Angular Position, Velocity and Acceleration
AP Physics C
Rigid Body
‐ An object or system of particles in which
distances between the particles are constant. There is no deformity in the object.
‐ Real object have some deformity.
Translation, Rotation, Rolling
• Translational motion: all particles in the object have the same instantaneous velocity (linear motion) • Rotational motion: all particles in the object have the same instantaneous angular velocity • Rolling motion: combination of translation and rotation
Polar Coordinates
• Radius r
• Angle θ measured counterclockwise from the + x axis
r
‐Angular position, θ is positive counterclockwise from the + x axis
O is the point
through which
the axis of
rotation
passes.
r
Angular Displacement
Δθ=θ−θο
(Final Angle-Initial Angle)
r
In this figure qo=0
Units for Angular Displacement
• Radian
– One full revolution is 2π radians.
– Radian is actually unitless. – The radian is used in the angular kinematics equations.
• Degree – 60 minutes in 1 degree, 60 seconds in 1 minute
– The degree is not used in the angular kinematics equations.
Radian
• Radian (rad) is the angle subtended by an arc length, s, equal to the radius.
• When s=r, θ=1 rad
s=r
r
How many radii lengths fit in an arc length s?
θ= s
r
Æ
s=θr
θ in radians
r meters
s meters • Why is the radian actually unitless?
1 rad = θ [rad] = = 57.3°
[degrees]
Average Angular Speed
θ f − θi
Δθ
=
ω=
t f − ti
Δt
SI unit : rad/sec
or 1/sec or sec‐1
• Note: common unit rpm (revolutions per minute) • Angular speed will be positive if θ is increasing (counterclockwise)
Instantaneous Angular Speed
ω≡
lim
Δt → 0
Δθ d θ
=
Δt
dt
Average Angular Acceleration
ω f − ωi
Δω
α=
=
t f − ti
Δt
Instantaneous Angular Acceleration
α≡
lim
Δt → 0
Δω d ω
=
dt
Δt
•
Units: rad/s² or s‐2 •
Angular acceleration is positive if an object rotating counterclockwise is speeding up or if an object rotating clockwise is slowing down.
Right Hand Rule
• ω, α are the magnitudes of the velocity and acceleration vectors.
• The directions are given by the right‐
hand rule.
Right Hand Rule
Wrap the four fingers in the direction of rotation. The thumb shows the direction
of the angular velocity vector.
+ angular velocity for counterclockwise rotation.
‐ angular velocity for clockwise rotation. Rotational Kinematic Equations
ω f = ωi + α t
1 2
θ f = θ i + ωi t + α t
2
ω = ω + 2α (θ f − θi )
2
f
2
i
1
θ f = θ i + (ω i + ω f ) t
2
Notes
–In solving rotational motion problems you must chose a rotational axis.
–The object may return to its original angular position.
Example #1
The angular position of a swinging door is described by θ= 5+10t+2t2, θ is in radians and t is in sec. Detremine the angular position, angular speed and angular acceleration of the door a)at t=0 and b) at t=3s.
Ans: a)5 rad, 10rad/s, 4 rad/s2 , b)53 rad, 22rad/s, 4 rad/s2
Example #6
A centrifuge in a medical lab rotates at an angular speed of 3600 rev/min. When switched off, it rotates 50 times before coming to rest. Find the constant angular acceleration of the centrifuge.
Ans: α= ‐2.26x10 2 rad/s2