Explanation Given information: The downward velocity ( v 0 ) is 5 m / s . The length of the rope ( l ) is 2 m . The acceleration due to gravity ( g ) is 9.81 m / s 2 . Calculation: Show the free body diagram of the sphere with the rope attached to the point O as in Figure (1). Calculate the initial kinetic energy ( T 1 ) using the formula: T 1 = 1 2 m v 0 2 Substitute 5 m / s for v 0 . T 1 = 1 2 m ( 5 2 ) = 12.5 m The expression for calculating the final kinetic energy ( T 2 ) as follows; T 2 = 1 2 m v 2 Calculate the work done ( U 1 − 2 ) using the formula: ( U 1 − 2 ) = m g l sin θ Substitute 2 m for l . ( U 1 − 2 ) = 2 m g sin θ Use work and energy principle which states that kinetic energy of the particle at a displaced point can be obtained by adding the initial kinetic energy and the work done on the particle during its displacement. The expression for the principle of work and energy for the earth is as follows; T 1 + U 1 − 2 = T 2 Substitute 12.5 m for T 1 , 2 m g sin θ for ( U 1 − 2 ) , and 1 2 m v 2 for T 2 . 12.5 m + 2 m g sin θ = 1 2 m v 2 m ( 12.5 + 2 g sin θ ) = 1 2 m v 2 12.5 + 2 g sin θ = v 2 2 v 2 = 25 + 4 g sin θ (1) Show the free body diagram sphere with the forces acting with the help of newton’s law as in Figure 2

Vector Mechanics for Engineers: Statics and Dynamics

11th Edition

ISBN: 9780073398242

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

Publisher: McGraw-Hill Education

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Chapter 13.1, Problem 13.39P

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Find the angle $(θ)$ at which the rope will break.

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Chapter 13.1, Problem 13.38P

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Vector Mechanics for Engineers: Statics and Dynamics

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