10 Questions
What do R, θ, and z represent in polar coordinates?
Radial, angular, and axial coordinates of a particle
What do eR, eθ, and ez represent in polar coordinates?
Unit vectors corresponding to radial, angular, and axial directions
What is the role of Coriolis acceleration in cylindrical coordinates?
To account for motion due to Earth's rotation
Which type of acceleration does the term ⃗ contain in cylindrical coordinates?
Centripetal acceleration
What aspect of motion does the derivative of unit vectors represent in polar coordinates?
Changing directions of motion
At which point during its trajectory does a basketball reach its peak amplitude?
@ Point A (the peak amplitude)
In what context would a basketball player need to calculate initial velocity and launch angle?
Shooting the basketball into the hoop as a projectile
What characteristic of a projectile's motion is significant when calculating initial velocity for a basketball shot?
Peak height attained by the projectile
What is the primary purpose of using cylindrical coordinates in analyzing particle motion?
To account for motion in three-dimensional space
What parameter does θ represent in polar coordinates?
Azimuthal angle from reference direction
Study Notes
Coordinates
- Polar coordinates use the angle θ
- Cartesian coordinates use x-z, where z=L.cos θ and x=L.sin θ
Cause-and-Effect Principle
- If there is an effect, there must be a cause behind it
- In a well-defined problem, the cause's or effect's magnitude will not be known until the problem is solved
Dynamics
- Dynamics has two parts: kinematics and kinetics
- Kinematics is the study of the geometry of motion
- Kinetics deals with the forces imparted on bodies and the response (motion) of the bodies to these forces
Rigid Bodies
- A rigid body is a continuous distribution of particles, hence a particle system for which the relative distances and orientations of the constituent particles remain fixed
Kinematics of Particles
- A particle is a body that has mass but no volume
- Kinematics is the study of the geometry of motion
- To locate a particle, we must specify its location with respect to some reference by considering coordinate systems
Coordinate Systems
- There are many coordinate systems, including:
- Cartesian Coordinates
- Path Coordinates (normal & tangential)
- Cylindrical Polar Coordinates
Cartesian Coordinates
- Basis vectors are constant in direction and magnitude
- Position, velocity, and acceleration equations are simplified
- Figure 1-6 shows Cartesian coordinates
Path Coordinates (normal & tangential)
- Used to describe motion along a curved path
- Normal (n) and tangential (t) coordinates are used
- Figure 1-7 shows normal & tangential coordinates
- Velocity vector is always tangent to the motion path (t-direction)
- Acceleration has two components: change in magnitude and change in direction
Cylindrical Polar Coordinates
- Used when a particle moves along a 3-D curve
- Radial (R), angular (θ), and axial (z) coordinates are used
- Figure 1.36 shows cylindrical polar coordinates
- Instantaneous velocity is obtained by taking time derivatives of the position vector
- Derivative of unit vectors is necessary to derive acceleration
Example Problem
- A basketball player wants to shoot the basketball into the hoop
- The ball travels as a projectile and reaches a height of 5h=4 before it begins its descent towards the basket
- Calculate the initial velocity v0 and the player's angle to launch the ball
Test your knowledge on possible generalized coordinates like polar and cartesian coordinates, and concepts such as the Cause-and-Effect Principle. Explore dynamics, including kinematics and kinetics, with a focus on a simple model of a vehicle (tricycle model).
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