Dueling Mass-Spring Systems

Consider the four block-and-spring(s) systems shown below. Each block moves on a horizontal, frictionless table. The blocks all have the same mass m, and all of the springs are identical and ideal, with spring constant k. At the instant shown, each block is released from rest a distance A to the right of its equilibrium position (indicated by the dashed line).

llllllll In case B , assume that each spring is at its equilibrium length when the block is at its equilibrium position. 3. Rank the cases according to magnitude ofthe net force on the block at the instant shown, from largest to smallest. (Hint: In case C, how far was the point connecting the two springs displaced when the block was displaced a distance A?) Explain. b. Use your answers above to rank the cases according to the time it takes the block to return to its equilibrium position. Explain. c. The total potential energy of a system of multiple springs is dened to be the sum of the potential energies stored in each of the springs. Rank the cases according to total potential energy at the instant shown. (Hint: In each case, consider how much each individual spring is extended.) Explain. 11. Suppose that in cases B, C, and D, each combination of springs were replaced by a single spring such that the motion of the block is the same as before the replacement. We will call the spring constant of this new spring the effective spring constant of the original combination. Quantitatively, the effective spring constant, kg\In case B , assume that each spring is at its equilibrium length when the block is at its equilibrium position. a. Rank the cases according to magnitude of the net force on the block at the instant shown, from largest to smallest. (Hint: In case C, how far was the point connecting the two springs displaced when the block was displaced a distance A?) Explain. b. Use your answers above to rank the cases according to the time it takes the block to return to its equilibrium position. Explain. c. The total potential energy of a system of multiple springs is dened to be the sum of the potential energies stored in each of the springs. Rank the cases according to total potential energy at the instant shown. (Hint: In each case, consider how much each individual spring is extended.) Explain. dc Suppose that in cases B, C, and D, each combination of springs were replaced by a single spring such that the motion of the block is the same as before the replacement. We will call the spring constant of this new spring the effective spring constant of the original combination. Quantitatively, the effective spring constant, keff, may be dened by the relationship przackyspring : keff5 ,where EFblac/cymmg is the sum of all forces on the block by the springs. and ii is the position of the block with respect to the equilibrium position, i. Rank the four cases according to keff, from largest to smallest. Explain. ii. Use your ranking above and the relationship T : 27r / k to rank the cases according to period of oscillation. Explain your ranking. iii. Use your, ranking from part d i and the relationship Em\"; = keAz to rank the cases according to total energy. Explain