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 = +4.0 cm, we must apply a force of magnitude 490 N to hold it there. We pull the block to x = +14.5 cm and then release it. (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. (b) The block is displaced by , and the spring is stretched by a positive amount x. Note the restoring force s exerted by the spring. (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 (a) xi = 6.5 cm to x = 4.5 cm 13.5 J (b) xi = 6.5 cm to x = -4.5 cm 13.5 J (c) xi = 6.5 cm to x = -6.5 cm 0 J (d) xi = 6.5 cm to x = -14.5 cm correct answer for part d) -103 J This is a practice version for my homework I cannot get part D. a-c all solve the same way by integrating kinetic energy and positions but d doesn't please help

Explanation / Answer

An appropriate way to find the work done in such cases is to determine the potential energy as a function in displacement. Then the work will be the difference of the initial and final potentials. As the spring force is F=-k*x, the potential W should be W=(1/2)*k*x^2, as to have F=-dW/dx. Knowing that F=380N causes a pull of 3cm, obviously k=380/3[N/cm]. Thus, the work the spring does on the block when the block moves from the first point given to the second point given is: a) (1/2)*380/3*9^2 - (1/2)*380/3*4.5^2 = 3847.5 N.cm b) (1/2)*380/3*9^2 - (1/2)*380/3* (-4.5)^2 = 3847.5 N.cm It’s the same as in a), because the works done in the sections (+4.5, 0) and (0, -4.5) cm the are of the same quantity and opposite in sign, compensating each other. c) the work is 0 by the reasons, mentioned above d) (1/2)*380/3*9^2 - (1/2)*380/3* (-10)^2 = -1203.33 N.cm. Negative value here means that an external force must do work to compress the spring.

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