Module 3:
Plane Motion
Dynamics 24859
Sheridan Institute of Technology and Advanced Learning
Outline
Review of Module 6 – Kinematics of Angular Motion
Module 7 – Plane Motion
• Relative Motion
• The Rolling Motion
• Instantaneous Center of Rotation
• Summary of Module 7.
• Assessment/Homework
• Next Class
Plane Motion
Critical Learning Outcomes:
Upon completion of the lesson the students will be able to:
• Solve for Linear values of Displacement, Velocity or Acceleration in
either Absolute or Relative Terms.
• Define and Locate an Instantaneous Center of an Object or Mechanism.
• Determine both Linear and Angular Velocities of Various Mechanisms
by means of Instantaneous Centers.
Review of Kinematics
of Angular Motion
Angular Velocity:
ω = Δθ/Δt
when “n” is in rpm: ω = 2πn/60
Angular Acceleration:
Rectilinear
α = Δω/Δt
Angular
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S = V0t+½at2
V
θ = ω0t+ ½ αt2
= v0 + at
ω = ω0 + αt
V2 = v02 + 2as
ω2 = ω02 + 2αθ
Review of Kinematics
of Angular Motion
Relationship between Rectilinear and Angular Motion:
S=rθ
V=rω
and
a =r α
Tangential Acceleration = Velocity Change (Magnitude)
Normal Acceleration = Velocity Change (Direction)
Relative Motion – Real Life Applications
General Plane Motion
General Plane Motion
Plane Motion
• What is Relative Motion?
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A body in motion may have
Displacement
Velocity
Acceleration
Plane Motion
• What is absolute velocity?
• Since the earth is considered stationary, a velocity
measured with respect to earth is an absolute velocity.
• What is Relative Velocity?
• When a velocity of one object is related to that of another
reference object that is also moving, then the velocity is
known as relative velocity.
• A relative velocity has no meaning unless the reference or
point to which the velocity is relative is stated.
Plane Motion
• Notations to be used:
• VA = Velocity of A
• VB = Velocity of B
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VB/A = Velocity of B with respect to A
VA/B = Velocity of A with respect to B
and
Relationships:
VB = VA + VB/A
VA = VB + VA/B
Plane Motion
• What is a plane motion?
• Let us look at the relative motion between two points on the
same object. This will occur when an object moves with
general plane motion: simultaneous translation and rotation.
• Example Fig 12-5, 12-6 and 12-7
• Example-Plane Motion
• Plane Motion = Translation + Rotation
• Translation: A moving downward and B moving to the
Right.
• Rotation: Bar rotates counter clockwise direction about its
centre.
Plane Motion
Plane Motion – Example 12-2
Solving Problems
by Relative Velocity Method
• We employ relative velocity method to solve problems:
• Relative displacement SA/B = rθ
• Relative velocity VA/B = rω
• “r” = AB
• While solving problems, we will employ vector additiontip-to-tail.
• Relative velocity is always perpendicular to the rotating
link.
Solving Problems
by Relative Velocity Method
• While solving problems, we will employ
vector addition- tip-to-tail.
• VA = VB + VA/B
•
In the above equation there are three magnitudes and three
directions, out of which four of them must be known before the
Problem could be solved.
Plane Motion – Example 12-2
Plane Motion – Example 12-2
Plane Motion – Example 12-4
Plane Motion – Example 12-4
Plane Motion – Example 12-4
Plane Motion – Example 12-5
Plane Motion – Example 12-5
Plane Motion – Problem 12-10
Plane Motion – Problem 12-13
Plane Motion – Problem 12-16
Plane Motion – Problem 12.21
Plane Motion - The Rolling Wheel
 If a wheel is rolled from one position to another, it
has both rotational and translational motion.
 When there is No Sliding, we have Pure Rolling.
Rolling Wheel
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If the wheel is pivoted at its centre, then the motion
would have only rotational.
For visualization:
A wheel is off the ground:
It is rotating at 8 radian/sec clockwise.
Velocity of A is the velocity with respect C.
vA/C = rω = 0.5(8)= 4m/s.
Rolling Wheel
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In translational motion, every point on the wheel must be moving
with same velocity.
For visualization:
Wheel lowered to the ground:
Point A stationary, but there is still relative velocity between A and C.
vC/A = 4 m/s
Rolling Wheel
For visualization:
In these diagrams, rotational (Fig. 12-38), translational (Fig. 12-39) components. (Fig
12-40) shows two motions superimposed: the result is Plane Motion.
.
Plane Motion – Example 12-9
Plane Motion – Example 12-9
Rolling Wheel – Example 12 -10
Rolling Wheel – Example 12 -10
Rolling Wheel – Problem 12-32
The cylinder shown below is rolling to the right with a velocity of 6
m/s. For the instant shown, determine (a) the angular velocity of the
cylinder and (b) the linear velocity of point B.
Rolling Wheel – Problem 12-33
The cylinder shown below is rolling to the right with a velocity of 8
m/s. Determine the velocity of point B at the position shown.
Rolling Wheel – Tutorial P12-34
A cord is wound in a slot of cylinder A. Mass B moves downwards
with a velocity of 6 m/s. Assume no slipping of the cylinder and
determine the velocities of points E and C on cylinder A. If B
drops 4 m, how far does cylinder A move to the right?
Instantaneous Centre of Rotation
Consider a rigid body with plane motion consisting of movement
downward and counter clockwise rotation. Points A, B and C have
absolute velocities.
The body will appear to have pure rotation if it is viewed from a point at which all the
velocities are tangential velocities.
Construct a radius arm perpendicular to each velocity.
From point O and instant shown, the body would appear to have pure rotation.
The point O about which all velocities appear as tangential velocities is called the
instantaneous Center of rotation and zero velocity.
Instantaneous Centre of Rotation
In reference to the figure aside:
For the bar to remain in contact with the wall and floor,
A and B must have the velocities as shown.
Draw perpendicular from VA and VB, their intersecting
point becomes the instantaneous centre.
Instantaneous Centre of Rotation
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For any point on the bar,
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V = rω or
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ω = v/r
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ω = VA/AO = VB/OB = VC/OC
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ω = ωAB,
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ωOA = ωOB = ωAB
ωAB = VA/B/AB
Instantaneous Centre of Rotation – Example
12-11
Instantaneous Centre of Rotation – Example
12-11
Instantaneous Centre of Rotation – Example 12-11
Instantaneous Centre of Rotation – Example 12-12
Instantaneous Centre of Rotation – Example 12-12
Instantaneous Centre of Rotation – Problem 12-49
Practice Questions
Practice Questions
Practice Questions
Practice Questions
SUMMARY: Plane Motion
Relative Motion
Rolling Wheel
Instantaneous
Center of
Rotation
SUMMARY: Plane Motion
Plane Motion
Did we Achieve our Goal?
Critical Learning Outcomes:
Upon Completion of the Lesson Students were be able to:
• Solve for Linear values of Displacement, Velocity or Acceleration in
either Absolute or Relative Terms.
• Define and Locate an Instantaneous Center of an Object or Mechanism.
• Determine both Linear and Angular Velocities of Various Mechanisms
by means of Instantaneous Centers.
Next Lecture
Module 8: Kinetics of Angular Motion
• Chap 13 page 444 -472 (Walker’s text)
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