(a) Explanation Given information: The radius ( c ) of sphere is 40 mm. The cord forms an angle ( β ) with vertical is 30 ° . The sphere precesses ( ϕ ˙ ) at the constant rate is 5 rad/s . The sphere has no spin ( ψ ˙ ) . Calculation: Consider principal axes xyz with origin at O . Sketch the system with dimensions as shown in Figure (1). Refer Figure (1), Write the equation of mass moment of inertia ( I ) about the axis BC . I = 2 5 m c 2 Here, m is the mass of the sphere. Write the equation of mass moment of inertia ( I ′ ) about the axis AD . I ′ = 2 5 m c 2 Find the difference between mass moment of inertia: I ′ − I = 2 5 m c − 2 5 m c = 0 Sketch the free body diagram of the cone as shown in Figure 2. Refer Figure (2), The angular velocity ( ω y ) along the x -axis is zero. Write the equation of angular velocity ( ω x ) along the y -axis: ω x = − ϕ ˙ sin θ Write the equation of angular velocity ( ω z ) along the z -axis: ω z = ψ ˙ + ϕ ˙ cos θ Find the equation of angular velocity ( ω ) of the cone. ω = ω x i + ω y j + ω z k Substitute − ϕ ˙ sin θ for ω x , 0 for ω y , and ψ ˙ + ϕ ˙ cos θ for ω z . ω = ( − ϕ ˙ sin θ ) i + ( 0 ) j + ( ψ ˙ + ϕ ˙ cos θ ) k = − ϕ ˙ sin θ i + ( ψ ˙ + ϕ ˙ cos θ ) k Write the equation of angular momentum about G ( H G ) . H G = I ′ ω x i + I ′ ω y j + I ω z k Substitute − ϕ ˙ sin θ for ω x , 0 for ω y , and ψ ˙ + ϕ ˙ cos θ for ω z . H G = I ′ ( − ϕ ˙ sin θ ) i + I ′ ( 0 ) j + I ( ψ ˙ + ϕ ˙ cos θ ) k = − I ′ ϕ ˙ sin θ i + I ( ψ ˙ + ϕ ˙ cos θ ) k Write the equation of vector form of angular velocity ( Ω ) of the reference frame Gxyz . Ω = − ϕ ˙ sin θ i + ϕ ˙ cos θ k Find the equation of rate of change of angular momentum about G ( H G ) using the equation: H ˙ G = Ω × H G 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 . H G = ( − ϕ ˙ sin θ i + ϕ ˙ cos θ k ) × [ − I ′ ϕ ˙ sin θ i + I ( ψ ˙ + ϕ ˙ cos θ ) k ] Use vector cross product. H G = | i j k − ϕ ˙ sin θ 0 ϕ ˙ cos θ − I ′ ϕ ˙ sin θ 0 I ( ψ ˙ + ϕ ˙ cos θ ) | = − j [ − I ϕ ˙ sin θ ( ψ ˙ + ϕ ˙ cos θ ) + I ′ ϕ ˙ 2 sin θ cos θ ] = − [ I ′ ϕ ˙ 2 sin θ cos θ − I ψ ˙ ϕ ˙ sin θ − I ϕ ˙ 2 sin θ cos θ ] i = [ I ψ ˙ ϕ ˙ sin θ − ( I ′ − I ) ϕ ˙ 2 sin θ cos θ ] i (1) Refer Figure (2), Find the equation of acceleration about mass center G (b) To determine The angle θ that BC forms with the vertical if the sphere spins at the rate of 30 rad/s. (c) To determine The angle θ that BC forms with the vertical if the sphere spins at the rate of − 30 rad/s .

Vector Mechanics for Engineers: Statics and Dynamics

12th Edition

ISBN: 9781259638091

Author: Ferdinand P. Beer, E. Russell Johnston Jr., David Mazurek, Phillip J. Cornwell, Brian Self

Publisher: McGraw-Hill Education

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Chapter 18.3, Problem 18.116P

(a)

To determine

The angle θ that BC forms with the vertical if the sphere has no spin.

(b)

To determine

The angle θ that BC forms with the vertical if the sphere spins at the rate of 30 rad/s.

(c)

To determine

The angle θ that BC forms with the vertical if the sphere spins at the rate of −30 rad/s.

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Chapter 18.3, Problem 18.115P

Chapter 18.3, Problem 18.117P

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Vector Mechanics for Engineers: Statics and Dynamics

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The angle θ that BC forms with the vertical if the sphere has no spin.

Solution Summary: The author illustrates the equation of mass moment of inertia (I) about the axis BC.

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The angle θ that BC forms with the vertical if the sphere has no spin.

The angle θ that BC forms with the vertical if the sphere spins at the rate of 30 rad/s.

The angle θ that BC forms with the vertical if the sphere spins at the rate of −30 rad/s.

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Chapter 18 Solutions