FACULTY OF SCIENCE, AUTUMN 2007
PHYS 124 LEC A1 : Particles and Waves
(Instructor: Marc de Montigny)
Walker, Physics, Chapter 11: Rotational Dynamics and Static Equilibrium
- Section 11-1: Torque
- Appendix, P. APP-8 and APP-9: Cross product
- Figures
P. 316, Figure 11-1 and
P. 318, Example 11-1 help to illustrate
that three variables determine the effect of a force is causing a change of angular
velocity:
- magnitude F of the force : obviously, a greater effect for a greater
F
- distance r between the axis of rotation and the point where the force
is applied : for a given force, a greater effect will result from a greater r
- direction of the force : the more perpendicular the force is, with respect
to the line between the axis of rotation and the point where the force is applied,
the greater effect will result.
- P. 317, Eq. 11-2 : Definition of Torque (τ = rF sinθ).
- The vector r starts at the (chosen) axis of rotation, or pivot point, and
ends at the point where the force F is applied.
- Consider the torque τ caused by F on p. 317,
Figure 11-3.
There are three ways to compute the torque :
- To bring the vectors F and r to the same location,
as in this Figure,
and simply use τ = rF sinθ
- Use τ = r⊥F, where r⊥
is defined as in p. 317, Figure 11-3: it is the
projection of r onto a line perpendicular to F.
- Use τ = F⊥r, where F⊥
is defined as in this Figure: it is the
projection of F onto a line perpendicular to r.
- Section 11-2: Torque and Angular Acceleration
- P. 320, Eq. 11-4: ∑τ = Iα (Newton II for Rotational
Motion)
- Read P. 320, Example 11-2
- P. 357 Problem 99
- Section 11-3: Zero Torque and Static Equilibrium
- P. 323, Eqs. 11-5 and 11-6: ∑ Fx = 0;
∑ Fy = 0; ∑ τ = 0
- Read P. 325, Example 11-4
- Read P. 328, Example 11-5
- P. 352 Problem 30
- P. 352 Problem 32
- P. 355 Problem 85
- Section 11-4: Centre of Mass and Balance [OMITTED IN 2007]
- Read P. 330-332. The method accompanying P. 331,
Figure 11-9
is quite interesting.
- Section 11-5: Dynamic Applications of Torque
- Nothing new here, just more examples of the theory of Sections 11-2 and 11-3.
- Read P. 333, example pertaining to Figure 11-10.
Note, however, that for their discussion to make sense, the acceleration a should
be pointing upward on Figure 11-10 (a).
- Section 11-6: Angular Momentum
- P. 335, Eq. 11-11: L = Iω (Definition of the Angular Momentum L)
- Eq. 11-12 of P. 335 is obtained by considering a point particle: I = mr²
and v = rω, so that L = rmv = rp.
- P. 336, Eq. 11-13: L = rp sinθ (Angular Momentum L for
a Point Particle)
- The vector r starts at the (chosen) origin of the coordinate system and
ends at the location of the particle (of momentum p).
- SI Units: kg⋅m²/s
- Section 11-7: Conservation on Angular Momentum
- An object with angular momentum Li acted on by a torque τ during
a period of time Δt, will have a final angular momentum
Lf = Lf + ΔL, where
τ = ΔL / Δt
Note the similarity with F = Δp / Δt.
- P. 339, Eq. 11-15: Lf = Li if
τnet, ext = 0 (Conservation of Angular Momentum)
- P. 339, Example 11-9
- P. 341, Rotational Collisions, Figure 11-14
- P. 342, Active Example 11-5 illustrates
a mixed use of L = Iω and L = rp sinθ.
- Section 11-8: Rotational Work and Power [OMITTED IN 2007]
- Eq. 343, Eq. 11-17: W = τ Δθ (Work Done by Torque)
- Eq. 343, Eq. 11-18: W = ΔK = Kf
- Ki (Work-Energy Theorem)
- Eq. 343, Eq. 11-19: P = ΔW / Δt
= τ Δθ / Δt = τ ω (Power Produced by Torque)
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