Angular Kinematics

Question 1

Angular Motion

Answer 1

Rotation about a point

Within or outside body

Question 2

3 Units to Measure Angles

Answer 2

Degrees (°)
-Most common


Revolutions or Rotations

• 1 revolution = 360°
• Qualitative analysis

Radian

• 1 radian = 57.3°
• Most appropriate

Question 3

What is a Radian?

Answer 3

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

Question 4

Radians

Answer 4

Determined in multiples of pi (pi = 3.14)
1 radian = 57.3 degrees
pi radians = 180 degrees
2pi radians = 360 degrees

Question 5

Converting to Radians

Answer 5

How many radians are in 72 degrees?

72xrad/57.3 = 1.26 rad

Question 6

Angular Distance

Answer 6

Total of all angular changes during its path of motion

-Analagous to linear distance

Question 7

Angular Displacement

Answer 7

Difference between the initial and final positions of a rotating object

• Analogous to linear displacement
• Indicate direction

Question 8

Angular Speed

Answer 8

Angular distance traveled per unit of time

Question 9

Angular Velocity (w)

Answer 9

Rate of change of angular displacement per unit of time
Direction depends on rotational direction

w= Δ θ/t = θf - θi/t

Question 10

Angular Acceleration (a)

Answer 10

Rate of change in angular velocity
α = Δω/t
α =ωf - ωi/t

Question 11

Relationship between Linear and Angular Motion

Answer 11

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

Question 12

Linear Distance & Angular Displacement

Answer 12

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

Question 13

Linear and Angular Velocity

Answer 13

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

Question 14

Linear & Angular Acceleration

Answer 14

Acceleration of a body in angular motion may be resolved into components

• Radial (a.k.a. centripetal)
• Tangential

Question 15

Tangential Acceleration (at)

Answer 15

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

Question 16

Relationship between tangential acceleration & angular acceleration

Answer 16

at = αr

at = instantaneous tangential acc.
α = angular acceleration (rad/s/s)

α must be expressed in rad/s/s

Question 17

Radial / Centripetal Acceleration (ar)

Answer 17

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?

Question 18

Radial / Centripetal Acceleration (ar) (2)

Answer 18

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)

Question 19

Radial Acceleration Skaters/Skiiers

Answer 19

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

Question 20

Linear & Angular Acceleration Throwing

Answer 20

• 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

Question 21

Direction of Rotation

Answer 21

Counter-clockwise = positive
Clockwise = negative

Right Hand Rule
Direction fingers curl = positive
Direction thumb points = positive