Angular Kinematics Flashcards
Angular Motion
Rotation about a point
Within or outside body
3 Units to Measure Angles
Degrees (°)
-Most common
Revolutions or Rotations
- 1 revolution = 360°
- Qualitative analysis
Radian
- 1 radian = 57.3°
- Most appropriate
What is a Radian?
Ratio of the arc length (r) of a circle to the radius length
-Number of radii in an arc length
Radians = arc length/r
Arc length & radius are units of m
Radian is a dimensionless measure
Radians
Determined in multiples of pi (pi = 3.14)
1 radian = 57.3 degrees
pi radians = 180 degrees
2pi radians = 360 degrees
Converting to Radians
How many radians are in 72 degrees?
72xrad/57.3 = 1.26 rad
Angular Distance
Total of all angular changes during its path of motion
-Analagous to linear distance
Angular Displacement
Difference between the initial and final positions of a rotating object
- Analogous to linear displacement
- Indicate direction
Angular Speed
Angular distance traveled per unit of time
Angular Velocity (w)
Rate of change of angular displacement per unit of time
Direction depends on rotational direction
w= Δ θ/t = θf - θi/t
Angular Acceleration (a)
Rate of change in angular velocity
α = Δω/t
α =ωf - ωi/t
Relationship between Linear and Angular Motion
Motion of any point on a rotating body can be explained in linear terms (d)
Need to Know:
- Axis of rotation
- Radius of rotation (r)
- -Distance from axis to point of rotation
- Kinematics in RADIANS
Linear Distance & Angular Displacement
d = θ r
θ must be expressed in the units of radians for this expression to be valid
Note: radians are “unit-less”
Increase radius of rotation = increase in linear displacement
Linear and Angular Velocity
vt = ω r
w must be expressed in the units of radians/s for this expression to be valid
Linear velocity of rotating object is tangential (perpendicular) to path of motion
Linear & Angular Acceleration
Acceleration of a body in angular motion may be resolved into components
- Radial (a.k.a. centripetal)
- Tangential
Tangential Acceleration (at)
Represents the change in linear velocity for a body traveling on a curved path
Directed tangential to curved path
at = Vt2 - Vt1 / Δt
Vt1 = linear velocity at t1 Vt2 = linear velocity at t2
Tangential acceleration is tangential to path of motion
Relationship between tangential acceleration & angular acceleration
at = αr
at = instantaneous tangential acc. α = angular acceleration (rad/s/s)
α must be expressed in rad/s/s
Radial / Centripetal Acceleration (ar)
Newton stated in “Principia” (the book where he first published his laws):
“A centripetal force is that by which bodies are drawn or impelled, or any way tend, towards a point as to a center”
…essentially saying – an object must be forced to follow a curved path
Change in direction = change in velocity
- Change in velocity = acceleration!
- Therefore, acceleration is required to change direction and keep an object traveling along a curved path – but where does it come from?
Radial / Centripetal Acceleration (ar) (2)
Directed toward the center, along the curved path
Represents change in direction
Linear speed of an object traveling a curved path remains constant
However, direction of object traveling a curved path is constantly changing
ar = ω2r
OR
ar = Vt2/r
Radial acceleration is also perpendicular to path of motion (center seeking)
Radial Acceleration Skaters/Skiiers
Skaters or skiers on a curve must force themselves to change directions.
Changes in direction result in changes in velocity – even if velocity magnitude is constant. Why?
Changes of velocity result in radial accelerations
Radial acceleration is caused by ground reaction forces directed toward the center of the turn
Linear & Angular Acceleration Throwing
- Ball follows curved path
- -Due to restraining force of arm
-Restraining force (arm) causes ar toward the center of the curved path of motion
- At release, ar no longer exists
- -No longer a restraining force
- Thus, ball follows path of tangent to curve at instant of release
- -Timing of release is critical
Direction of Rotation
Counter-clockwise = positive Clockwise = negative
Right Hand Rule
Direction fingers curl = positive
Direction thumb points = positive
