Q92CP - University Physics with Modern Physics

Question

The Effective Force Constant of Two Springs. Two springs with the same unstretched length but different force constants \({k_1}\)  and \({k_2}\) are attached to a block with mass m on a level, frictionless surface. Calculate the effective force constant \({k_{eff}}\) in each of the three cases (a), (b), and (c) depicted in Fig. P14.92. (The effective force constant is defined by \(\sum {F_x} =  - {k_{eff}}x\) ) (d) An object with mass m, suspended from a uniform spring with a force constant k, vibrates with a frequency \({f_1}\). When the spring is cut in half and the same object is suspended from one of the halves, the frequency is \({f_2}\). What is the ratio \({f_1}/{f_2}\)?

Step-by-Step Solution

Answer

(a) \({k_{eff}} = {k_1} + {k_2}\)

Step 1: Given Data

Force constants = \({k_1}\;{\rm{and}}\;{k_2}\) 

Mass = m

Step 2: Concept

Force constant is defined as the spring’s resisting force directed towards the equilibrium. 

Step 3(a): Calculate the effective force constant

Both springs are in parallel because on displacing block the amount of deformation is same in both springs,

\({k_{eff}} = {k_1} + {k_2}\) 

Hence the effective force constant is \({k_{eff}} = {k_1} + {k_2}\)