Review of Planar Kinematics and Kinetics
General Features of Planar (2-D) motionof a rigid body
1. Translation (No rotation)
, , and : 3-D vectors but only 2 components change
2. Rotation
Motion of any point P in a rigid body: Restricted on a circle
Directions of and : Fixed (Normal to the plane of rotation)
3. General motion = Translation + Rotation
Kinematics
Define the object’s position:
Find the velocity (Time derivative of displacement)
Find the acceleration (Time derivative of velocity)
Kinetics
Find all forces acting on the object.
These forces generate the acceleration along the direction of force
or
1. 2D Kinematics of a rigid body
- How to determine Velocity and Accel. of a point in the body
Translation
Position
- = Position vector of point A (B) in the body
- = Relative-position vector of B with respect to A
Velocity
Acceleration
Rotation about a fixed axis (Polar coordinate system)
(1) Position of a point P in the body:
(2) Velocity of a point P
where (angular speed)
Direction
(3) Acceleration of a point P
where (angular acceleration)
Direction (Acceleration) or (Deceleration)
1.Tangential comp. (Faster and slower rotation)
2.Normal comp. (Centripetal)
General Plane Motion(= Translation + Rotation)
Analysis Method:
Step 1. Set a Fixed reference frame (Origin O)
Step 2. Set a Translating reference frame (Origin Ain the body)
Step 3. Separate General motion of a point B of interest into
= Translation of A + Relative motion (Rotation) of B about A
(1) Position of B: (Arbitrary point in the body)
(2) Velocity of B:
= Translation of A + Rotation of B about A
(3) Acceleration of B:
2. 2D Kinetics of a rigid body
- How to establish Newton’s equations of motion
Equations of motion
(1) Translation – Effect of Forces [Mass (m) and Acceleration ()]
: 2 equations (2D planar motion)
(2) Rotation – Effect of Moment (torque)
[Moment of inertia (I) and angular acceleration ()]
: 2 equations
Finding Moment of inertia (I )
-Dependant to the Body shape & the Axis of rotation.
(Discrete) or I = or (Continuous)
Parallel-Axis Theorem:
where IG = Moment of inertia about the axis passing through the mass center G
d = Perpendicular distance between two parallel axes
(See the back cover of textbook for typical examples of I.)
Work and Energy
Kinetic energy:
Potential energy: (= Angle between and )
= Negative of Work of a Force ()
Special examples
: Constant force
: Gravitational force
: Spring force
Principle of Work and Energy
: Total work done by all the external forces on the body
= Difference in Kinetic energy before and after applying the force.
Conservation of (Mechanical) Energy (For a conservative force)
or or
Impulse (How fast does the momentum change?)
Momentum
Linear momentum:
Angular momentum:
(about an axis passing through G)
Principle of Impulse and Momentum
→ (Linear impulse)
→ (Angular impulse)
Conservation of momentum
If = 0 =
If = 0 =
For a momentum change;
Over a short (long) time period
Large (small) force felt by a body
e.g. Egg falling on hard floor or carpet