We can add two springs (with different spring constants k_ and k_2) together so

Question

We can add two springs (with different spring constants k_ and k_2) together so that both exert forces on a mass. There are two common ways to do this. The first is in series, as shown in Fig. 5 and the second is in parallel as shown in Fig. 6. In both cases we can, conceptually, replace the two springs with one spring with a spring constant k_eff that depends on k_1 and k_2. Our goal is to figure out what k_eff is. (a) Use Newton's 2nd law and Hooke's law to figure out a relationship between k_eff and the two individual spring constants for the series case. (b) Use Newton's 2nd law and Hooke's law to figure out a relationship between k_eff and the two individual spring constants for the parallel case. (c) Go through the work we did in class to convince yourself that the motion of the mass should look like x(t) = x_m cos (squareroot k_eff/M t).

Explanation / Answer

a) Let two springs of sprok constant k1 and k2 are connected in series and each is elongated by x1 and x2 when a force F is applied

F = k1 x1 and F = k2 x2

let us replace the two springs by a single spring of effective sprong constant keff

F = (x1+x2)* keff

x1 = F/k1 and x2 = F/k2

(x1 + x2) = F/keff = F/k1 + F/k2

1/Keff = 1/k1 + 1/k2

b) When the springs are connected in parallel, each spring has a elongation of x

F = (k1 + k2) x

when replaced by a single spring of effective spring constant keff

F = keff *x = (k1 + k2) x

keff = k1 + k2

Note: Hooks law states that

when the deformation is small

stress/strain = Y, constant

stress = Force /unit area

strain = x /L   , x, is the elongation of the spring

The spring formula F = kx , is a result of hooks law and the force

F = ma is from Newton's second law

c) is omething related to your class work , you ave to figure out.

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