General Physics IGeneral Physics I

Lecture 10: Rolling Motion and Lecture 10: Rolling Motion and Angular MomentumAngular Momentum

Prof. WAN, Xin （万歆）万歆）

xinwan@zju.edu.cnhttp://zimp.zju.edu.cn/~xinwan/

OutlineOutline

● Rolling motion of a rigid object: center-of-mass motion + rotation around

● Generic rotational motion in the vector language− Kinematics/Energy/Torque

● Angular momentum of− A particle− A rotating rigid object

● Conservation of angular momentum

A CycloidA Cycloid

One light source at the center of a rolling cylinder and another at one point on the rim illustrate the different paths these two points take. The center moves in a straight line (green line), whereas the point on the rim moves in the path called a cycloid (red curve).

Pure Rolling MotionPure Rolling Motion

condition for pure rolling motion

Frictional force?

Translation + RotationTranslation + Rotation

Same Angular SpeedSame Angular Speed

At any instant, the part of the rim that is at point P is at rest relative to the surface because slipping does not occur.

In other words, all points rotate about P. But, how to calculate the moment of inertia about P?

Parallel-Axis TheoremParallel-Axis Theorem

r '=r−rCM

Parallel-Axis TheoremParallel-Axis Theorem

● Define r '=r−rCM

K =12∑i

mi(ω × r i)⋅(ω × r i)

=12∑i

mi [ω × (r i ' + rCM)]⋅[ω × (r i ' + rCM)]

=12∑i

mi [ω × r i ']⋅[ω × r i ']

+ ∑imi[ ω × r i ']⋅[ω × rCM ]

+12∑i

mi[ω × rCM ]⋅[ω × rCM ]

(1)

(2)

(3)

Parallel-Axis TheoremParallel-Axis Theorem

● Term (1): The rotational energy about the parallel axis through the center of mass

● Term (2):

● Term (3): as if the object shrink to the center of mass

K ' =12∑i

mi[ω × r i ']⋅[ω × r i ']

∑imi[ ω × r i ']⋅[ω × rCM ]=0

KCM =12 (∑i

mi ) [ ω × rCM ]⋅[ω × rCM ]

∑imi r i '=∑i

mi r i−(∑imi ) rCM=0

Parallel-Axis TheoremParallel-Axis Theorem

● Suppose the moment of inertia about an axis through the center of mass of an object is ICM. The parallel-axis theorem states that the moment of inertia about any axis parallel to and a distance D away from this axis is

Kinetic EnergyKinetic Energy

● The total kinetic energy of the rolling cylinder

● rotational kinetic energy about the center of mass

● translational kinetic energy of the center of mass

Example: Rolling SphereExample: Rolling Sphere

● For the solid sphere shown below, calculate the linear speed of the center of mass at the bottom of the incline and the magnitude of the linear acceleration of the center of mass.

● Accelerated rolling motion is possible only if a frictional force is present between the sphere and the incline to produce a net torque about the center of mass.● Despite the presence of friction, no loss of mechanical energy occurs because the contact point is at rest relative to the surface at any instant.

Moment of InertiaMoment of Inertia

Now, try it yourself.

Appendix: Spherical ShellAppendix: Spherical Shell

● First we calculate the moment of inertia of a spherical shell about a diagonal axis.

I z=∫dm(x2+y2)

By the rotational symmetry,

I x=I y=I z=13

(I x+ I y+ I z)

=23∫dm(x2+y2

+z2)=23∫dmR2

=23MR2

Appendix: Solid SphereAppendix: Solid Sphere

● By dimension analysis, the moment of inertia of a solid sphere (density r) is

I (R) = cM R2∝ cρ

4π

3R5

Since a spherical shell can be written as the difference of two solid spheres

I (R+ΔR)−I (R) =23

(ρ4πR2ΔR)R2

dI =8πρ

3R4dRΔR→0

I =8πρ

15R5

=25MR2

Conservation of EnergyConservation of Energy

Kinematic equation

CommentsComments

● All homogeneous solid spheres experience the same speed and acceleration on a given incline, independent of

− mass, and − the radius of the sphere

● Try hollow sphere, solid cylinder, or hoop, you will find:− Only the dimensionless factor would differ. − The factors depend only on the moment of inertia

about the center of mass for the specific body. ● In all cases, the acceleration of the center of mass is less

than the value the acceleration would have if the incline were frictionless and no rolling occurred.

