Explanation Given: The rope pulled with the constant velocity of v . The radius of the gear is r G = 2 r . The radius of the inner spool is r S = r . Draw the free body diagram of gear as shown in Figure 1. Write the relation between the linear velocity and angular velocity. ω = v r (I) v = ω r (II) Here, ω is the angular velocity, v is the linear velocity and r is the radial distance from the instantaneous center or permanent center to the point where the linear velocity acting on the object. Refer Figure (1). Write the equation of relative acceleration of point A with respect to the point G . a A = a G + α × r A / G − ω 2 r A / G (III) Write the equation of relative acceleration of point B with respect to the point G . a B = a G + α × r B / G − ω 2 r B / G (IV) Here, a A is the acceleration vector of the point A , a B is the acceleration vector of the point B , a G is the acceleration vector of the point G α is the angular acceleration, r A / G is the position vector directed from G to A , r B / G is the position vector directed from G to B and ω is the angular velocity. Conclusion: Refer Equation (I). Calculate the angular velocity of the gear. ω = v r Refer Figure (1a). The instantaneous center located at where the rope leaves the gear and shown in Figure (1a). From Figure (1a). r I C − B = 4 r Calculate the r I C − A . Consider the triangle A C ( I C ) and use Pythagoras theorem. r I C − A = ( 2 r ) 2 + ( 2 r ) 2 = 8 r 2 = 2 2 r Refer Equation (II). Calculate the linear velocity of the gear at point A . v A = ω r I C − A Substitute v r for ω and 2 2 r for r I C − A . v A = v r ( 2 2 r ) = 2 v 2 Thus, the velocity of the point A is 2 v 2 . Refer Equation (II). Calculate the linear velocity of the gear at point B . v B = ω r I C − B Substitute v r for ω and 4 r for r I C − B . v B = v r ( 4 r ) = 4 v Thus, the velocity of the point B is 4 v . Here, the rope is pulled with constant acceleration. Hence, the acceleration of the gear is zero. a G = 0 Refer Figure (1b). Write the position vector ( r A / G ) directed from G to A

Engineering Mechanics: Statics & Dynamics (14th Edition)

14th Edition

ISBN: 9780133915426

Author: Russell C. Hibbeler

Publisher: PEARSON

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Chapter 16.7, Problem 115P

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The velocity and acceleration of points A and B .

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Chapter 16.7, Problem 114P

Chapter 16.7, Problem 116P

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Engineering Mechanics: Statics & Dynamics (14th Edition)

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The velocity and acceleration of points A and B .

Solution Summary: The author illustrates the relationship between the linear velocity and angular velocity by drawing the free body diagram of gear.

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The velocity and acceleration of points A and B .

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