Kinematics of Particles and Bodies Flashcards
What is the acceleration of a particle with respect to its tangential unit vector and its normal unit vector?
v = v u(t) where u is the unit vector
a = dv/dt * u(t) + v * du(t) / dt
du(t) / dt = d theta / dt * u(n)
a = dv/dt u(t) + v d theta / dt * u(n)
u(t) is the motion tangentially u(n) is the motion perpendicular
What is the equation for a(small t) and what does it represent?
It is equal to dv / dt and it represnets the variation of the magnitude of velocity
What is a (small n) and what does it represent?
It is equal to v^2 / radius and it represents the variation of direction of velocity
What is the tangential unit vector of curvilinear motion using Cartesian Co-ordinates
cos(x) i + sin(x) j where x is the angle between the x axis and the direction of tangential motion
What is the normal unit vector of curvilinear motion using Cartesian Co-ordinates?
-sin(x) i + cos(x) j where x is the angle between the x axis and the direction of tangential motion
What is the length of an arc (s) on a circle with a radius R and an angle x?
s = Rx
What is the angular velocity of a particle, and how do we derive it?
v = ds / dt = R * d theta / dt = R * Omega
What is the angular acceleration relative to the angular velocity?
a = R * Omega ^ 2
What are the two elementary motions of a rigid body?
- Translation
- Rotation
What is translation?
It’s motion when a line attached to the rigid body stays parallel to itself during the motion
What is the velocity and acceleration of all points in a rigid body under translation?
Velocity and acceleration is constant for all particles under translation
What is the velocity and acceleration of all points in a rigid body under translation?
Velocity and acceleration is constant for all particles under translation
What is the name of motion when the paths are curves other than straight lines?
Curvilinear motion
What is the name of motion when the paths of all points in the body are circles?
Circular motion
What is Rotation?
The motion when two points of a rigid body remain fixed during the motion. If two points remain fixed during the motion, then all points on the line connecting them are also fixed
What is the equation for angular velocity?
It is the differentiation of the angle travelled with respect to time
d theta / dt
If points A and B are both points within a rigid body, what is the relative velocity of B when compared to A?
We know that from the origin, r(B) = r(A) + r(B/A)
If we differentiate these, we can then find the velocities of these points
v(B) = v(A) + v(B/A)
Using the equations of circular motion, we can then calculate v(B/A) using the circular velocity
v(B/A) = dr(B/A) / dt = Omega X r(B/A)
Now we can do a final substitution to find v(B)
v(B) = v(A) + Omega X r(B/A)
The first part of the equation represents the translation, and the second part of the equation represents the rotation
What is the Instantaneous Centre of Rotation?
The point which, at an instant of time has zero velocity
How can you locate the Instantaneous Centre of Rotation geometrically?
Draw lines perpendicular from the velocities vectors of two points in the solid body, and see where they intersect
How do you locate the Instantaneous Centre of Rotation if point B is on the perpendicular to the velocity of point A?
We can still use a geometric solution, but we need to know the magnitudes of the velocities of the two points
Where is the Instantaneous Centre of Rotation if velocities A and B are equal?
It is located at infinity. This also means that the motion is a translation
