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Unformatted text preview: Rotational Motion
Chapter Outline GENERAL PHYSICS
PHY 302K Angular Quantities
Constant Angular Acceleration Chapter 8: Rotational Motion
Rolling Motion
Torque Maxim Tsoi Rotational Dynamics: Torque and Rotational Inertia
Rotational Kinetic Energy
Angular Momentum and Its Conservation Physics Department,
The University of Texas at Austin
http://www.ph.utexas.edu/~tsoi/
302K: Ch.8 302K: Ch.8 Angular Position, Velocity, and Acceleration Angular Position, Velocity, and Acceleration Rigid object Angular position • It is convenient to represent position of P with its
Analysis of a rotational motion can be greatly simplified by
assuming that an object in motion is rigid polar coordinates (r, ) compact disc rotating
about a fixed axis
through O r distance from the origin (r=const) measured counterclockwise from a reference line • A rigid object is one that is nondeformable – that is, the relative
locations of all particles of which the object is composed remain • P moves through an arc of length s: constant s r s
r is measured in radians (rad) 180o= rad is angular position of the rigid object
• We can associate the angle with the entire object
as well as with an individual point
302K: Ch.8 302K: Ch.8 Angular Position, Velocity, and Acceleration Angular Position, Velocity, and Acceleration Angular displacement and speed • Angular displacement: f i
• Average angular speed: f i
t f ti t • Instantaneous angular speed: t 0 t A particle on a rotating
rigid object moves from
A to B in time interval t Righthand rule • Angular velocity vector
• For rotation about a fixed axis, the
only rotation that uniquely specifies
the rotational motion is the direction
along the axis of rotation
• Righthand rule: when the four
fingers of the right hand are wrapped
in the direction of rotation, the
extended right thumb points in the
direction of lim units radians per second (rad/s) 302K: Ch.8 d
dt 302K: Ch.8 1 Angular Position, Velocity, and Acceleration Rotational Motion with Constant Acceleration Angular acceleration Kinematic equations for constant angular acceleration • If the instantaneous angular speed of an
object changes from I to f in the time
interval t the object has an angular
acceleration A particle on a rotating
rigid object moves from
A to B in time interval t • Using definitions of the average angular acceleration and angular
velocity: • Average angular acceleration: f i
t f ti t t t f i i t 1 t 2
2 2 i2 2 f i f f i t • Instantaneous angular acceleration : f i i t t
1
2 lim
t 0 t f i t f i 1 i f t
2 2 2
2
v xf v xi 2a x x f x i v xf v xi a x t x f xi units radians per second squared (rad/s2)
302K: Ch.8 1
2 v xi v xf t x f x i v xi t 1 a x t 2
2
Example 1 302K: Ch.8 Angular and Linear Quantities Rotational Kinetic Energy Relationships between angular and linear speed and acceleration Rigid object rotates about a fixed axis every particle of the object Moment of inertia Kinetic energy of an object is the energy associated with its motion through space moves in a circle whose center is the axis of rotation
• Relation of the linear v (tangential) speed to the
angular velocity:
• Relation of the tangential
acceleration to the angular
acceleration: mi v i2 m i ri 2
2 2 s r r
t
t Ki K R K i 1 2 mi ri2
2 v r
at r
t
t i • Kinetic energy of an
individual particle
• Total kinetic energy of
the rotating rigid object i I mi ri2 • Moment of inertia i • Centripetal acceleration in ac terms of the angular speed: • The total linear acceleration: 2 v r 2
r measure of the resistance
of an object to changes in
its rotational motion K R 1 I 2
2 a at ac at2 ac2 r 2 4 302K: Ch.8 Example 2 302K: Ch.8 Calculation of Moments of Inertia Rolling Motion of a Rigid Object Homogeneous rigid objects with different geometries In general – very complex motion Moment of inertia: I lim mi 0 Cylinder rolling on a straight path r m
2 i i i • Density =m/V mass per unit volume
• Homogeneous object • Trajectory of a particle on the rim (red) cycloid =const • Trajectory of the center of mass straight line
If an object rolls without slipping on the surface (pure rolling motion) a simple relationship exists between its rotational and translational motions 302K: Ch.8 Example 5 302K: Ch.8 2 Rolling Motion of a Rigid Object Rolling Motion of a Rigid Object Condition for pure rolling motion Combination of translational and rotational motion • As cylinder rotates through an angle , its
center of mass moves a linear distance s=R vCM s R R
t
t • The magnitude of the linear
acceleration of the center of mass aCM vCM R R
t
t linear velocities of various points
on and within the cylinder 302K: Ch.8 302K: Ch.8 Rolling Motion of a Rigid Object Rolling Motion of a Rigid Object Kinetic energy of rolling cylinder Using energy methods K
I 2
K CM
2 2
MvCM
K
2 I CM Mv 2
2
2 2
CM 2 2
I v MvCM CM CM 2 R 2 I CM 2 1 2 M vCM
2 R Conservation of mechanical energy: I 2
K P
2 K f U f Ki Ui 2
1 I CM M vCM 0 0 Mgh
2
2 R I P I CM MR 2
302K: Ch.8 2 gh
vCM 2
1 I
CM MR 1 2 302K: Ch.8 Relationship between and Torque
Torque and moment arm Rotational analog of Newton’s second law • When a force is exerted on a rigid Ft ma t object pivoted about an axis, the
object tends to rotate about that axis mr r mr I
2 • The tendency of a force to rotate an
object about some axis is measured by
a vector quantity called torque • For a rigid object: vector point to the line of action of F • If two or more forces tend to produce rotating in a circle of radius r
under the influence of a
tangential force Ft and a radial
force Fr Torque acting on a particle is proportional to its angular acceleration! rF sin Fd d – moment arm  distance from pivot • Consider a particle of mass m Ft r ma t r 1 Ft ma t r Ft rat m r 2 m 2 r m I
2 F1d1 F2 d 2 rotation of a rigid object net torque
302K: Ch.8 302K: Ch.8 Example 78 3 Angular Momentum Conservation of Angular Momentum Angular momentum of a rotating rigid object 3rd conservation law • Consider a rigid object rotating about a fixed axis • The total angular momentum of a system is constant in both magnitude
and direction if the external torque acting on the system is zero I I I I t I t 0 t 0 angular momentum of the whole object: Ltot const L I • Change of the object’s angular I momentum is equal to the net external
torque L
0
t or Li L f • L=I=const if mass of the rotating system undergoes redistribution in some way L
t system’s moment of inertia I changes angular speed changes
• Example spinning figure skater 302K: Ch.8 302K: Ch.8 SUMMARY
Rotational Motion
Angular acceleration: t
t
• Kinematic equations for rotational motion: • Angular speed: f i t f i i t 1 t 2
2 2 i2 2 f i f • Relation between linear and angular quantities: s r
• Moment of inertia of a rigid object: f i 1 i f t
2
v r a t r I m i ri2
i • Rotational kinetic energy:
• Torque associated with force F: K R 1 I 2
2 rF sin • Newton’s 2nd law for rotational motion: I • Angular momentum of a rigid object rotating about a
fixed axis (z): Lz I • If net external torque acting on a system is zero total angular momentum of the system is conserved Li L f 302K: Ch.8 4 ...
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