rectilinear Motion
• Relative Motion along a Line

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Dynamics
Up until now we have cover the concept of
statics, where the rigid body is in equilibrium
and the sum of the forces and moments are zero.
When a body is in motion these equations are no
longer zero and the study of the results are
known as dynamics.
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Newtons Laws
Law 1
A particle at rest will remain at rest. A
particle originally moving with a constant
velocity will continue to move with a constant
velocity along a straight line unless the
particle is acted on by an unbalanced force.
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Newtons Laws
Law 2
When a particle is acted on by an unbalanced
force, the particle will be accelerated in the
direction of the force. The magnitude of
acceleration will be proportional to the force
and inversely proportional to the mass of the
particle.
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Newtons Laws
Law 3
For every action there is an equal and opposite
reaction. The forces of action and reaction
between contacting bodies are equal in magnitude
opposite in direction and collinear.
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Definition
Understand the definitions inertia - the
property of matter that causes resistance to
a change in motion particle - is an object whose
size and shape can be ignored when studying
its motion. rigid body - is a collection of
particles that remain at fixed distance from
each other at all times and under all
conditions of loading.
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Definitions
Introduction kinematics - study of objects
move kinetics - study of the relationship
between the motion and the forces that cause
the motion.
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Definition- Motion
Rectilinear motion - all of the y and z
components of displacement, velocity and
acceleration are zero for all time. Plane
curvilinear motion - a particle is not moving in
rectilinear motion but a coordinate system can be
found such that z-components of position,
velocity and acceleration are zero. General
curvilinear motion - no system of coordinates can
be found that at least one component of the
position, velocity, and acceleration are zero
for all time.
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Displacement
Given a path of particle relative to a coordinate
system, O
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Displacement
Given a path of particle relative to a coordinate
system, O
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Displacement
One can rewrite the equations to describe the
motion
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Displacement
The displacement is defined as the change in the
location The change of location is independent
of coordinate systems.
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Velocity
The velocity is defined as the change in the
location with respect to time
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Acceleration
The acceleration is defined as the change in the
location with respect to time
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Rectilinear Motion
In rectilinear motion, the y and z components
(displacement, velocity and acceleration) are
zero.
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Rectilinear Motion
Given x(t) find the velocity and acceleration
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Rectilinear Motion
Given v(t) find x(t)
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Rectilinear Motion
Given a(t) find v(t)
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Rectilinear Motion
Given a(x) find v(x) using the chain rule we can
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Example Problem I
Consider a particle moving in a straight line,
and assume that its position is defined by the
equation, x 6t2 t3 where x is expressed
in meters and t in seconds. Determine the
velocity and acceleration and plot the results.
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Example Problem I
Compute the velocity of the particle Find out
where the velocity is equal to zero. So a local
maximum or minimum occur at 0 and 4 seconds.
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Example Problem I
Compute the acceleration of the particle Find
out where the acceleration is equal to zero. So
a local maximum or minimum for the velocity
occur at 2 seconds.
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Example Problem I
Compare the values
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Class Problem I
The position of a particle which moves along a
straight line is defined by the relationship, x
t3 6t2 15t 40 where x is expressed in
feet and seconds. Determine the time at which
the velocity will be zero (b) the position and
distance traveled by the particle at that time
(c) the acceleration of the particle at that
time, the distance traveled by the particle from
t 4s and t 6s.
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Example Problem II
The brake mechanism used to reduce recoil in
certain types of guns consists essentially of a
piston which is attached to the barrel and may
move in a fixed cylinder filled with oil. As the
barrel recoils with an initial velocity v0, the
piston moves and oil is forced through orifices
in the piston, causing the piston and the barrel
to decelerate at a rate proportional to their
velocity, a kv . Express(a) v in terms of t
and (b) x in terms of t and (c) v in terms of x
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Example Problem II
Determine the velocity in terms of t.
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Example Problem II
Determine the x in terms of t.
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Example Problem II
Determine the v in terms of x.
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Relative Motion
One can use the relative motion or constraints to
find the actual velocity of the particle relative
to another particle.
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Relative Motion
One can use the relative motion or constraints to
find the actual velocity of the particle relative
to another particle.
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Dependent Relative Motion
The members are dependent on one another and a
constraint condition is applicable.
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Example III Problem
A ball is thrown vertically upward from the 40-ft
level in an elevator shaft, with an initial
velocity of 50 ft/s. At the same instant an open
platform elevator passes the 10 ft. level, moving
upward with a constant velocity of 5 ft/s.
Determine (a) when and where the ball will hit
the elevator,(b) the relative velocity of the
ball with respect to the elevator when the ball
hits the elevator.
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Example III Problem
Calculate the movement of the ball.
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Example III Problem
Calculate the movement of the elevator.
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Example III Problem
Determine when the ball and the elevator are at
the same location.
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Example III Problem
Determine the location
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Example III Problem
Determine the relative velocity of the elevator
and the ball.
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Example IV Problem
Two blocks A and B are connected by a cord
passing over three pulleys C, D and E. Pulleys C
and E are fixed, while D is pulled downward with
a constant velocity of 1.5 m/s. At t 0 block A
starts moving downward from the position K with a
constant acceleration and no initial velocity.
Knowing the velocity of the block A is 6 m/s as
it passes through point L, determine the change
in elevation, the velocity and the acceleration
of block B when A passes through L.
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Example IV Problem
Find the acceleration and time of block A
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Example IV Problem
Determine the deflection of pulley D in order to
find deflection in B.
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Example IV Problem
Look at the entire cord and we find that the
length is a constant. So that
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Example IV Problem
The change in B will be
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Example IV Problem
The change in Bs velocity will be
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Example IV Problem
The change in Bs acceleration will be
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Class Problem III
Two blocks A and B are connected by a cord
passing over two pulleys C and D and E and F. At
t 0 block A starts moving downward from the
position with a constant acceleration of 0.4 m/s2
and no initial velocity. What the velocity of
the block A after it has gone 5 m down, determine
the change in elevation, the velocity and the
acceleration of block B when A at that instant.
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Homework (Due 4/9/03)
Problems
13-1, 13-3, 13-4, 13-9, 13-25, 13-52, 13-70,
13-74