In order to test the resistance of a chain to impact, the...

In order to test the resistance of a chain to impact, the chain is suspended from a 240 -lb rigid beam supported by two columns. A rod attached to the last link is then hit by a $60-\mathrm{lb}$ block dropped from a 5-ft height. Determine the initial impulse exerted on the chain and the energy absorbed by the chain, assuming that the block does not rebound from the rod and that the columns supporting the beam are (a) perfectly rigid, ( $b$ ) equivalent to two perfectly elastic springs.

In order to test the resistance of a chain to impact, the chain is suspended from a 240 -lb rigid beam supported by two columns. A rod attached to the last link is then hit by a $60-\mathrm{lb}$ block dropped from a 5-ft height. Determine the initial impulse exerted on the chain and the energy absorbed by the chain, assuming that the block does not rebound from the rod and that the columns supporting the beam are
(a) perfectly rigid, ( $b$ ) equivalent to two perfectly elastic springs.

Key Concepts

Impulse-Momentum Relationship

This concept relates impulse, which is the product of force and the time interval over which it acts, to the change in momentum of a system. It is fundamental when analyzing short-duration impacts where forces large enough to produce a rapid change in motion are applied.

Gravitational Potential Energy Conversion

This refers to the conversion of stored energy due to an object's height in a gravitational field into kinetic energy during free fall. Understanding this conversion is crucial for determining the energy available at impact in collision scenarios.

Work-Energy Principle

The work-energy principle establishes that the work done on an object is equal to its change in kinetic energy. This is important in impact analysis for calculating how much energy is absorbed by a structure or material during a collision.

Elastic and Rigid Support Behavior

This concept examines how different types of supports—rigid versus elastic—respond to applied loads. Rigid supports resist deformation, leading to higher localized forces, while elastic supports yield and deform, thereby influencing the energy absorption and load distribution during impact events.

Transcript

00:02 In this problem, we have a chain which is suspended from a rigid beam supported by two columns.

00:08 A rod is attached to the last link and then hit by a block which is dropped from a given height.

00:14 We want to calculate the initial impulse exerted on the chain and the energy absorbed by the chain, assuming that the block does not rebound from the rod.

00:24 And we want to determine this under two situations.

00:26 Firstly, for the columns supporting the beam to be perfectly rigid, and secondly for the columns to be perfectly elastic, perfectly elastic springs.

00:37 So the first thing we'll do is find the velocity of the block just before impact.

00:56 So initially the block is at rest, so its initial kinetic energy is zero.

01:02 Its initial potential energy is due to gravity.

01:06 So its gravitational potential energy is its weight times its height above the rod.

01:10 And this is 60 pounds times the distance or a height of 5 feet.

01:20 And so this is equal to 300 pound feet.

01:27 Now the final kinetic energy is a half mv squared, where v is the velocity at the block just before impact.

01:39 And v2 at the rod will be zero.

01:46 So using the conservation of energy, t1 plus v1 must equal to t2 plus v2.

01:54 So essentially all the initial gravitational potential energy 300 is equal to the final kinetic energy, a half, and 60 over g, 32 .2 feet per square second, xx2.

02:12 So from here we can see that we can rearrange this and solve for the velocity of the block just before it strikes the rod.

02:22 When we get v is equal to 14 .94 feet per second.

02:34 So for part a we'll assume that the columns are rigid and the momentum impulse diagram shows us that the block folds with initial velocity mv initial momentum mv and then there's an impulse when the block strikes the rod f delta t and the final momentum of the block is zero since the block comes to rest.

03:02 So if we take the upper direction as positive and rewrite this in mathematical form, we get that minus mv plus f delta t...

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