ENGR 2030 Dynamics
All Lecture Examples
Instructor: Dr. Ning Fang
12.1
12.2
Introduction
Rectilinear kinematics: continuous motion
Example 1: A rocket
Section 12.1-12.2
Example 2: An FE Exam Sample
A motorcyclist travels along a
straight road at a speed of 27 m/s.
When the brakes are applied, the
motorcycle decelerates at a rate of
-6t m/s2. What is the distance the
motorcycle travels before it stops?
a)
b)
c)
d)
34 m
44 m
54 m
64 m
Section 12.1-12.2
Chapter 12 lecture examples, page 1
12.3
Rectilinear kinematics: erratic motion
Example #1: Car Acceleration
If a car starts from rest and accelerates
according to the graph shown, the car’s
velocity at t = 20 second is:
A) 200 m/s
B) 100 m/s
C) 0 m/s
D) 20 m/s
Section 12.3
Example #2
The a-s graph for a race car moving along a straight
track has been experimentally determined. If the car
starts from rest at s = 0, determine its velocity v when
s = 50 ft, 150 ft, and 200 ft. respectively.
Turn the graph
into math first.
Section 12.3
Chapter 12 lecture examples, page 2
Chapter 12 lecture examples, page 3
12.4
12.5
12.6
General curvilinear motion
Curvilinear motion: rectangular components
Motion of a projectile
Ex#1: Projectile Motion
Given:
Find:
Initial Velocity V0 = 10 m/s
Initial angle q = 60o
Total travel time Ttotal
Maximum height hmax
Maximum distance Smax
V0
q
Sections 12.4-12-6
Ex#2: Projectile Motion
Given: Skier leaves the ramp at qA = 25o
and hits the slope at B.
Find: The skier’s initial speed vA
Critical thinking:
•
•
•
•
How do you establish the
coordination system?
Which equation should I use?
Apply that equation in X- or Ydirection, or both?
Check if:
the number of unknowns =
the number of equations?
Sections 12.4-12-6
Chapter 12 lecture examples, page 4
12.7
Curvilinear motion: normal and tangential components
Example 1: A Motorboat
Given: Starting from rest, a motorboat travels
around a circular path of r = 50 m at a speed
that increases with time, v = (0.2 t2) m/s.
Find: The magnitudes of the boat’s velocity
and acceleration at the instant t = 3 s.
Section 12.7
Example 2: Beaver Mountain Skiing
Section 12.7
Chapter 12 lecture examples, page 5
12.8
Curvilinear motion: cylindrical components
Example #1
Given:
r = 4t2 (m/s)
q = 8t3 +6 (rad)
Find:
the magnitude of total
velocity and acceleration a
at time t = 2 seconds.
Section 12.8
Example #2
For a short time the position of the roller-coaster
car along its path is defined by r = 30 m,
q = (0.4t) rad, and z = (-10 cosq) m, where t is in
seconds. Determine the magnitude of the car’s
velocity and acceleration when t = 5s.
Section 12.8
Chapter 12 lecture examples, page 6
12.9
12.10
Absolute dependent motion analysis of two particles
Relative-motion analysis of two particles using translating axes
Example #1: Lift a Block
Given: In the figure on the left,
the cord at A is pulled down
with a speed of 8 ft/s.
Find: The velocity of block B.
Sections 12.9-12.10
Example #2: Relative Motion
Given: vA = 600 km/hr
vB = 700 km/hr
Find: vB/A
Sections 12.9-12.10
Chapter 12 lecture examples, page 7
13.1
13.2
13.3
13.4
Newton’s lows of motion
The equation of motion
Equation of motion for a system of particles
Equation of motion: rectangular coordinates
Classic Example #1
Sections 13.1-13.4
Classic Example #2
At a given instant, the 10-lb block A is moving
downward with a speed of 6 ft/s. Determine its speed
2 seconds later. Block B has a weight of 4 lb, and the
coefficient of kinetic friction between it and the
horizontal plane is mk=0.2. Neglect the mass of the
pulleys and cord.
