Set 8: Introduction to Springs
MC pg 349  #11,13; Exercises pg 350 #37,38,40

11. A spring with a large elastic force constant can be said to be fairly (a) soft (b) rigid (c) long (d) short (a) none of these.

• B

13. A piece of wire is stretched by a certain amount and then allowed to return to its original configuration. It is then stretched twice that initial amount, without exceeding its elastic limit. Compared to the first stretching, the second elongation stored (a) twice as much energy (b) four times as much energy (c) half as much energy (d) no additional energy (e) none of these.

• B

PEe = ½ks₁²

PEe = ½k(2s₁)²
PEe = ½k(4s₁²)
PEe = 4 [½ks₁²]

37. A helical spring is 55 cm long when a load of 100 N is hung from it and 57 cm long when the load is 110 N. Find its spring constant.

• 500 N/m

We do not know how far the spring has been stretched in each of the situations. We only know its total length. Therefore we will set up simultaneous equations using F = ks that involve both k and the equilibrium position, x_o.

F = ks

110 = k (0.57 - x_o)  or  110 = 0.57k - kx_o
100 = k (0.55 - x_o)  or  100 = 0.55k - kx_o

Subtracting the two equations from each other yields

10 =  0.02k
k = 500 N/m

38. Two identical helical springs are attached to one another, end-to-end, making one long spring. If k = 500 N/m is the elastic constant of each separate (essentially weightless) spring, what is the constant for the combination?

• 250 N/m

k_system = [1/k₁ + 1/k₂]^-1
k_system = [1/500 + 1/500]^-1
k_system = [2/500]^-1
k_system = 250 N/m

40.Two identical wires each of length L are attached to one another producing a single long wire. If k = 500 N/m was the elastic constant of each separate (essentially weightless) wire, what is the constant for the combination?

• 250 N/m

This is exactly the same problem as #38. Wires are elastic media.