(a) Explanation Given information: The mass ( m A ) of block A is 20 kg. The mass ( m B ) of block B is 25 kg. The length of cable ( r ) is 0.8 m. The angle of cable ( θ ) with horizontal is 30 ° . The velocity ( v A ) of block A is 2 m/s. Calculation: Consider r and θ re the polar coordinates of block A and y B is the position coordinate (positive downward, origin at the pulley). The motion of block B is rectilinear. Sketch the polar coordinates of velocity block A as shown in Figure (1). Refer Figure (1), Find the transverse component of velocity ( v θ ) of block A . v θ = − v A sin θ Substitute 2 m/s for v A and 30 ° for θ . v θ = − 2 sin 30 ° = − 1.00 m/s Find the velocity ( θ ˙ ) of cable using the equation: θ ˙ = v θ r Substitute 1.00 m/s for v θ and 0.8 m for r . θ ˙ = 1.00 0.8 = 1.25 rad/s Write total length of cable connecting block A and block B . r + y B = constant (1) Differentiate Equation (1) with respect to t to write velocity of the blocks. r ˙ + v B = 0 (2) Here r ˙ is the velocity of cable and v B is the velocity of the block B . Differentiate Equation (2) with respect to t to write acceleration of the blocks. r ¨ + a B = 0 r ¨ = − a B (3) Here, r ¨ is the acceleration of cable and a B is the acceleration of block B . Sketch the free body diagram of block A as shown in Figure (2). Refer Figure (2). Consider equilibrium along x -axis: Σ F x = m A a A T cos θ = m A a A T = m A a A cos θ (4) Sketch the free body diagram of block B as shown in Figure (3). Refer Figure (3), Consider equilibrium along y -axis: Σ F y = m B a B m B g − T = m B a B − T = m B a B − m B a B (5) Add Equation (4) and (5). T − T = m A a A cos θ + m B a B − m B a B m B g = m A a A sec θ + m B a B (6) Sketch the polar coordinate of acceleration of block A with triangle construction as shown in Figure (4) (b) To determine Find the acceleration of block A . (c) To determine Find the acceleration of block B .

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 12.1, Problem 12.72P

(a)

To determine

The tension in the cable.

(b)

To determine

Find the acceleration of block A.

(c)

To determine

Find the acceleration of block B.

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Chapter 12.1, Problem 12.71P

Chapter 12.2, Problem 12.74P

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Vector Mechanics for Engineers: Statics and Dynamics

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