A spring and block are in the arrangement of the figure below. When the block is

Question

A spring and block are in the arrangement of the figure below. When the block is pulled out to x = +3.5 cm, we must apply a force of magnitude 350 N to hold it there. We pull the block to x = +11.5 cm and then release it. Figure (a): A spring in its relaxed state. The origin of an x axis has been placed at the end of the spring that is attached to a block. Figure (b): The block is displaced by d, and the spring is stretched by a positive amount x. Note the restoring force F, exerted by the spring. Figure (c): The spring is compressed by a negative amount x. Again, note the restoring force. For each of the following, find how much work the spring does on the block when the block moves from the first point given to the second point given the following. (a) x_i = 8.0 cm to x = 3.5 cm (b) x_i = 8.0 cm to x = -3.5 cm (c) x_i = 8.0 cm to x = -8.0 cm (d) x_i = 8.0 cm to x = -11.5 cm

Explanation / Answer

k = 350/0.035 = 10000 N/m.
Work by spring w = k(xi^2-x^2)/2. Note that displacements of equal magnitude and opposite sign have equal amounts of associated PE. This means that if x = -xi, w = 0.
a. w = 10000*(0.08^2-0.035^2)/2 = 25.875 J
b. w = 10000*(0.08^2-(-0.035^2))/2 = 38.125 J
c. w = 0 J
d. w = 10000*(0.08^2-(0.115^2))/2 = -34.125 J

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