A number of identical springs are joined in four arrangements. Which arrangement

Q#2 (Past Exam Paper – June 2016 Paper 11 Q20)

A number of identical springs are joined in four arrangements.

Which arrangement has the same spring constant as a single spring?

Solution:

Answer: C.

Let the spring constant of a single spring = k

When 2 springs are joined end-to-end (in series),

Effective spring constant = (1/k + 1/k)^-1 = k/2                        [Choice A]

When 2 springs are joined side-by-side (in parallel),

Effective spring constant = k + k = 2k                                   [Choice B]

Choice C may be considered to be as if the arrangement in B is in series with another similar one [or equivalently, the arrangement in A is in parallel with another similar one]. 

Considering the former case,

Effective spring constant = k/2 + k/2 = k

This has the same spring constant as a single spring.

Considering the latter case,

Effective spring constant = (1/2k + 1/2k)^-1 = 2k/2 = k

Again, it has the same spring constant as a single spring.