Chapter 10
• w = w = lowercase omega
• Rotational Motion
o Given an object that can pivot and rotate
o
o Angular displacement
§ When the point of interest is rotated by angle q
§
§Arc length: ℓ = 𝑅𝜃
• Length of the partial circular path
• In meters
§ Circumference: ℓ = 2𝜋𝑟
§ 2pradians = 360° = 1 revolution
o Modeled same as linear motion
§ Angular displacement: ∆𝜃 = 𝜃! − 𝜃"
• Radians, degrees, or revolutions
∆$
§ Average angular velocity: 𝑤+ = ∆%
• Rad/sec; deg/sec; rev/sec
• Can also get RPM (rev/min)
&$
§ Instantaneous angular velocity: w = &%
∆$
• 𝑤 = lim
∆%→( ∆%
• 𝑤 = omega (lowercase)
• 𝜃(𝑡) = 𝑐𝑡 !
&$
o &% = 2𝑐𝑡
o Implies angular acceleration
∆) ) *)
§ Angular acceleration (Average): 𝛼5 = ∆% = %! *% "
! "
• Rad/sec2
, &)
§ Instantaneous acceleration: 𝛼 = &%
∆)
• 𝛼 = lim
∆%→( ∆%
o Include path of a point as it circles around the center
§
§ Path length: ℓ = 𝑟∆𝜃
∆ℓ ,∆$
§ Change in path length: ∆% = ∆% = 𝑟𝑤
§ Tangential velocity: 𝑣% = 𝑟𝑤
• m*rad/sec à m/s
o Any time you multiply radians * anything, radians
disappear
• As radius increases, vt increases
•
&/ 0&)
§ Tangential acceleration: 𝑎%-. = &%# = &% = 𝑟𝛼
• Another form of acceleration for objects rotating around a pivot
point
•
/ ! (,))!
• Centripetal acceleration: 𝑎, = ,# = ,
o Always pointing radially inwards
o If magnitude does not change, direction does, so
acceleration is radially inwards
o Rotational Kinematics
§ Conversions
• 𝑥 = 𝑟𝜃
, • 𝑣 = 𝑟𝑤
• 𝑎 = 𝑟𝛼
§ Kinematic Equations
• 𝜃 = 𝜃( + 𝑤 +𝑡
• 𝑤 = 𝑤( + 𝛼𝑡
"
• 𝜃 = 𝜃( + 𝑤( 𝑡 + ! 𝛼𝑡 !
• 𝑤 ! = 𝑤( ! + 2𝛼(𝜃 − 𝜃( )
o Torque: force provided at a radius to make something spin
§ A vector
o Ex: A carousel of radius 2.5m is initially at rest, so 𝜃( = 0 and 𝑤( = 0. At t=0, the
,-&
angular acceleration is 𝑎 = 0.06 345 ! for 8 seconds. What is the angular velocity
at 8 seconds? If there is a child located r=2.5m from the center, what is the
tangential velocity? Tangential acceleration? Centripetal acceleration? And the
total acceleration of the child?
§ Angular velocity
• 𝑤 = 𝑤( + 𝛼𝑡
• 𝑤 = 0 + (0.06)(8)
• 0.48 rad/sec
§ Tangential velocity
• 𝑣% = 𝑟𝑤
• 𝑣% = (2.5)(0.48)
• 1.2 m/s
§ Tangential acceleration
• 𝑎%-. = 𝑟𝛼
• 𝑎%-. = (2.5)(0.06)
• 0.15 m/s2
§ Centripetal acceleration
(,))!
• 𝑎, = ,
".!!
• 𝑎, = !.8
• 0.576 m/s2 inward
§ Total acceleration
•
• 𝑎 ! = 𝑎% ! + 𝑎, !
• 𝑎! = (𝑟𝑎)! + (𝑟𝑤 ! )!
• 𝑎 = 𝑟√𝑎! + 𝑤 9
, • 𝑎 = B(𝑎%-. )! + (𝑎, )!
o Use values of ar and atan from above
• 𝑎 = B(0.15)! + (0.576)!
• 0.6 m/s2
o Rotational frequency: number of revolutions per second
,-& ,4/ )
§ Rotational frequency: 𝑓 = F 345 G F!:,-& G = !:
"
• Frequency = 345 = Hertz (Hz)
" !:
§ Period of rotation: 𝑇 = ; = )
• Sec
§ Ex: Given angular velocity as a function of time, a disk of radius r=3m
rotates at w=(1.6+1.2t) rad/sec. At t=2 sec, determine the instantaneous
values for angular acceleration, tangential velocity, tangential
acceleration, and centripetal acceleration?
