At its natural resting length, a muscle is close to its optimallength for producing force. as the muscle contracts, the maximumforce it can deliver decreases. when a muscle is at approximately80\% of its natural length, it cannotcontract much more and the force it can produce drops drastically. for a muscle stretched beyond its natural length, the same is true. at about 120\% of its natural length, the forcethat a muscle can exert again drops drastically.

this muscle length to force relationship can be demonstrated bydoing a chin-up. as you hang from the bar, your biceps muscles arestretched and can produce only a relatively small force. as you getclose to the bar, your biceps muscles contract substantially, andyou again experience difficulty. the easiest part of the chin-upoccurs somewhere in between, when your muscles are close to theirnatural length.

imagine hanging from a chin-up bar and beginning a chin-up. which of the following velocity versus time graphs best representsthe first part of your motion (from being at rest to beingapproximately halfway to the bar)? assume you are trying to do thechin-up as quickly as possible.

Figure E is the correct representation of the first part of the motion. When in a hanging position from the chin-up bar, the bicep muscles are stretched beyond their normal length already. So at this point they are at the peak of their capacity and you are at rest (this corresponds to the velocity v = 0 at t = 0). On contracting the bicep muscles and pulling your whole body up, you begin to gain speed and v increases. This increase in velocity is exponential. Soon the bicep muscles contract up to 80% their normal length reducing the force they can produce to keep you rising up to zero. The velocity change happens because the body is accelerating and the muscles can still supply a net force to lift you up. The acceleration is present because of this net force. The moment this force reduces to zero, the acceleration too reduces to zero. (From Newton's second law of motion). This reduction in acceleration is responsible for the reduction of the curvature of the v curve in figure E above. The point where the velocity becomes horizontal corresponds to the point where the muscles reach their maximum contraction unit and can supply no more net force and as a result no acceleration. This further results inba constant velocity which is the flat nature of the curve seen in diagram E.

Thank you for reading.

Explanation:

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find the surface area of the sphere to the nearest square unit. use a calculator.

a. 254 in.2

b. 64 in.2

c. 127 in.2

d. 1,018 in.2