Sections 13.1-13.4
Chapter 13 lecture examples, page 1
13.5
Equation of motion: normal and tangential coordinates
Example: Motion of a Swing
Given: At the instant q = 60°, the boy’s
center of mass G is momentarily at
rest. The boy has a weight of 60 lb.
Neglect his size and the mass of the
seat and cords.
Find: The boy’s speed and the
tension in each of the two supporting
cords of the swing when q = 90°.
Kinetics (force) + kinematics (motion)
Section 13.5
Chapter 13 lecture examples, page 2
Chapter 13 lecture examples, page 3
13.6
Equation of motion: cylindrical coordinates
Chapter 13 lecture examples, page 4
14.1
14.2
14.3
The work of a force
Principle of work and energy
Principle of work and energy for a system of particles
Example #1:
The Work Done by a Spring Force
If a spring force is Fs = 5 s3 N/m and the spring is
compressed by s = 0.5 m, the work done (by the
spring force) on a particle attached to the spring
will be
A) 0.625 N · m
B) – 0.625 N · m
C) 0.0781 N · m
D) – 0.0781 N · m
Sections 14.1__14.3
Chapter 14 lecture examples, page 1
14.4
Power and efficiency
Example #1: Power of a Train Engine
The diesel engine of a 400-Mg train increases
the train’s speed uniformly from rest to 10 m/s
in 100 seconds along a horizontal track.
Determine the average power developed.
Section 14.4
Example #2: Motion of a Car
Given: A sports car has a mass of 2 Mg and an engine
efficiency of e = 0.65. Moving forward, the wind creates a
drag resistance on the car of Fr = 1.2v2 N, where v is the
velocity in m/s. The car accelerates at 5 m/s2, starting
from rest. Neglect the friction force between the wheels
and the ground.
Find: The engine’s input power when t = 4 s.
Section 14.4
Chapter 14 lecture examples, page 2
Chapter 14 lecture examples, page 3
14.5
14.6
Conservative forces and potential energy
Conservation of energy
Chapter 14 lecture examples, page 4
15.1
15.2
Principle of linear impulse and momentum
Principle of linear impulse and momentum for a system of particles
Example 1: Golf Game
V2
Given: A 40 g golf ball is hit over a time interval of 3 ms by a
driver. The ball leaves with a velocity of 35 m/s, at an angle of
40°. Neglect the ball’s weight while it is struck.
Find: The average impulsive force exerted on the ball.
Sections 15.1-15.2
Example 2: Lifting a Bucket
The winch delivers a horizontal
towing force T to its cable at A
which varies as shown in the graph.
The mass of bucket B is 70 kg and
it originally it is moving upward at
v1= 3 m/s.
T
T
Determine the speed of the bucket
B when t = 18 s.
t = 18 s
Sections 15.1-15.2
Chapter 15 lecture examples, page 1
15.3
Conservation of linear momentum for a system of particles
Example #1
The 20 g bullet is fired horizontally at
1,200 m/s into the 300 g block resting on
a smooth surface. If the bullet becomes
embedded in the block, what is the
velocity of the block immediately after
impact?
1,200 m/s
Sections 15.3
Example #2
Given: Two rail cars with masses of mA = 15 Mg and
mB = 12 Mg and velocities as shown.
Find: The speed of the cars after they meet and
connect. Also find the average impulsive force
between the cars if the coupling takes place in 0.8 s.
Section 15.3
Chapter 15 lecture examples, page 2
15.4
Impact
Example #1
The two balls A and B each weight 0.5 lb and
have a coefficient of restitution of e = 0.85. If
ball A is released from rest and strikes ball B,
determine the velocity of balls A and B after the
impact. Neglect friction.
Section 15.4
Example #2
A ball A strikes the frictionless surface B with a
velocity of 40 m/s. If the coefficient of restitution
in the y direction ey = 0.8, determine the
velocity of A after the collision.
y
A
v
30
40 m/s

x
B
Section 15.4
Chapter 15 lecture examples, page 3
15.5
15.6
15.7
Angular momentum
Relation between moment of a force and angular momentum
Angular impulse and momentum principles
Example #1
Sections 15.5-15.7
Example #2
Given: Two 0.4 kg masses with initial velocities of
2 m/s experience a constant moment of 0.6 N·m.
