An irresponsible person has left a wooden beam standing vertically near an expen

Question

An irresponsible person has left a wooden beam standing vertically near an expensive wall. Murphy's law dictates that the beam falls over and hits the wall, causing an unsightly dent. You (a physics student) and a friend in a Pivots materials science engineering course want to understand the strength of the wall material by analyzing the dent. With your combined knowledge, you think you can do it! The engineer asks you for several quantities. Based on the quantity she receives, she will then attempt to determine the 'strength' of the wall material, whatever that means. What you know: The beam has mass m & length L, is a horizontal distance a away from the wall, and pivots about the bottom. The initial angular velocity of the beam is so small you can treat it as zero. 1. What is the moment of inertia of the rotating beam? [If you already feel comfortable with how to calculate the moment of inertia the real way (via an integral), you can just look the answer up in the textbook. Otherwise, get practice doing the integral. It might be useful.] 2. What is the rotational kinetic energy just before the beam hits? You may use I in your answer. 3. What is the angular velocity just before the beam hits? Answer in terms of given quantities and/or quantities you've already calculated. 4. Immediately after the collision, the wall will slow the beam down to zero angular velocity. Let's assume the angular acceleration during the collision is constant. This means you can use the angular kinematic equations discussed in Knight 4e sections 4.4-4.6 and 12.1. With that in mind: Using a tape measure is easy, so you can easily measure the small angle Delta theta_ dent that the beam travels during the collision by examining the dent. Based on this new measurement delta theta_ dent, what is the magnitude of the angular acceleration of the beam during the collision? Answer in terms of delta theta_ dent and your answer to #3. This should be a one or two-step problem depending on how good you are at subtracting and dividing at the same time. 5. Finally, what is the force exerted by the wall? Answer in terms of given quantities and or quantities you've already calculated. Assume that the force supplied by the wall is perpendicular to the axis of the beam. (This assumption should make dealing with the sine term easier.) Your friend will use this force in her material-science analysis.

Explanation / Answer

3.) moment of inertia of beam = I

Now initial potential energy of beam = mgL/2 (take the zero level of potential energy at floor )

when it hits its potenrial energy = mg(L/2)[(1 - a2/L2)]

and when it hits let suppose its angular speed be . So its kinetic energy KE = 1/2I2

applying law of consevation of energy we have

mgL/2 = mg(L/2)[(1 - a2/L2)] + 1/2I2

or 1/2I2 =  mgL/2 - mg(L/2)[(1 - a2/L2)]

or 1/2I2 =  mgL/2[1 - (1 - a2/L2)]

or 2 =  (2mgL/2)/I[1 - (1 - a2/L2)]

or = [ (2mgL/2)/I[1 - (1 - a2/L2)]]

4.) let the angular acceleration be

then 0 - 2 = 2dent

(0 = final angular velocity as the wall slows down the angular velocity of the beam to zero imediately after it hits the wall... and = initial angular velocity when the beam hits the wall)

or =  - 2/2dent

This concludes the answers. If you find anything lacking please let me know.. I will resolve your query without delay....

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