Still Prefer the Newton's Laws?Still Prefer the Newton's Laws?

The frictional force does not change the total kinetic energy, but it contributes to F and decreases the acceleration of the center of mass.

TorqueTorque

● The torque t involves the two vectors r and F, and its direction is perpendicular to the plane of r and F. We can establish a mathematical relationship between t, r, and F, using a new mathematical operation called the vector product, or cross product

Will discuss more next week.

Angular Momentum: MotivationAngular Momentum: Motivation

Newton's second law

v×(mv ) = 0

Angular momentum(rotational analog)

Angular Momentum: DefinitionAngular Momentum: Definition

● The instantaneous angular momentum L of the particle relative to the origin O is defined as the cross product of the particle’s instantaneous position vector r and its instantaneous linear momentum p.

L is zero when r is parallel or antiparallel to p.

Magnitude:

SI Unit:

● The net torque acting on a particle is equal to the time rate of change of the particle’s angular momentum.

● Both the magnitude and the direction of L depend on the choice of origin.

● The expression is valid for any origin fixed in an inertial frame, as long as L and t are measured about the same origin.

Dependence of the OriginDependence of the Origin

A System of ParticlesA System of Particles

● The time rate of change of the total angular momentum of a system about some origin in an inertial frame equals the net external torque acting on the system about that origin.

The moment arm d from O to the line of action of the internal forces is equal for both particles.

Recall

A Rotating Rigid ObjectA Rotating Rigid Object

The Scalar VersionThe Scalar Version

● Now extend to a rigid object of arbitrary shape rotating about a fixed axis.

I: The moment of inertia

What’s Further?What’s Further?

L = ∑iri×mi v i = ∑i

r i×mi(ω×r i)

ri = x i x+ y i y+ zi z ω = ω x x+ω y y+ω z z

a×(b×c) = b(a⋅c) − c(a⋅b)Identity:

= ∑i[mi ri

2ω−mi ri(r i⋅ω ) ]

Lx = ∑imi( y i

2+ zi

2)ω x−∑i

mi x i y iω y−∑imi xi ziω z

L y = −∑imi y i x iω x+∑i

mi(zi2+xi

2)ω y−∑i

mi y i ziω z

Lz = −∑imi zi xiω x−∑i

mi zi y iω y+∑imi(x i

2+ y i

2)ω z

Let

Moment of Inertia is a MatrixMoment of Inertia is a Matrix

Lx = ∑imi( y i

2+ zi

2)ω x−∑i

mi x i y iω y−∑imi xi ziω z

L y = −∑imi y i x iω x+∑i

mi(zi2+xi

2)ω y−∑i

mi y i ziω z

Lz = −∑imi zi xiω x−∑i

mi zi y iω y+∑imi(x i

2+ y i

2)ω z

(LxL yLz

) = (I xx I xy I xzI yx I yy I yzI zx I zy I zz

) (ω x

ω y

ω z )

I = ∑i [mi r i⋅r i−mi r i r i ]

dyad product

Now you should be ready to advance by yourself.

rank-2 tensor

Symmetry Tells UsSymmetry Tells Us

I = (I 0 0 00 I 0 00 0 I 0

)

I = (I x 0 00 I 0 00 0 I 0

)

ExerciseExercise

A rigid rod of mass M and length l is pivoted without friction at its center. Two particles of masses m

1 and m

2 are

connected to its ends. The combination rotates in a vertical plane with an angular speed w. (a) Find an expression for the magnitude of the angular momentum of the system. (b) Find an expression for the magnitude of the angular acceleration of the system when the rod makes an angle q with the horizontal.

SolutionSolution

Conservation of Angular MomentumConservation of Angular Momentum

● The total angular momentum of a system is constant in both magnitude and direction if the resultant external torque acting on the system is zero.

If the system is an object rotating about a fixed axis, such as the z axis,

Figure Skating Figure Skating

Anything wrong?

Kepler’s Second LawKepler’s Second Law

● When a force is directed toward or away from a fixed point and is function of r only, it is called a central force.

Kepler’s Second LawKepler’s Second Law

● The radius vector from the Sun to a planet sweeps out equal areas in equal time intervals.

Isolated SystemsIsolated Systems

● The (mechanical) energy, linear momentum, and angular momentum of an isolated system all remain constant.

Example: Ball and StickExample: Ball and Stick

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