Find: The speed of blocks A and B when t = 3 s.
Sections 15.5-15.7
Chapter 15 lecture examples, page 4
Additional Example: Which Tool to Use?
The girl has a mass of 40 kg and the center of mass at
G. If she is swinging to a maximum height defined by
 = 60o, determine the force developed along each of the
four supporting posts such as AB at the instant
 = 0o. The swing is centrally located between the posts.
Chapters 14 and 15 Review
Chapter 15 lecture examples, page 5
16.1
16.2
16.3
Rigid-body motion
Translation
Rotation about a fixed axis
Example #1
Given: the angular velocity of the disk w = (5t2+2) rad/s,
where t is in seconds.
Find: The velocity of and acceleration of point A on the
disk when t = 0.5 s.
V
w
a
Sections 16.1-16.3
Chapter 16 lecture examples, page 1
Chapter 16 lecture examples, page 2
16.4
Absolute general plane motion analysis
Example #1:
Let Us Start From a Simple Case
For the bar shown, how to relate the position
x of end B to angular position q ?
A) x = h tanq
B) x = l cosq
C) x = h cosq
D) x = h/l
Section 16.4
Chapter 16 lecture examples, page 3
16.5
Relative-motion analysis: velocity
Example #1
Given: The block C moves downwards at 4 ft/s.
Find: The angular velocity of bar AB at the instant shown.
Section 16.5
Example #2
Given: The velocity of block A is 2 m/s downward.
Find: The velocity of block B at the instant q = 45o.
j
i
Section 16.5
Chapter 16 lecture examples, page 4
16.6
Instantaneous center of zero velocity
Example
Given: The velocity of the block vD is 3 m/s.
Find: The angular velocities of links AB and BD.
Section 16.6
Another Example
Given VB = 0.6 m/s, VA = 0.3 m/s, the radius of wheel is 0.125 m.
The angular velocity of the wheel will be approximately
A)
B)
C)
D)
0.6 rad/s
3.6 rad/s
6.6 rad/s
9.6 rad/s
VA
VB
Chapters 16 and 17 Review
Chapter 16 lecture examples, page 5
16.7
Relative-motion analysis: acceleration
Example #1
Given: The ball rolls without slipping.
The acceleration of point O is 2 ft/s2
to the left.
Find: The accelerations of point A
at this instant.
Section 16.7
Example #2
Given: At a given time instant, the top A of the ladder
has an acceleration aA = 2 ft /s2 and a velocity of VA = 4
ft/s, both acting downward.
Find: The acceleration of the bottom B of the ladder,
and the ladder’s angular acceleration at this instant.
A
B
Section 16.7
Chapter 16 lecture examples, page 6
More examples for Chapter 16
Example #1
Determine normal acceleration at point A, in
terms of scale form an, and vector form an,
respectively.
A
w = 3 rad/s
a = 4 rad/s2
r=5m
O
Example #2: determine acceleration at point A for the
following two cases
A
A
O
O
ao = 20 m/s2
Same r = 5 m
Same w = 3 rad/s
Same a = 4 rad/s2
Chapter 16 lecture examples, page 7
17.1
Moment of inertia
Example #1
Given: The volume shown with density r = 5 slug/ft3.
Find: The mass moment of inertia of this body
about the y-axis: Iy
Section 17.1
Example #2
Given: Two rods assembled as shown, with each
rod weighing 10 lb.
Find: 1) The location of the center of mass G: y
and 2) moment of inertia about an axis passing
through G of the rod assembly: IG
Section 17.1
Chapter 17 lecture examples, page 1
17.2
17.3
Planar kinetic equations of motion
Equations of motion: translation
Example #1
Given: The uniform crate has a mass m and rests on a
rough pallet.
Find: 1) The acceleration of the pallet ap that causes
the crate to tip, and 2) prove that at the same time, the
relationship ms = b/h holds, where ms is the coefficient of
static friction between the crate and pallet.
