Math 222, Spring 2016.
Present your work in an organized fashion. Make sure that your work is algebraically
correct and logically sound. Show all your work. Discussion (if necessary) with others is
encouraged, while copying others solution is a violation
Math 222, Spring 2016.
Present your work in an organized fashion. Make sure that your work is algebraically
correct and logically sound. Show all your work. Discussion (if necessary) with others is
encouraged, while copying others solution is a violation
15 Solving 2nd-Order ODEs
P
Recall the case of a rod leaning against a wall
and sliding under its own weight.
Suppose it is released from rest when = 0.
G
From AMB/P, we showed that = (t) is the
solution to the following initial value problem:
3g
=
sin
2
6 Force & Motion
6.0 Learning Objectives
Distinguish between friction in a static situation and a kinetic situation.
Determine direction and magnitude of a frictional force.
Solve problems related to drag force and terminal speed
Solve problems relate
9 Systems of Particles
9.0 Learning Objectives
Given the positions of several particles along an axis or a plane, determine
the location of their center of mass.
Locate the center of mass of an extended, symmetric object by using the
symmetry.
Apply Ne
7 Work & Kinetic Energy
7.0 Learning Objectives
Relate a particles kinetic energy, mass, speed, and the work done on a
particle by an external force.
If multiple forces act on a particle, calculate the net work done by them.
Calculate the work done by
2 Motion along a Straight Line
2.0 Learning Objectives
Relate position, displacement, average velocity, instantaneous velocity, and
acceleration of a particle
Solve problems related to constant and free-fall acceleration.
2.1 Position, Displacement, and
4 2D Rectilinear Motion
4.0 Learning Objectives
Use vector representations to relate position, velocity, and acceleration
For each dimension of motion, apply the constant-acceleration equations to
relate acceleration, velocity, position, and time.
On a
0 Course Overview
0.1 Background
Mechanics = science describing the
condition of bodies under the action
of forces
Physical & Applied
Science
describes
physical
phenomenon
experimentally
validated laws
and theories
used in
engineering
practice
Mechanics
R
24-351
HW 9: Double Pendulum
Due: 12/3/15
As illustrated below, a double pendulum is composed of two bars of length = 1 m, mass
m = 1 kg, and mass moment of inertia IG = m2/12.
Here are the governing equations for the dynamical state variables cfw_1, 1, 2
Mass-spring-damper System
Balance law and governing equation:
F = kx cx m
x
x + 2 n x + 2n x = 0
Define natural frequency and damping
ratio:
k
n =
m
Solution:
c
=
2m n
x = A1e 1t + A 2 e 2t
cfw_
1 = n + 2 1
Find A1 and A2 by applying ICs:
cfw_
2
Example 1 (From textbook)
Length L
Mass m
Moment of Inertia IG = mL2/12
Solution (work in groups)
1. Relate kinematic variables and find expression for aG = aO + aO/G
2. Perform AMB about O
3. Solve for
4. Integrate d = d to get
er
a G = ai + a G O
B
e
Work-Energy Balance
The energy balance is the same as before:
U!12 = T + V
Vg = gravitational potential energy
Ve = elastic potential energy
U = all other sources of work
For general plane motion T =
1
1
mv 2 + I G 2
2
2
Solution Method:
1. Free-body d
Energy Conservation: Learning Objectives
Relate a particles kinetic energy, mass, speed, and the work done on a
particle by an external force.
Apply the relationship (Hookes law) between the force on an object by a
spring, the stretch or compression of
Planar Kinetics: Learning Objectives
Use vector representations to relate position, velocity,
acceleration, and force
Apply the constant-acceleration equations to 2D motion
Solve problems related to uniform circular motion
Solve problems related to re
Vectors: Learning Objectives
Given the components of a vector, draw the vector
and determine its magnitude and orientation.
Add and substract vectors by drawing them in headto-tail arrangements, applying the commutative and
associative laws.
Solve prob
Solving ODEs in MATLAB
First Order ODEs
An ordinary differential equation (ODE) contains one or more derivatives of a
dependent variable y with respect to a single independent variable t, usually
referred to as time. The derivative of y with respect to t
23-351
Midterm II Practice
A1
A2
A3
11/12/15
23-351
Midterm II Practice
B1
B2
B3
11/12/15
23-351
Midterm II Practice
C1
C2
C3
Sample Problem B1
11/12/15
24-351 Midterm II
1 Two massless rods of length L are used to connect a cylinder
C to a motor A. The rods are connected by a frictionless hinge
at point B. Let 6 denote the angular orientation of rod AB.
Suppose (o = dB/dt is constant.
a) Express the leng
23-351
Midterm I
10/15/15
Name:
1 A crate (mass mC) is sitting on a truck bed with friction coefficients k = 0.7 and s = 0.8. Suppose that
the truck (mass mT > mC) is moving with velocity v0 but then suddenly brakes (at t = 0). Assume the truck
has a kine
18 Forced Vibration
18.0 Learning Objectives
Solve problems related to forced harmonic vibration of a mass-springdamper system
Understand definition of magnification factor and phase angle
Understand analogy between mechanical and electrical systems
D
16 Double Pendulum
A double pendulum is composed of two bars of
length = 1 m, mass m = 1 kg, and mass moment
of inertia IG = m2/12.
The governing equations for the dynamical state
variables cfw_1, 1, 2, 2 are derived by
performing the LMB and AMB on each
23~351 Midterm I Practice 10/9/15
A2 A particie of mesa m is attached to one and 9f 3 a ~.—r _ k a
light slander rocl Which pimts about a hcrizonbal ) '11: ’ 0 “Ta ’0 z “B3 '5 k0?“ z)
axis threugh poiat O. The Spring constant (V3350 (Va); : 31:1,“ Ma
14 Impulse-Momentum for Rigid Bodies
14.0 Learning Objectives
Extend principles of impulse-momentum from particles to planar motion of
rigid bodies and interconnected systems
Relate forces, moments, and kinematic variables for linear and angular
momentu
13 Work-Energy for Rigid Bodies
13.0 Learning Objectives
Apply work-energy relationships to determine change in translational and/or
rotational velocity of rigid bodies
Identify problems where energy methods should be used instead of Newtons
equations
1