A 130 g block is dropped onto a relaxed vertical spring that has a spring consta

Question

A 130 g block is dropped onto a relaxed vertical spring that has a spring constant of k = 2.1 N/cm (see the figure). The block becomes attached to the spring and compresses the spring 11 cm before momentarily stopping. While the spring is being compressed, what work is done on the block by (a) the gravitational force on it and (b) the spring force? (c) What is the speed of the block just before it hits the spring? (Assume that friction is negligible.) (d) If the speed at impact is doubled, what is the maximum compression of the spring? http://edugen.wileyplus.com/edugen/courses/crs4957/art/qb/qu/c07/fig07_42.gif

Explanation / Answer

Consider the block to fall a distance X before making first contact with the spring. It therefore drops a total distance of X+0.13 before stopping Then (X + 0.13) 0.210 x 9.81 J is the energy absorbed by the spring = 470 x 0.13 / 2 J. Therefore X = (470 x 0.13)/{2(9.81 x 0.210)} - 0.13 = 14.7m block's energy when it meets spring is 0.21 x 14.7 x 9.81 = 30.28 J a) work done on block by gravity while it compresses the spring = 0.13 x 0.21 x 9.81 = 0.268 J b) total energy absorbed by spring = 30.28 + 0.268 = 30.55 J (Note: this checks using the spring compression calculation, energy = 0.5 x deflection x spring constant = 0.5 x 0.13 x 470 = 30.55) c) kinetic energy of block = 0.5 x M x v^2 where M is its mass (0.21) and v its speed. As it meets the spring its energy is = 30.28J (see above) Therefore v^2 = 30.28/(0.5 x 0.21) = 288.4 so v = 16,98 m/s at instant of meeting spring. d) Double the speed at impact will mean 4 times the energy at that point. Therefore the total energy to be absorbed by the spring is : 30.28 x 4 + Y x 9.81 x 0.21 = 470 x Y /2 where Y = spring deflection in m. Solving for Y gives: Y = 4 x 30.28 / (235 - 2.1) = 121.1/ 232.9 = 0.52 m which assumes the spring behaves linearly over that distance. This is 4 x 0.13 which is the "obvious" answer. The system behaves linearly after all !

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