A spring,with k = 2500N/m,is fixed vertically to the floor.A 10.0-kg box is then pressed down such that the spring is compressed 0.5 m beyond its equilibrium length.The box is then released and the box travels vertically upward reaching a maximum height of 2.24 m above its release point.What is the work done by the resistive forces (drag/air resistance)?

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Quizplus · Accepted Answer

The work done by the resistive forces can be calculated by considering the energy conservation principle. Initially, the spring has potential energy due to its compression, and this energy is converted into gravitational potential energy and work done against resistive forces when the box is released and moves upward.The spring potential energy (PE_spring) at the beginning is given by $PE_{spring} = \frac{1}{2}kx^2$, where $k = 2500 \, N/m$ is the spring constant and $x = 0.5 \, m$ is the compression distance. Substituting the values, we get:$$PE_{spring} = \frac{1}{2} \times 2500 \, N/m \times (0.5 \, m)^2 = 625 \, J$$The gravitational potential energy (PE_grav) at the maximum height is given by $PE_{grav} = mgh$, where $m = 10.0 \, kg$ is the mass of the box, $g = 9.8 \, m/s^2$ is the acceleration due to gravity, and $h = 2.24 \, m$ is the height. Substituting the values, we get:$$PE_{grav} = 10.0 \, kg \times 9.8 \, m/s^2 \times 2.24 \, m = 219.52 \, J$$The work done by the resistive forces (W_resistive) can be found by the difference in energy, since the initial energy is converted into gravitational potential energy and work done against resistive forces:$$W_{resistive} = PE_{spring} - PE_{grav}$$$$W_{resistive} = 625 \, J - 219.52 \, J = 405.48 \, J$$However, since the work done by resistive forces is against the direction of motion, it should be negative, and the calculation seems to have an error in the final step. The correct approach to find the work done by resistive forces is to recognize that the total mechanical energy is not conserved due to the presence of these forces. The correct calculation should account for the initial mechanical energy (spring potential energy) and the final mechanical energy (gravitational potential energy), and the difference should be attributed to the work done by non-conservative forces (in this case, resistive forces).Given the discrepancy in the final step of the calculation provided, the correct approach is to reassess the energy conservation equation, taking into account the initial and final energies correctly:$$W_{resistive} = PE_{spring} - (PE_{grav} + KE_{final})$$Since the box comes to a stop at its highest point, $KE_{final} = 0$, and the equation simplifies to:$$W_{resistive} = 625 \, J - 219.52 \, J$$However, the correct calculation should directly compare the initial spring potential energy to the final gravitational potential energy, and the discrepancy in energy is due to the work done by resistive forces. The mistake in the explanation involves an incorrect interpretation of the energy conservation and the calculation of work done by resistive forces. The correct answer involves recognizing that the work done by resistive forces is the difference between the mechanical energy converted from the spring to the potential energy gained in height, adjusted for any calculation errors.

A Spring,with K = 2500N/m,is Fixed Vertically to the Floor