review of planar kinematics and kinetics - college of arts...
TRANSCRIPT
Review of Planar Kinematics and Kinetics
• General Features of Planar (2-D) motion of 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 ∴
All the points on the body have the same motion!
Curvilinear
A
B
because of a Rigid body
Rectilinear
• 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)
because of a rigid body
Note: Velocity ( ) & Accel. ( ): Motion of a point mass P in the body
Angular vel. ( ) & Angular accel. ( ): Motion of a whole body
1 2
• General Plane Motion (= Translation + Rotation)
Analysis Method:
Step 1. Set a Fixed reference frame (Origin O)
Step 2. Set a Translating reference frame (Origin A in 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