Sections 17.2-17.3
Example #2
Given: The pipe has a mass of m = 800 kg and is being
towed behind a truck. The angle q = 30o. The coefficient
of kinetic friction between the pipe and the ground is mk
= 0.1.
Find: The acceleration atruck of the truck and the
tension T in the cable.
atruck
Sections 17.2-17.3
Chapter 17 lecture examples, page 2
17.4
Equation of motion: rotation about a fixed axis
Example
Given: A rod with mass of 20 kg is rotating at
5 rad/s at the instant shown. A moment of 60
N·m is applied to the rod.
Find: The angular acceleration a and the
reaction force R at pin O when the rod is in
the horizontal position.
Section 17.4
Chapter 17 lecture examples, page 3
17.5
Equation of motion: general plane motion
Example
Given: A spool has a mass of 8 kg and a radius
of gyration (kG) of 0.35 m. Cords of negligible
mass are wrapped around its inner hub and outer
rim. There is no slipping.
Find: The angular acceleration (a) of the spool.
Step #1:
FBD (forces + moments)
KD (maG + IGa)
Step#2:
Build bridges (XYM equations that connect FBD
& KD)
Step #3:
Review equations: How many unknowns?
Hopefully: No. of equations = No. of unknowns
Section 17.5
Group Problem-Solving
Given: A 50 lb driving-wheel has a radius of gyration kG
= 0.7 ft. While rolling, the wheel slips with mk = 0.25.
Find: The acceleration aG of the mass center G.
Section 17.5
Chapter 17 lecture examples, page 4
Chapter 17 more examples:
Chapter 17 lecture examples, page 5
18.1
18.2
18.3
18.4
Kinetic energy
The work of a force
The work of a couple
Principle of work and energy
Example
The spool has a weight of 150 lb and a radius of gyration of
KG = 2.25 ft. A horizontal force of P = 40 lb is applied to a
cable wrapped around its inner core. If the spool is
originally at rest, determine its angular velocity after the
mass center G has moved 10 ft to the right. The spool rolls
without slipping. Neglect the mass of the cable.
Sections 18.1-18.4
Chapter 18 lecture examples, page 1
18.5
Conservation of energy
Example 1
Given: An automobile tire has a mass of 7 kg
and radius of gyration G of KG = 0.3 m. The tire is
released from rest at A on the incline and rolls
without slipping. Neglect friction.
Find: Its angular velocity when the tire reaches
the horizontal plane.
Section 18.5
Example 2
Given: The 50-lb wheel has a radius of gyration about its
center of gravity G of KG = 0.7 ft. The wheel is released
from rest. It rolls without slipping. The spring AB has a
stiffness k = 1.20 lb/ft and an unstretched length of 0.5 ft.
Find: Its angular velocity when the wheel has rotated
clockwise 90o from the position shown.
Section 18.5
Chapter 18 lecture examples, page 2
19.1
19.2
Linear and angular momentum
Principle of impulse and momentum
Example #2
Sections 19.1-19.2
Chapter 19 lecture examples, page 1
19.3
19.4
Conservation of momentum
Eccentric impact
Example #1
Given: A slender rod (Wr = 5 lb) has a
wood block (W w = 10 lb) attached. A bullet
(W b = 0.2 lb) is fired into the block at 1000
ft/s. The pendulum is initially at rest and
the bullet embeds itself into the block. The
moment of inertia at A (including the rod,
woodblock, and the embedded bullet ) is
IA= 2.239 slug·ft2.
Find: The angular velocity of the pendulum
just after impact.
Sections 19.3-19.4
Example #2:
Given:
• Bullet mass = 0.004 kg
• Rod mass = 5 kg
• Mass moment of inertia of
the rod (around its mass
center, does not include the
embedded bullet) = 0.417
kg.m2
700
Find: angular velocity of the
rod after impact
Chapter 19 lecture examples, page 2
22.1
Undamped free vibration
Example #1
Section 22.1
Example #2
Determine the period of vibration t for
the simple pendulum. The bob has a
mass m and is attached to a cord of
length l. Neglect the size of bob.
Section 22.1
Chapter 22 lecture examples, page 1
More lecture examples, page 1
More lecture examples, page 2