(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 .

11th Edition

ISBN: 9781259633133

Author: BEER

Publisher: MCG

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Question

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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