INTRODUCTION
1.7 REVIEW OF DYNAMICS
1.7.2 KINETICS
The basic law for kinetics of particles is Newton’s second law of motion
(1.32) where the sum of the forces is applied to a free-body diagram of the particle. A rigid body is a collection of particles. Writing an equation similar to Equation (1.32) for each particle in the rigid body and adding the equations together leads to
(1.33) where is the acceleration of the mass center of the body and the forces are summed on a free-body diagram of the rigid body. Equation (1.33) applies to all rigid bodies.
A moment equation is necessary in many problems. The moment equation for a rigid body undergoing planar motion is
(1.34) where Gis the mass center of the rigid body and is the mass moment of inertia about an axis parallel to the zaxis that passes through the mass center.
I aMG = Ia
a
aF = ma aF = ma
aB = |rB>A|ait - rv2in aB = aA + aB>A vB/A = |rB>A|v
rB/A r៝B
r៝A
A B y
x y
x y
x
(a)
vA vA A B
(b)
vA
⎪rB/A⎪α aA
aA A B
(c)
⎪rB/A⎪ω2
⎪rB/A⎪ω FIGURE 1.13
(a) The triangle rule for vector addition is used to define the relative position vector. (b) For a rigid body undergoing planar motion, the velocity ofBviewed from Ais that of a particle moving in a circular path centered at A. (c) The relative acceleration is that of a particle moving in a circular path centered atA.
Equations (1.33) and (1.34) can be used to solve rigid-body problems for planar motion. In general, the force equation of Equation (1.33) yields two independent equa- tions, and the moment equation of Equation (1.35) yields one. If the axis of rotation is fixed, Equation (1.33) may be replaced by
(1.35) where IOis the moment of inertia about the axis of rotation. In Figure 1.14(a), Ois a fixed axis of rotation, and Equation (1.35) is applicable. In Figure 1.14(b), link BChas does not have a fixed axis of rotation, and Equation (1.35) is not applicable.
Recall that a system of forces and moments acting on a rigid body can be replaced by a force equal to the resultant of the force system applied at any point on the body and a moment equal to the resultant moment of the system about the point where the resultant force is applied. The resultant force and moment act equivalently to the original system of forces and moments. Thus Equations (1.33) and (1.34) imply that the system of exter- nal forces and moments acting on a rigid body is equivalent to a force equal to applied at the body’s mass center and a resultant moment equal to . This latter resultant system is called the system of effective forces. The equivalence of the external forces and the effec- tive forces is illustrated in Figure 1.15.
The previous discussion suggests a solution procedure for rigid-body kinetics problems.
Two free-body diagrams are drawn for a rigid body. One free-body diagram shows all exter- nal forces and moments acting on the rigid body. The second free-body diagram shows the
Ia
ma aMO = IOa
(a)
A C
B O
(b)
FIGURE 1.14
(a) Rotation about a fixed axis atO. (b)ABhas a fixed axis of rotation at A, butBCdoes not have a fixed axis of rotation.
G
= G FB
Iα ma
M1 M2
F4 F3
F2
F1
FIGURE 1.15
The system of external forces and moments acting on a rigid body undergoing planar motion is equivalent to the system of effective forces, a force equal to applied at the mass center, and a moment equal toIa.
ma
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effective forces. If the problem involves a system of rigid bodies, it may be possible to draw a single free-body diagram showing the external forces acting on the system of rigid bodies and one free-body diagram showing the effective forces of all of the rigid bodies. Equations (1.33) and (1.34) are equivalent to
(1.36) and
(1.37) taken about any point Oon the rigid body. Equations (1.36) and (1.37) are statements of D’ Alembert’s principle applied to a rigid body undergoing planar motion.
aMOext = aMOeff aFext = aFeff
E X A M P L E 1 . 4
The slender rod ACof Figure 1.16(a) of mass mis pinned at B and held hor- izontally by a cable at C. Determine the angular acceleration of the bar immediately after the cable is cut.
S O L U T I O N
Immediately after the cable is cut, the angular velocity is zero. The bar has a fixed axis of rotation at B. Applying Equation (1.35)
(a) to the FBD of Figure 1.16(b) and taking moments as positive clockwise, we have
(b) The parallel-axis theorem is used to calculate IBas
(c) Substituting into Equation (b) and solving for yields
(d)
A L T E R N A T I V E M E T H O D
Free-body diagrams showing effective and external forces are shown in Figure 1.16(c). The appropriate moment equation is
(e) leading to
(f) and a =
12g 7L. mg L
4 = 1
12mL2 + amL 4ab aL
4b 1gMB2ext = 1gMB2eff
a = 12g
7L
IB = I + md2 = 1
12mL2 + maL 4b2 =
7 48mL2 mg L
4 = IB a aMB = aIB a
1I = 1
12mL22
Determine the angular acceleration of the pulley of Figure 1.17.
S O L U T I O N
Consider the system of rigid bodies composed of the pulley and the two blocks. If is the counterclockwise angular acceleration of the pulley, then, assuming no slip between the pulley and the cables, block A has a downward acceleration of rA and block B has an upward acceleration of rB.
Summing moments about the center of the pulley, neglecting axle friction in the pulley, and using the free-body diagrams of Figure 1.17(b) assuming moments are positive counterclockwise yields
Substituting given values leads to ⫽7.55 rad/s2. mA grA - mB grB = IPa + mB r2A a + mB r2B a
gMO
ext
= gMO
eff
B C
A
3L 4 L
4
L 4
1 12 (a)
G
R (b)
mg
G
=
R
(c)
mg m
mL2α α
FIGURE 1.16
(a) System of Example 1.4 where the slender rod is pinned atBand held by the cable at C. (b) FBD of bar immediately after cable is cut. The problem involves rotation about a fixed axis at B, so
(c) FBD’s showing external forces and effective forces immediately after cable is cut.
aMB = IBa.
E X A M P L E 1 . 5
rA rA = 30 cm
rB = 20 cm IP = 0.6 kg · m2 mA = 5 kg mB = 3 kg
mA mB
rB
(a)
mAg
mPg
mBg R
External forces
(b)
mArAα mBrBα IPα
External forces
=
FIGURE 1.17
(a) System of Example 1.4. (b) FBDs showing external forces and effective forces.
Copyright 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).