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