Explanation Given Information: The three dimensional motion of a particle is defined by the position vector is r = ( A t cos t ) i + ( A t 2 + 1 ) j + ( B t sin t ) k . The curve described by the particle lies on the hyperboloid is ( y A ) 2 − ( x A ) 2 − ( z B ) 2 = 1 . The value of A and B are 3 and 1 respectively. Calculation: Write the three dimensional motion of a particle position vector equation. r = ( A t cos t ) i + ( A t 2 + 1 ) j + ( B t sin t ) k (1) Here, x is ( A t cos t ) i , y is ( A t 2 + 1 ) j and z is ( B t sin t ) k . Consider x : x = ( A t cos t ) cos t = x A t (2) Consider y : y = ( A t 2 + 1 ) t 2 + 1 = y A t 2 = ( y A ) 2 − 1 (3) Consider z : z = ( B t sin t ) sin t = z B t (4) Calculate the t 2 using the relation: cos 2 t + sin 2 t = 1 Substitute x A t for cos t and z B t for sin t . ( x A t ) 2 + ( z B t ) 2 = 1 t 2 = ( x A ) 2 + ( z B ) 2 Check whether the position vector equation satisfied the curve equation or not. Substitute ( x A ) 2 + ( z B ) 2 for t 2 in equation (3). ( x A ) 2 + ( z B ) 2 = ( y A ) 2 − 1 ( x A ) 2 + ( z B ) 2 − ( y A ) 2 = − 1 − ( y A ) 2 + ( x A ) 2 + ( z B ) 2 = − 1 ( y A ) 2 − ( x A ) 2 − ( z B ) 2 = 1 Hence, the equation is satisfied. Rewrite the Equation (1). Substitute 3 for A and 1 for B in Equation (1). r = ( 3 t cos t ) i + ( 3 t 2 + 1 ) j + ( 1 t sin t ) k Write the expression for velocity using the relation: v = d r d t Substitute ( A t cos t ) i + ( A t 2 + 1 ) j + ( B t sin t ) k for r . v = d ( 3 t cos t ) i + ( 3 t 2 + 1 ) j + ( 1 t sin t ) k d t = 3 ( cos t − t sin t ) i + 3 t t 2 + 1 j + ( sin t + t cos t ) k (5) Calculate velocity vector ( v ) when time is 0 sec: Substitute 0 for t in Equation (5). v = 3 ( cos ( 0 ) − ( 0 ) sin ( 0 ) ) i + 3 ( 0 ) ( 0 ) 2 + 1 j + ( sin ( 0 ) + ( 0 ) cos ( 0 ) ) k = 3 ( 1 − 0 ) i + 0 j + ( 0 ) k = 3 i + 0 j + 0 k Here, v x is 3 , v y is 0 and v z is 0

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 11.5, Problem 11.152P

To determine

The radius (ρ) of curvature of the path described by the particle when time (t) is 0 sec.

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Chapter 11.5, Problem 11.151P

Chapter 11.5, Problem 11.153P

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Vector Mechanics for Engineers: Statics and Dynamics

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The radius ( ρ ) of curvature of the path described by the particle when time ( t ) is 0 sec.

Solution Summary: The author calculates the three dimensional motion of a particle defined by the position vector equation.

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The radius (ρ) of curvature of the path described by the particle when time (t) is 0 sec.

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Chapter 11 Solutions