Engineering Mechanics - Dynamics
Chapter 20
Problem 20-1 The ladder of the fire truck rotates around the z axis with angular velocity 1 which is increasing at rate 1. At the same instant it is rotating upwards at the constant rate 2. Determine the
Engineering Mechanics - Dynamics
Chapter 21
a
Iyz
0
2m a h
2
h( a
y)
h a y dy 2 2
1 mah 6
Iyz
1 mah 6
Problem 21-7 Determine by direct integration the product of inertia Ixy for the homogeneous prism. The density of the material is . Expre
Engineering Mechanics - Dynamics
Chapter 15
Problem 15-1 A block of weight W slides down an inclined plane of angle with initial velocity v0. Determine the velocity of the block at time t1 if the coefficient of kinetic friction between the block an
Engineering Mechanics - Dynamics
Chapter 14
Problem 14-1 A woman having a mass M stands in an elevator which has a downward acceleration a starting from rest. Determine the work done by her weight and the work of the normal force which the floor ex
Engineering Mechanics - Dynamics
Chapter 13
Problem 13-1 Determine the gravitational attraction between two spheres which are just touching each other. Each sphere has a mass M and radius r. Given: r Solution: F GM ( 2r) Problem 13-2 By using an in
Engineering Mechanics - Dynamics
Chapter 12
Problem 12-1 A truck traveling along a straight road at speed v1, increases its speed to v2 in time t. If its acceleration is constant, determine the distance traveled. Given: v1 Solution: a v2 t v1 t 1 2
ME 2580 Dynamics
Curvilinear Motion Rectangular Components
General Concepts:
Position, Velocity, and Acceleration
If a particle does not move in a straight line,
then its motion is said to be curvilinear. Given
r (t ) the position vector of a particle P,
ME 2580 Dynamics
Kinetics of Particles: Newtons Second Law
Until now we have been studying kinematics, that is, the study of motion without
regard to the forces that cause (or result from) the motion. Now we want to extend
our study to include the forces
ME 2580 Dynamics
Relative Motion of Two Particles
The figure shows the paths of motion of
two particles A and B. The vectors r A and
r B represent the position vectors of A and
B relative to a fixed point O, and the vector
r A/ B represents the position v
ME 2580 Dynamics
Curvilinear Motion Normal and Tangential Components
Normal and Tangential Components
Normal and tangential components refer to
components that are normal and tangential to
the path of P. These directions are defined by
the unit vectors e
ME 2580 Dynamics
Curvilinear Motion Radial and Transverse Components
Radial and Transverse Components
Another way to describe the motion of P as it
moves along a curved path is to use radial and
transverse components. Here, we define the unit
vector e r t
ME 2580 Dynamics
Rectilinear (Straight Line) Motion
General Concepts:
Position, Velocity, and Acceleration
A particle P has rectilinear motion when it moves in a straight line. As shown in the
figure, define the direction of motion as the X-axis along whi
ME 2580 Dynamics
Acceleration Profiles
a (t )
2
a (t ) (m/s )
Given: A car has an acceleration profile as
shown. Its initial position and velocity are
(0) (0)
zero. ( s= v= 0 )
0.3
15
5
v(t ) = a (t )dt and s (t ) = v(t )dt
20
10
t (sec)
-0.6
Find: (a) Th
ME 2580 Dynamics
Acceleration Profiles A Second Example
Given: A car has the acceleration profile
shown where a (t ) is in (m/sec).
Find:
v(t ) and s (t ) the speed and position
of the car as functions of time for
0 t 30 (sec) . Also, find the speed
and p
ME 2580 Dynamics
Point Moving on a Rigid Body (Sliding Contact)
The relative motion of two points fixed on a rigid body may be calculated using the
relative velocity and relative acceleration equations. These equations may be used to
analyze many systems
ME 2580 Dynamics
Relative Velocity of Two Points Fixed on a Rigid Body
The figure depicts a rigid body moving in two
dimensions. The two points P and Q are fixed on the body.
At any instant of time, the position vector of P may be
written as
r P r Q r P/
ME 2580 Dynamics
Relative Acceleration of Two Points Fixed on a Rigid Body
The figure depicts a rigid body moving in two
dimensions. The two points P and Q are fixed on the
body. At any instant of time, the position vector of P may
be written as
rP r Q r
ME 2580 Dynamics
Pure Rotational Motion (Fixed Axis Rotation) in Two-Dimensions
A body B has pure rotational motion when one point of the
body is fixed and all the other points of the body rotate around it.
The body is rotating about an axis that passes t
ME 2580 Dynamics
Center of Mass in Two Dimensions
Definition of Mass Center
The figure depicts a rigid body in two
dimensions. The mass center of the body is
defined as that point where
r dm 0
B
x dm 0
B
y dm 0
B
A more practical definition of the loca
ME 2580 Dynamics
Introduction to Rigid Body Kinematics (2D)
Rigid Body Motion
The diagram illustrates a rigid body B in two
dimensions acted on by a set of forces
Fi i 1, , N . There are three basic types of
motion that B can undergo pure translation, pur
ME 2580 Dynamics
Instantaneous Centers of Zero Velocity
An instantaneous center is a point of a rigid body (or rigid body extended) that has
zero velocity at a given instant of time. The acceleration of that point is generally not
zero. The concept of ins
ME 2580 Dynamics
Power and Efficiency
The work done by a force F as a particle moves
from position 1 to position 2 is
U12
t2
F dr F v dt .
t1
The power generated by F at any instant is P F v dU dt . The average power
generated by F over an interval of t
ME 2580 Dynamics
Conservation of Energy for Particles
Conservative and Nonconservative Forces
Consider a particle that moves from position 1 to position
2 along one path (forward path) and back again to position 1
along a second path (return path) as show
Three Kinematic Equations for Constant Acceleration
Velocity as a Function of Time
Position as a Function of Time
2
a=
dv d s
=
dt dt 2
v
v=
then
s
t
dv= a c dt
v0
ds
dt
Velocity as a Function of Position
a ds=v dv
then
v
t
ds= (v 0 +a c t)dt
yields
so