Chapter 27, Problem 1PQ

CASE STUDY Concept Exercise 27.1 (page 829), we ignored air resistance acting on the ring launched by the experimenters in the case study. How would including air resistance change your answer to the question about the size of the spring? Does air resistance require that more, less, or the same amount of electric potential energy be stored in the spring? Explain your answers.

To determine

The size of the spring and the amount of potential energy when air resistance is considered.

Answer to Problem 1PQ

The size of the spring will be more compressed. And more potential energy is needed to attain the same final height.

Explanation of Solution

Write the equation for energy for a spring system.

  $\frac{1}{2}k{x}^2=mgh$                                                                                                (I)

Here, $k$ is the spring constant, $x$ is the stretched or compressed distance by spring, $m$ is the mass, $g$ is the gravitational acceleration and $h$ is the height.

If the air resistance is considered, the travelled height will be less. Again, the square of compressed distance is directly proportional to the height. Thus, the spring will be more compressed.

Due to air resistance, there will be a dissipation of mechanical energy. Thus, the total energy of the system will be reduced. To maintain that, there will be needed more amount of potential energy for the ring to attain the equal final height.