88.1 Work done by variable force Example 1: A spring hangs vertically in its relaxed state. a block of mass m is attached to the spring, but the block is held in place so tha the spring at first does not stretch. Now the hand holding block is slowly lowered, so that the block descends at constant speed until it reaches the point at which it hangs at equilibrium with the hand removed. At this point the spring is measured to have stretched a distance d from its previous relaxed length. Find the work done on the block in this process by(a)gravity,(b)the spring, (c) the hand. 88.1 Work done by variable force Solution: (a)gravity--a constant force tF W,=F2 4F=-mgj(0-d)j=mgd F, (b)the spring-a variable force mg F=-kyj kd=mg k g Ws= Fs dyj=l-kvj dyj272 d 2 (c)the hand-a variable force F+F.+F=0 F-F =刚yj+mgj 66 Example 1: A spring hangs vertically in its relaxed state. A block of mass m is attached to the spring, but the block is held in place so that the spring at first does not stretch. Now the hand holding block is slowly lowered, so that the block descends at constant speed until it reaches the point at which it hangs at equilibrium with the hand removed. At this point the spring is measured to have stretched a distance d from its previous relaxed length. Find the work done on the block in this process by (a)gravity, (b)the spring, (c) the hand. §8.1 Work done by variable force Solution: (a)gravity—a constant force W F r mgj d j mgd g = g ⋅ = − ⋅ − =ˆ (0 ) ˆ r r ∆ (b)the spring—a variable force W F y j kyj y j kd mgd s s 2 1 2 1 ˆ d ˆ ˆ d 2 -d 0 -d 0 = ⋅ = − ⋅ = − = − ∫ ∫ r F kyj s ˆ = − r d mg kd = mg k = (c)the hand —a variable force F F F kyj mgj F F F h s g s h g ˆ ˆ 0 = − − = + + + = r r r v r r mg r Fs r Fh y r §8.1 Work done by variable force