VECTOR MECH...,STAT.+DYNA.(LL)-W/ACCESS
VECTOR MECH...,STAT.+DYNA.(LL)-W/ACCESS
11th Edition
ISBN: 9781259633133
Author: BEER
Publisher: MCG
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Chapter 18.3, Problem 18.142P

(a)

To determine

The rate of precession ϕ˙0 of the sphere in its initial position.

(a)

Expert Solution
Check Mark

Answer to Problem 18.142P

The rate of precession ϕ˙0 of the sphere in its initial position is 15g11a_.

Explanation of Solution

Given information:

The position of the sphere β is zero.

The largest value of β in the ensuing motion is 30°.

Calculation:

Conservation of angular momentum about the Z and z axes:

The only external forces are acting in homogenous sphere is weight of the sphere and reaction at A. Hence, the angular momentum is conserved about the Z and z axes.

Choose the principal axes Axyz with taking y horizontal and pointing into the paper.

Write the expression for the angular velocity ω.

ω=ϕ˙cosβi+β˙j+(ψ˙ϕ˙sinβ)k

The principal moment of inertia are  Ix=Iy=m(25a2+(2a)2) and Iz=25ma2.

Draw the Free body diagram of homogeneous sphere and the forces acting on it as in Figure (1).

VECTOR MECH...,STAT.+DYNA.(LL)-W/ACCESS, Chapter 18.3, Problem 18.142P

Write the expression for the angular momentum about point A.

HA=Ixωxi+Iyωyj+Izωzk

Substitute m(25a2+(2a)2) for Ix, ϕ˙cosβ for ωx, m(25a2+(2a)2) for Iy, β˙ for ωy, , 25ma2 for Iz, and (ψ˙ϕ˙sinβ) for ωz.

HA=m(25a2+(2a)2)(ϕ˙cosβ)i+m(25a2+(2a)2)β˙j+25ma2(ψ˙ϕ˙sinβ)k=225ma2ϕ˙cosβi+225ma2β˙j+25ma2(ψ˙ϕ˙sinβ)k

Consider Hz=constant or HAK=constant.

The scalar value of iK=cosβ, jK=0, and kK=sinβ.

Determine the conservation of angular momentum about fixed Z axis HAK.

HAK=constant

Substitute 225ma2ϕ˙cosβi+225ma2βj+25ma2(ψ˙ϕ˙sinβ)k for HA, cosβ for (iK), 0 for (jK), and cosβ for (kK).

{225ma2ϕ˙cosβ(iK)+225ma2β˙(jK)+25ma2(ψ˙ϕ˙sinβ)(kK)}=constant{225ma2ϕ˙cosβ(cosβ)+225ma2β˙(0)+25ma2(ψ˙ϕ˙sinβ)(sinβ)}=constant{225ma2ϕ˙cosβ(cosβ)+25ma2(ψ˙ϕ˙sinβ)(sinβ)}=constant (1)

Substitute ϕ˙0 for ϕ˙, 0 for ψ˙, and 0 for β in Equation (1).

{225ma2ϕ˙0cos0(cos0)+25ma2(0ϕ˙0sin0)(sin0)}=constantconstant=225ma2ϕ˙0

Substitute 225ma2ϕ˙0 for constant in Equation (1).

225ma2ϕ˙cosβ(cosβ)+25ma2(ψ˙ϕ˙sinβ)(sinβ)=225ma2ϕ˙025ma2[11ϕ˙cos2β(ψ˙ϕ˙sinβ)sinβ]=25×11ϕ˙011ϕ˙cos2β(ψ˙ϕ˙sinβ)sinβ=11ϕ˙0 (2)

Determine the constant value using the angular momentum along z–axis.

Hz=constantIzωz=constant

Substitute 25ma2 for Iz and (ψ˙ϕ˙sinβ) for ωz.

25ma2(ψ˙ϕ˙sinβ)=constant (3).

Substitute ϕ˙0 for ϕ˙, 0 for ψ˙, and 0 for β in Equation (3).

25ma2(0ϕ˙0sin(0))=constantconstant=0

Substitute 0 for constant in Equation (3).

25ma2(ψ˙ϕ˙sinβ)=0ψ˙ϕ˙sinβ=0

Substitute 0 for (ψ˙ϕ˙sinβ) in Equation (2).

