Explanation Given information: The weight ( W ) of the gear A is 2 lb. The length ( L ) of the axle AD is 20 inch. The radius ( r ) of gear A is 4 inch. The angular velocity ( ω 1 ) of shaft DE is 4 rad/s. The angular velocity of axle AD is ( ω 2 ) . Calculation: Let point D be the common point of rotation ω 1 and ω 2 and consider principal axes x , y , and z with origin at point O . Sketch the axes of the system as shown in Figure (1). Refer Figure (1), Write the vector equation of angular velocity ( ω 1 ) of shaft DE using kinematics: ω 1 = ω 1 sin β i + ω 1 cos β j Write the vector equation of angular velocity ( ω 2 ) of axle AD using kinematics: ω 2 = ω 2 j Find the vector equation of angular velocity ( ω ) of system using kinematics: ω = ω 1 + ω 2 Substitute ω 1 sin β i + ω 1 cos β j for ω 1 and ω 2 j for ω 2 . ω = ω 1 sin β i + ω 1 cos β j + ω 2 j = ω 1 sin β i + ( ω 1 cos β + ω 2 ) j (1) Write the magnitude of x -component of angular velocity ( ω x ) from equation (1). ω x = ω 1 sin β Write the magnitude of y -component of angular velocity ( ω y ) from equation (1). ω y = ω 1 cos β + ω 2 Write the equation of length ( r G / D ) of axle AD corresponding to point D . r G / D = L sin β i Find the equation of velocity ( v G ) of the system at mass center G . v G = ω 1 × r G / D Substitute L sin β i for r G / D . v G = ω 1 j × L sin β i = − ω 1 L sin β k Find the equation of acceleration ( a G ) of the system at mass center G . a G = ω 1 × v G Substitute − ω 1 L sin β for v G a G = ω 1 × ( − ω 1 L sin β ) = − ω 1 2 L sin β = ω 1 2 L sin β ( ← ) The velocity at C is zero. Find the magnitude of y -component of angular velocity ( ω y ) in terms of ω 1 using the equation: v C = v G + ω × r C / G Here, r C / G the distance between mass center and point C . Substitute 0 for v C , − ω 1 L sin β k for v G , ( ω x i + ω y j ) for ω , and − r i for r C / G . 0 = − ω 1 L sin β k + ( ω x i + ω y j ) × ( − r i ) 0 = − ω 1 L sin β k + 0 + ω y r k ω y r k = ω 1 L sin β k Equate the k -vector coefficients. ω y r = ω 1 L sin β ω y = ω 1 L r sin β Write the equation of angular momentum of system about G ( H G ) . H G = I ¯ x ω x i + I ¯ y ω y j Write the angular velocity of reference frame ( Ω ) . Ω = ω 1 Substitute ω 1 sin β i + ω 1 cos β j for ω 1 . Ω = ω 1 sin β i + ω 1 cos β j The rate of change of angular momentum ( H ˙ G ) G x y z about the reference frame is 0. Write the equation of rate of change of angular momentum ( H ˙ G ) of the system. H ˙ G = ( H ˙ G ) G x y z + Ω × H G Substitute 0 for ( H ˙ G ) G x y z , ω 1 sin β i + ω 1 cos β j for Ω , and I ¯ x ω x i + I ¯ y ω y j for H O . H ˙ G = 0 + ( ω 1 sin β i + ω 1 cos β j ) × ( I ¯ x ω x i + I ¯ y ω y j ) = 0 + I ¯ y ω y ω 1 sin β k − I ¯ x ω x ω 1 cos β k = ( I ¯ y ω y sin β − I ¯ x ω x cos β ) ω 1 k (2) Find the mass moment of inertia ( I ¯ x ) of the system about x -axis using the equation: I ¯ x = 1 4 m r 2 Here, m is the mass of the gear A

12th Edition

ISBN: 9781260663778

Author: BEER

Publisher: MCG

See similar textbooks

Concept explainers

Conservation of Energy

In physics, the rule of the conservation of energy states that energy doesn’t disappear from the atmosphere; instead, it changes its form from one type to another. Energy is one of the most important physical quantities in nature, and it cannot be create…

Videos

Question

Chapter 18.2, Problem 18.89P

To determine

The force F exerted by gear B on gear A.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 18.2, Problem 18.88P

Chapter 18.2, Problem 18.90P

Students have asked these similar questions

7. The rotor of a turbine installed in a boat with its axis along the longitudinal axis of the boat makes 1500 rp.m. clockwise when viewed from the stern. The rotor has a mass of 750 kg and a radius of gyration of 300 mm. If at an instant, the boat pitches in the longitudinal vertical plane so that the bow rises from the horizontal plane with an angular velocity of I rad/s, determine the torque acting on the [Ans. 10.6 kN-m] boat and the direction in which it tends to turn the boat at the instant.

Arm ACB rotates about Point Cwith an angular velocity of 39 radis counterclockwise. Two friction disks A and Bare pinned at their centers to arm ACB as shown. Know that the disks roll without slipping at the surfaces of contact. 2.4 in. 1.2 in. 0.9 in. 1.5 in. 0.6 in. D Determine the angular velocity of disk B. The angular velocity of disk B is radis . Determine the angular velocity of disk A. The angular velocity of disk A is [ rad/s O Determine the angular velocity of disk A. The angular velocity of disk A is rad/s O.

Parvinbhai

Chapter 18 Solutions

VECTOR MECH. FOR EGR: STATS & DYNAM (LL

Ch. 18.1 - A thin, homogeneous disk of mass m and radius r...Ch. 18.1 - Prob. 18.2P Ch. 18.1 - Prob. 18.3P Ch. 18.1 - A homogeneous disk of weight W = 6 lb rotates at...Ch. 18.1 - A homogeneous disk of mass m = 8 kg rotates at the...Ch. 18.1 - A solid rectangular parallelepiped of mass m has a...Ch. 18.1 - Prob. 18.8P Ch. 18.1 - Determine the angular momentum HD of the disk of...Ch. 18.1 - Prob. 18.10P Ch. 18.1 - Determine the angular momentum HO of the disk of...

Knowledge Booster

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, mechanical-engineering and related others by exploring similar questions and additional content below.

The force F exerted by gear B on gear A .

Solution Summary: The author explains the force F exerted by gear B on gear A. The angular velocity of shaft DE is 4 rad/s.

Concept explainers

Videos

The force F exerted by gear B on gear A.

Want to see the full answer?

Chapter 18 Solutions