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# Suddenly Lifted Bar

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Mechanics Corner
A Journal of Applied Mechanics and Mathematics by DrD, # 51

Suddenly Lifted Bar

Introduction

In the Forum section, under the Machine Dynamics and Design Considerations heading, Mr. Negar Rahbar has posted a topic, Bending Moment. It is a problem he found in an older textbook (one that I might have studied back in my college days, although in this case, I did not), a book titled Theory and Problems of Machine Design by Allen S. Hall, 1961. It is an excellent problem, and I thank him for posting it for us. The problem statement from the textbook is shown in the PDF that follows.

The questioner says that the book provides an answer for the maximum bending moments as 592 in-lb. He is confused because he sees the system as unconstrained and asks for help understanding the solution. This is a great problem, one that really requires us to think about the dynamics of the system.
Before attempting to calculate anything, it is well to simply think through the expected sequence of events. Initially the bar is lying undisturbed on the table top, presumably in intimate contact at every point along the length. When the force P is suddenly applied, the bar acting as a rigid body begins to lift. Contact between the bar and the table top is ended at every point along the length, except at the left end. The left end remains in contact, and an upward force R on the bar develops at that location. Now we are only interested in the first instant after the bar lifts off the table, so the geometry is essential the same as it was before except that the bar is now supported by two point forces, unknown force R at the left end and known force P at the right end. It is a uniform bar, so the weight of the bar acts at the middle of the length. What can we say about the motion?

The questioner says that the book provides an answer for the maximum bending moments as 592 in-lb. He is confused because he sees the system as unconstrained and asks for help understanding the solution. This is a great problem, one that really requires us to think about the dynamics of the system.
Before attempting to calculate anything, it is well to simply think through the expected sequence of events. Initially the bar is lying undisturbed on the table top, presumably in intimate contact at every point along the length. When the force P is suddenly applied, the bar acting as a rigid body begins to lift. Contact between the bar and the table top is ended at every point along the length, except at the left end. The left end remains in contact, and an upward force R on the bar develops at that location. Now we are only interested in the first instant after the bar lifts off the table, so the geometry is essential the same as it was before except that the bar is now supported by two point forces, unknown force R at the left end and known force P at the right end. It is a uniform bar, so the weight of the bar acts at the middle of the length. What can we say about the motion?

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