Explanation Given: The velocity of the car A is v A = 25 m/s . The constant deceleration of the car A is a A = 2 m/s 2 . The velocity of the car C is v C = 15 m/s . The constant acceleration of the car C is a C = 3 m/s 2 . Draw the free body diagram of system as shown in Figure 1. Refer Figure (1). Write the formula for velocity of the car A measured from the x , y , z reference. v A = v C + Ω × r A / C + ( v A / C ) x y z (I) Here, v A velocity vector of car A measured from the X , Y , Z reference, v C is the velocity vector measured from the X , Y , Z reference, ( v A / C ) x y z is the velocity vector of the car A with respect to car C measured by an observer attached to the rotating x , y , z reference, Ω is the angular velocity of the x , y , z reference measured from the X , Y , Z reference and r A / C is the position vector of the car A with respect to car C . Write the formula for acceleration of the car measured from the x , y , z reference a A = a C + Ω ˙ × r A / C + Ω × ( Ω × r A / C ) + 2 Ω × ( v A / C ) x y z + ( a A / C ) x y z (II) Here, a A acceleration vector of car A measured from the X , Y , Z reference, a C is the acceleration vector of the car C measured from the X , Y , Z reference, ( v A / C ) x y z , ( a A / C ) x y z is the velocity and acceleration vector of the car A with respect to car C measured by an observer attached to the rotating x , y , z reference, Ω ˙ is the angular acceleration due to the rotation of x , y , z reference, Ω is the angular velocity of the x , y , z reference measured from the X , Y , Z reference and r A / C is the position vector of the car A with respect to car C . Conclusion: Refer Figure (1). Consider the rotating x y z reference is attached to the car C and also coincides with the fixed reference frame X Y Z . Here, the car travels in a circular road of radius ρ = 250 m . Hence the car C experience the normal acceleration of ( a C ) n = v C 2 ρ and the tangential acceleration of ( a C ) t . The normal acceleration of the car C is ( a C ) n = v C 2 ρ Substitute 15 m/s for v C and 250 m for ρ . ( a C ) n = ( 15 m/s ) 2 250 m = 0.9 m/s 2 The given acceleration of the car C is acting tangent to circular road. Hence the tangential acceleration of the car C is ( a C ) t = 3 m/s 2 Express the vector coordinate for all the velocity and acceleration components of the system. The vector coordinate of the acceleration of the car C . Refer Figure (1). Resolve the normal and tangential acceleration along x , y axes. a C = − ( a C ) n cos 45 ° i + ( a C ) n sin 45 ° j − ( a C ) t cos 45 ° i − ( a C ) t sin 45 ° j = − ( a C ) n cos 45 ° i − ( a C ) t cos 45 ° i + ( a C ) n sin 45 ° j − ( a C ) t sin 45 ° j Substitute 0.9 m/s 2 for ( a C ) n and 3 m/s 2 for ( a C ) t . a C = − 0.9 cos 45 ° i − 3 cos 45 ° i + 0.9 sin 45 ° j − 0.3 sin 45 ° j = − 0 .6364i − 2.1213 i + 0 .6364j − 2.1213 j = − 2.7577 i − 1.4849 j The vector coordinate of the velocity of the car C . Refer Figure (1). Resolve the velocity along x , y axes. v C = − 15 cos 45 ° i − 15 sin 45°j = − 10 .6066i − 10.6066 j The vector coordinate of the acceleration of the car A . a A = 0 i − 2 j = − 2j The vector coordinate of the velocity of the car A . v A = 0 i + 25 j = 25j Since the car C is attached to the moving reference x y z . Calculate the angular velocity of the moving reference is Ω = v C ρ Substitute 15 m/s for v C and 250 m for ρ . Ω = 15 m/s 250 m = 0.06 rad/s The vector coordinate of the angular velocity of the moving reference x y z . Ω = − Ω k = − 0 .06k (counter clockwise) Calculate the angular acceleration of the moving reference is Ω ˙ = a C ρ Substitute 3 m/s 2 for a C and 250 m for ρ . Ω ˙ = 3 m/s 2 250 m = 0.012 rad/s 2 The vector coordinate of the angular acceleration of the moving reference x y z . Ω ˙ = − Ω ˙ k = − 0 .012k (counter clockwise) Refer Figure (1b). Consider the triangle D C O . cos 45 ° = O D 250 m O D = 250 cos 45 ° O D = 176.7767 m When, θ = 45 ° . O D = C D = 176.7767 m From Figure (1b). O D + D A = 450 m Substitute 176.7767 m for O D

Engineering Mechanics: Statics & Dynamics (14th Edition)

14th Edition

ISBN: 9780133915426

Author: Russell C. Hibbeler

Publisher: PEARSON

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Chapter 16.8, Problem 148P

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The velocity and acceleration of the car A with respect to car C at the instant shown.

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Chapter 16.8, Problem 147P

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Engineering Mechanics: Statics & Dynamics (14th Edition)

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The velocity and acceleration of the car A with respect to car C at the instant shown.

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The velocity and acceleration of the car A with respect to car C at the instant shown.

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