11ϕ˙cos2β(ψ˙ϕ˙sinβ)sinβ=11ϕ˙011ϕ˙cos2β0(sinβ)=11ϕ˙0ϕ˙=11ϕ˙011cos2βϕ˙=ϕ˙0sec2β

Conservation of energy:

Determine the value of kinetic energy T.

T=12(Ixωx2+Iyωy2+Izωz2)

Substitute m(25a2+(2a)2) for Ix, ϕ˙cosβ for ωx, m(25a2+(2a)2) for Iy, β˙ for ωy, , 25ma2 for Iz, and (ψ˙ϕ˙sinβ) for ωz.

T={12(m(25a2+(2a)2)(ϕ˙cosβ)2+m(25a2+(2a)2)(β˙)2+25ma2(ψ˙ϕ˙sinβ)2)}=12(225ma2ϕ˙2cos2β+225ma2(β˙)2+25ma2(ψ˙ϕ˙sinβ)2)

Select the datum at β=0.

Determine the value of conservation of energy using the relation.

T+V=constant

Here, E is the constant and V is the potential energy.

Substitute 12(225ma2ϕ˙2cos2β+225ma2(β)2+25ma2(ψ˙ϕ˙sinβ)2) for T and 2mgasinβ for V.

{12(225ma2ϕ˙2cos2β+225ma2(β˙)2+25ma2(ψ˙ϕ˙sinβ)2)2mgasinβ}=constant (4)

Substitute ϕ˙0 for ϕ˙, 0 for β, 0 for β˙, and 0 for ψ˙ in Equation (4).

{12(225ma2ϕ˙02cos20+225ma2(0)2+25ma2(0ϕ˙0sin0)2)2mgasin0}=constantconstant=115ma2ϕ˙02

Substitute 115ma2ϕ˙02 for constant in Equation (4).

{12(225ma2ϕ˙2cos2β+225ma2(β˙)2+25ma2(ψ˙ϕ˙sinβ)2)2mgasinβ}=115ma2ϕ˙0212×25ma2((11ϕ˙2cos2β+11(β˙)2+(ψ˙ϕ˙sinβ)2)10gasinβ)=115ma2ϕ˙02((11ϕ˙2cos2β+11(β˙)2+(ψ˙ϕ˙sinβ)2)10gasinβ)=11ϕ˙02 (5)

Consider β˙=0 for the maximum value of β.

Substitute ϕ˙0sec2β for ϕ˙, 0 for (ψ˙ϕ˙sinβ), and 0 for β˙ in Equation (5).

11(ϕ˙0sec2β)2cos2β+11(0)210gasinβ=11ϕ˙0211ϕ˙02sec4β×1sec2β10gasinβ=11ϕ˙0211ϕ˙02sec2β11ϕ˙02=10gasinβ11ϕ˙02(1cos2βcos2β)=10gasinβ

ϕ˙02(sin2βcos2β)=1011gasinβϕ˙02=1011gasinβ×cos2βsin2βϕ˙02=1011gacos2βsinβ (6)

Substitute 30° for β in Equation (6).

ϕ˙02=1011gacos230°sin30°ϕ˙0=15g11a(43)

Therefore, the rate of precession ϕ˙0 of the sphere in its initial position is 15g11a_.

(b)

To determine

The rates of precession ϕ˙ and the spin ψ˙.

(b)

Expert Solution
Check Mark

Answer to Problem 18.142P

The rate of precession ϕ˙ is 220g33a_.

The rate of spin ψ˙ is 20g33a_.

Explanation of Solution

Given information:

The position of the sphere β is zero.

The largest value of β in the ensuing motion is 30°.

Calculation:

Determine the rate of precession ϕ˙ using the relation.

ϕ˙=ϕ˙0sec2β

Substitute 15g11a for ϕ˙0 and 30° for β.

ϕ˙=15g11asec230°=15g11a×1cos230°=15g11a43=15×16g11×9a=220g33a

Therefore, the rate of precession ϕ˙ is 220g33a_.

Determine the rate of spin ψ˙ using the relation.

ψ˙ϕ˙0sinβ=0

Substitute 220g33a for ϕ˙ and 30° for β.

ψ˙220g33asin30°=0ψ˙=20g33a

The rate of spin ψ˙ is 20g33a_.

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Chapter 18 Solutions

VECTOR MECH...,STAT.+DYNA.(LL)-W/ACCESS

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