• Instantaneous angular acceleration
o
&)
o 𝛼 = &%
&
o 𝛼 = &% (1.6 + 1.2𝑡)
o 1.2 rad/sec2
• Instantaneous tangential velocity
o 𝑣% = 𝑟𝑤
o 𝑣% = (3)(1.6 + (1.2 ∗ 2))
o 12 m/s
• Instantaneous tangential acceleration
o 𝑎%-. = 𝑟𝛼
o 𝑎%-. = (3)(1.2)
o 3.6 m/s2
• Instantaneous centripetal acceleration
/# !
o 𝑎, = ,
"!!
o 𝑎, = <
o 48 m/s2
o Angular Vectors and Torque
§ Two directions: positive and negative
• CCW is positive
• w = w = lowercase omega
• Rotational Motion
o Given an object that can pivot and rotate
o
o Angular displacement
§ When the point of interest is rotated by angle q
§
§Arc length: ℓ = 𝑅𝜃
• Length of the partial circular path
• In meters
§ Circumference: ℓ = 2𝜋𝑟
§ 2pradians = 360° = 1 revolution
o Modeled same as linear motion
§ Angular displacement: ∆𝜃 = 𝜃! − 𝜃"
• Radians, degrees, or revolutions
∆$
§ Average angular velocity: 𝑤+ = ∆%
• Rad/sec; deg/sec; rev/sec
• Can also get RPM (rev/min)
&$
§ Instantaneous angular velocity: w = &%
∆$
• 𝑤 = lim
∆%→( ∆%
• 𝑤 = omega (lowercase)
• 𝜃(𝑡) = 𝑐𝑡 !
&$
o &% = 2𝑐𝑡
o Implies angular acceleration
∆) ) *)
§ Angular acceleration (Average): 𝛼5 = ∆% = %! *% "
! "
• Rad/sec2
, &)
§ Instantaneous acceleration: 𝛼 = &%
∆)
• 𝛼 = lim
∆%→( ∆%
o Include path of a point as it circles around the center
§
§ Path length: ℓ = 𝑟∆𝜃
∆ℓ ,∆$
§ Change in path length: ∆% = ∆% = 𝑟𝑤
§ Tangential velocity: 𝑣% = 𝑟𝑤
• m*rad/sec à m/s
o Any time you multiply radians * anything, radians
disappear
• As radius increases, vt increases
•
&/ 0&)
§ Tangential acceleration: 𝑎%-. = &%# = &% = 𝑟𝛼
• Another form of acceleration for objects rotating around a pivot
point
•
/ ! (,))!
• Centripetal acceleration: 𝑎, = ,# = ,
o Always pointing radially inwards
o If magnitude does not change, direction does, so
acceleration is radially inwards
o Rotational Kinematics
§ Conversions
• 𝑥 = 𝑟𝜃
, • 𝑣 = 𝑟𝑤
• 𝑎 = 𝑟𝛼
§ Kinematic Equations
• 𝜃 = 𝜃( + 𝑤 +𝑡
• 𝑤 = 𝑤( + 𝛼𝑡
"
• 𝜃 = 𝜃( + 𝑤( 𝑡 + ! 𝛼𝑡 !
• 𝑤 ! = 𝑤( ! + 2𝛼(𝜃 − 𝜃( )
o Torque: force provided at a radius to make something spin
§ A vector
o Ex: A carousel of radius 2.5m is initially at rest, so 𝜃( = 0 and 𝑤( = 0. At t=0, the
,-&
angular acceleration is 𝑎 = 0.06 345 ! for 8 seconds. What is the angular velocity
at 8 seconds? If there is a child located r=2.5m from the center, what is the
tangential velocity? Tangential acceleration? Centripetal acceleration? And the
total acceleration of the child?
§ Angular velocity
• 𝑤 = 𝑤( + 𝛼𝑡
• 𝑤 = 0 + (0.06)(8)
• 0.48 rad/sec
§ Tangential velocity
• 𝑣% = 𝑟𝑤
• 𝑣% = (2.5)(0.48)
• 1.2 m/s
§ Tangential acceleration
• 𝑎%-. = 𝑟𝛼
• 𝑎%-. = (2.5)(0.06)
• 0.15 m/s2
§ Centripetal acceleration
(,))!
• 𝑎, = ,
".!!
• 𝑎, = !.8
• 0.576 m/s2 inward
§ Total acceleration
•
• 𝑎 ! = 𝑎% ! + 𝑎, !
• 𝑎! = (𝑟𝑎)! + (𝑟𝑤 ! )!
• 𝑎 = 𝑟√𝑎! + 𝑤 9
, • 𝑎 = B(𝑎%-. )! + (𝑎, )!
o Use values of ar and atan from above
• 𝑎 = B(0.15)! + (0.576)!
• 0.6 m/s2
o Rotational frequency: number of revolutions per second
,-& ,4/ )
§ Rotational frequency: 𝑓 = F 345 G F!:,-& G = !:
"
• Frequency = 345 = Hertz (Hz)
" !:
§ Period of rotation: 𝑇 = ; = )
• Sec
§ Ex: Given angular velocity as a function of time, a disk of radius r=3m
rotates at w=(1.6+1.2t) rad/sec. At t=2 sec, determine the instantaneous
values for angular acceleration, tangential velocity, tangential
acceleration, and centripetal acceleration?
• Instantaneous angular acceleration
o
&)
o 𝛼 = &%
&
o 𝛼 = &% (1.6 + 1.2𝑡)
o 1.2 rad/sec2
• Instantaneous tangential velocity
o 𝑣% = 𝑟𝑤
o 𝑣% = (3)(1.6 + (1.2 ∗ 2))
o 12 m/s
• Instantaneous tangential acceleration
o 𝑎%-. = 𝑟𝛼
o 𝑎%-. = (3)(1.2)
o 3.6 m/s2
• Instantaneous centripetal acceleration
/# !
o 𝑎, = ,
"!!
o 𝑎, = <
o 48 m/s2
o Angular Vectors and Torque
§ Two directions: positive and negative
• CCW is positive
