Help me with this question?

  • Help me to solve the problem b?

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1 Answer
Apr 21, 2018

(b) #2 pi sqrt(m/(k_1+k_2))#

Explanation:

If the block is displaced through #x#, the spring on the right is compressed through #x#, while that on the left is extended by #x#. So, the former exerts a force of #k_2x# on the block, towards the left, while the latter exerts a force #k_1x#, again to the left! Thus, the net restoring force acting on the block is

#F = -k_1x-k_2x = -(k_1+k_2)x#

This is the same as that exerted by a single spring with spring constant #k_e = k_1+k_2#. This immediately leads to the time period:

#T=2 pi sqrt(m/k_e) = 2 pi sqrt(m/(k_1+k_2))#

Note: I have said above that the spring on the right is compressed, etc. All this is assuming that #x > 0#, i,e,, the block is deflected to the right. Had the block been deflected to the left , the restoring force would have been #k_1|x|+k_2|x|# to the right, but the expression #F = -(k_1+k_2)x# would still be valid.