Explanation Given information: The height of the cone is 12 in , the base diameter of the cone is 6 in , the angle of cord A B from vertical is 45 ° , the angle of the axis B C of the cone from vertical is 30 ° and the precision rate of cone is 8 rad / s . The below figure represent the free body diagram of the system. Figure-(1) Write the expression angular velocity of the cone. ω = − ϕ ˙ sin θ i + ( ψ ˙ + ϕ ˙ cos θ ) k ........ (I) Here, the precision rate is ϕ ˙ , the spin rate is ψ ˙ , the angle of the axis B C of the cone from vertical is θ and the unit vector in x direction is i and the unit vector in z direction is k . Write the expression of angular momentum about the centroid G . H G = I x ω x i + I y ω y j + I z ω z k ........ (II) Here, the moment of inertia about the x axis is I x , the moment of inertia about the y axis is I y , the moment of inertia about the z axis is I z , the angular velocity about the x axis is ω x , the angular velocity about the y axis is ω y , the angular velocity about the z axis is ω z and the unit vector in y direction is j . Write the expression of angular momentum about the centroid G . H G = − ( I ′ ϕ ˙ sin θ ) i + I ( ψ ˙ + ϕ ˙ cos θ ) k ........ (III) Here, the inertia about the reference frame is I ′ . Write the expression of angular velocity about the reference frame. Ω = − ϕ ˙ sin θ i + ϕ ˙ cos θ k ........ (IV) Write the expression of the total moment. H ˙ G = Ω × H G ......... (V) Substitute − ϕ ˙ sin θ i + ϕ ˙ cos θ k for Ω and − ( I ′ ϕ ˙ sin θ ) i + I ( ψ ˙ + ϕ ˙ cos θ ) k for H G in Equation (V). H ˙ G = ( − ϕ ˙ sin θ i + ϕ ˙ cos θ k ) × { − ( I ′ ϕ ˙ sin θ ) i + I ( ψ ˙ + ϕ ˙ cos θ ) k } = [ ( − ϕ ˙ sin θ i ) × ( − I ′ ϕ ˙ sin θ i ) + ( − ϕ ˙ sin θ i ) × { I ( ψ ˙ + ϕ ˙ cos θ ) k } + ( ϕ ˙ cos θ k ) × ( − I ′ ϕ ˙ sin θ i ) + ( ϕ ˙ cos θ k ) × { I ( ψ ˙ + ϕ ˙ cos θ ) k } ] = [ ( ϕ ˙ sin θ ) × { I ( ψ ˙ + ϕ ˙ cos θ ) j } − ( ϕ ˙ cos θ ) × ( I ′ ϕ ˙ sin θ ) j ] = I ϕ ˙ ψ ˙ sin θ j + I ϕ ˙ 2 sin θ cos θ j − I ′ ϕ ˙ 2 cos θ sin θ j H ˙ G = I ϕ ˙ ψ ˙ sin θ j − ϕ ˙ 2 sin θ cos θ ( I ′ − I ) j = { I ϕ ˙ ψ ˙ sin θ − ϕ ˙ 2 sin θ cos θ ( I ′ − I ) } j ........ (VI) Write the expression of acceleration about the mass centre. a ¯ = ( l sin β + c sin θ ) ϕ ˙ 2 ........ (VII) Here, the lengt6h of the cord is l and the distance between the centroid of the cone is c . Write the expression of equilibrium of force in y direction as shown in Figure-(1). T cos β = m g T = m g cos β ........ (VIII) Here, the tension on the cord is T and the mass of the cone is m . Write the expression of equilibrium of force in x direction as shown in Figure-(1). T sin β = m a ¯ ........ (IX) Substitute m g cos β for T and ( l sin β + c sin θ ) ϕ ˙ 2 for a ¯ in Equation (IX). ( m g cos β ) sin β = m ( l sin β + c sin θ ) ϕ ˙ 2 g tan β = ( l sin β + c sin θ ) ϕ ˙ 2 ......... (X) Write the expression of moment about point G as shown in Figure-(1). H ˙ G = T c sin ( β − θ ) ......... (XI) Substitute m g cos β for T in Equation (XI) To determine (b) The length of the cord A B .

12th Edition

ISBN: 9781259977305

Author: BEER, Ferdinand P. (ferdinand Pierre), Johnston, E. Russell (elwood Russell), Cornwell, Phillip J., SELF, Brian P.

Publisher: Mcgraw-hill Education,

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Chapter 18.3, Problem 18.114P

To determine

(a)

The rate of spin ψ˙ of the cone about its axis BC.

To determine

(b)

The length of the cord AB.

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Chapter 18.3, Problem 18.113P

Chapter 18.3, Problem 18.115P

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