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Answer :
The force constant of the spring can be calculated using the work-energy relationship k = 2W/x^2; the magnitude force is found with F = kx; work done to compress the spring is W = 1/2 kx^2. The force to stretch it by a certain distance is the same as in the initial stretch.
To find the force constant of a spring, we use the formula derived from Hooke's Law, which relates the force exerted on a spring (F) to the displacement (x) from its equilibrium position and the spring constant (k): F = kx. When work (W) is being done to stretch or compress a spring, the work is equal to the change in elastic potential energy stored in the spring, given by W = 1/2 kx2.
(A) To find the spring constant, we can rearrange the work-energy relationship to solve for k: k = 2W/x2. Using the information given that 16.0 J of work must be done to stretch the spring 8.00 cm (0.080 m), we find k = 2 * 16.0 J / (0.080 m)2.
(B) The magnitude of the force needed to stretch the spring 8.00 cm is found using Hooke's Law: F = kx. Here x is already given as 8.00 cm and k would be the constant we found in part A.
(C) The work needed to compress the spring 4.00 cm would also be found using the formula W = 1/2 kx2, with x now being 4.00 cm (0.040 m).
(D) The force needed to stretch it to 8.00 cm is the same as the force calculated in part B because the spring's force is linearly proportional to the displacement from its unstretched length.
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