ROTATIONAL DYNAMICS

ROTATIONAL DYNAMICS

Newton's 3rd Law ? A Closer Look

Consider 2 particles with action/reaction pair of forces:

? Newton's 3rd Law F 1 on 2 = - F 2 on 1

2

? Puts no restrictions on the direction of F 1 on 2

r 12

Symmetry Considerations

1

? Attempt to write a formula for the force F 1 on 2 ? No "universal" xyz directions What can F 1 on 2 depend on? ? Relative position vector r12 and relative velocity vector v12 only

If F 1 on 2 depends only on r 12 mathematically:

? Direction of r12 only "defined" direction in space ? F 1 on 2 must point in direction of r12 (or opposite direction) ? Forces of this type are called "central" forces

Central Forces and Torque 2

Mathematical definition of central force: r12 ? F 1 on 2 = 0 r12

? True in every reference frame!

y 1

Calculating in a particular reference frame S:

r 12 ? F 1 on 2 = 0

r 2 - r 1 ? F 1 on 2 = 0

The quantity r i ? F i is called the "torque" x (i) on the ith particle Internal central forces produce zero net torque

r 2 ? F 1 on 2 - r 1 ? F 1 on 2 = 0 on a system

r 2 ? F 1 on 2 r 1 ? F 2 on 1 = 0

Examples:

External forces and non-central internal forces can exert non-zero net torque on a system

? Central forces: gravity, electric ? Non-central force: magnetic (depends on position and velocity)

Static Equilibrium

"Equilibrium" Fnet = 0 aCM = 0

"Static Equilibrium" ai = 0 for every particle

? Examples: buildings, bridges (not perfectly static!)

? Requires: F net , external = 0 and net , external =

r i ? F i = 0

? Useful for calculating structural loads and sitresses

Examples:

? Shelf calculate tension in cable ? Calculate force of wall on plank

d m 3d/4

2m d

? Door of width w and height h draw direction of each Fhinge

Angular Momentum

Consider the net torque on a system of particles

net =

r i ? F i

i

net = i

r i

?

d pi dt

net

=

d dt

ri ? pi

i

? ri ? pi is referred to as "angular momentum"( Li ) of ith particle ? Internal, central forces exert zero net torque

? Net torque must be provided by external forces:

net , external

=

d Ltotal dt

(Similar to Newton's 2nd Law )

? If zero net external torque angular momentum is conserved

Both and L depend on choice of origin

? Unlike force and momentum (only depend on xyz directions) ? However, the equation above is true in all reference frames

 ................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download