Questions: To demonstrate springs in a two-dimensional lattice, you build the apparatus shown below that is composed of 8 parallel chains of springs, with each chain containing 2 identical low-mass springs in series. At the top and bottom, the springs are attached to a lightweight rod. A mass of 10 kg is attached to the bottom rod. Relaxed, the distance d from the bottom to the top of the apparatus is 0.12 m . (The top of the apparatus is attached to the ceiling in the lab which is not shown.) Each of the 16 springs has a stiffness of 48 N / m. If you support the bottom of the apparatus at rest and if d=0.503 m, what is the magnitude of the force by your hand on the apparatus? Fhand = i N

Solution Steps

Each chain consists of 2 identical springs in series. The equivalent spring constant \( k_{\text{chain}} \) for two springs in series is given by: \[ \frac{1}{k_{\text{chain}}} = \frac{1}{k} + \frac{1}{k} = \frac{2}{k} \] \[ k_{\text{chain}} = \frac{k}{2} \] Given \( k = 48 \, \text{N/m} \): \[ k_{\text{chain}} = \frac{48}{2} = 24 \, \text{N/m} \]

There are 8 parallel chains, each with a spring constant of \( 24 \, \text{N/m} \). The equivalent spring constant \( k_{\text{total}} \) for 8 parallel chains is given by: \[ k_{\text{total}} = 8 \times k_{\text{chain}} = 8 \times 24 = 192 \, \text{N/m} \]

The force exerted by the springs when the apparatus is stretched by \( d = 0.503 \, \text{m} \) is given by Hooke's Law: \[ F_{\text{springs}} = k_{\text{total}} \times d \] \[ F_{\text{springs}} = 192 \, \text{N/m} \times 0.503 \, \text{m} = 96.576 \, \text{N} \]

Final Answer