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10. [2D and 3D motion] Show that the potential energy U(r) ofFa particle of nass jat aa
distance r(> 員 from a planet of Inass 4 and radins 戶話
11. [The cross product] Caleulate the cross product A x B, assuming that A and B both
lie in the x-y plane and have the respective nonzero components 4,, 4, and 妖,, 且,
12. [Angular momentum] Show that 主r, the instantaneous position ofa particle of mass
mu with respect to a certain origin O, is parallel to its acceleration dv/吧, then the time
Tate of change of the angular Inoimentum L with respect to the given origin vanishes
13. [Theorem 了 Show that the torque-angular momentum formnla
_箇
7二
is also valid for a two-particle system 放 the origin lies at the center of mass
14. [Theorem IIH] Two particles of respective Inass ml and 2 are connected to the ends of
amassless rod and Inove in the uniform gravitational field g ofthe earth.
(a) What is the acceleration of their center of nass Telative to a Newtonian origin O?
(b) Show that the angular Inomentum of the two particles about their center of nass 生
aconstant of the motion.
15. [Moments of inertia] Calculate the moment of inertia ofathin, uniform rod of mass m
and length / about an axis perpendicular to the rod and at a distance q from one end.
16. [Rotation about a fixed axis] Consider the rotating cylinder in Figure, and suppose
that A7 sg, Fo 三 0.6 Nt, and @ 二 0.2m, Calculate
(a) The torque 7 acting on the cylinder
(b) Ti
ngular aceeleration q of the cylinder
17. [Work and kinet:
instant Totat
eenergy] A 40-kg homogeneous sphere of radius 10 cm is at acertain
about ashaft through its center at 600 rpm. Assuining that a constant fric
itude
ets so that the sphere comes to rest in 10 seconds, Calculate th
tional torque
ofthis torque.
18. [The physical pendl
end and allowed to oscillate freely in a horizontal plane (see Figure)
m] A uniform Tod of Inass ru and length / is suspended at one
(a) What is the period of this physic
lpendulum for small amplitudes?
(b) Ithe
velocity u0 of the mass ce
Hulum is initially released at rest in a horizontal position。what is the
er at a subsequent instant when the Tod is vertical?
(ec) Calculate the vertical and the horizontal components of the force on the rod at the
point of suspensic
19
and rotations] Analyze the motion of a homogeneous sphere of mass 和7
and Tadius dq, which rolls without slipping down a fixed inclined plane of angle #
20.
of angular momentum] A Inan has ainoment of inertia 五 about the =
axis. He is originally at Test and standing on asinall platform which can turn freely. Ifhe
is handed a wheel which is rotating at and has anoinent of inertia 了about its spinning
axis, determine his angular velocity 放 (a) he holds the wheel upright, (b) turns the wheel
out 9 二 90?, and (c) turns the wheel downward, 9 二 180?. Neglect the effect of holding
the wheel a distance d away from the z axis
