Home / Expert Answers / Advanced Physics / 4-consider-a-series-of-two-springs-with-spring-constants-k-and-k-respectively-connected-i-pa999

(Solved): 4. Consider a series of two springs with spring constants k and k, respectively, connected i ...



4. Consider a series of two springs with spring constants k? and k?, respectively, connected in series with
interconnected ma

4. Consider a series of two springs with spring constants k? and k?, respectively, connected in series with interconnected masses m? and m? as shown in the diagram below. Hooke's law states that a spring extended a length L experiences a contraction force equal to kL where k is the spring constant. (a) The two masses are located at distances ? and 2 away from a fixed point from which the springs hang under gravity. By considering the balance of forces on each of the two masses, justify, perhaps with a force diagram or an explanation, why ? and ? must satisfy m?? = k?x? +m?9+k?(x? ? ?), m?Ï? = ?k?(x2 ?1?) + m?g. (3) IL llllllllllllllllll leeeeeeeeeeeeeeeee m? I2 k? m? (b) Eliminate ? from the system of equations above to show that r2 must satisfy the fourth-order linear, non-homogeneous differential equation m?m?x) + (m?k? + m2k1 + m2k2) x(²) + (k?k2)x2 = g (m?k? + (m? + m?)k2). (c) From now on set m? = m? = 1. Show that the roots to the characteristic equation for (4) are purely imaginary numbers tiw? and tiw which you must determine in terms of k? and k?. Hence, find the general solution to (4). (d) Optional: Consider a mass m attached to a single spring of spring constant k hanging under gravity as shown below. Find a differential equation for x and compare this to the equation obtained by taking the limit in (4) as m? ?0. Hence, show that the addition of two springs of spring constant k? and k? attached in series behaves as a single spring whose spring constant is the harmonic mean k?k? k?+k? m Scree


We have an Answer from Expert

View Expert Answer

Expert Answer


---------------------------------------- Please let me know if you
We have an Answer from Expert

Buy This Answer $5

Place Order

We Provide Services Across The Globe