A mass is attached to a spring on a horizontal bench and it

A mass is attached to a spring on a horizontal bench, and it oscillates. The spring has a spring constant, k Nm, and the mass has mass m kg. The cost to purchase a spring is 6k$, and the cost to purchase a mass is 4m$. The position of the mass is given by

What is the expression to calculate the total cost of the configuration in terms of k and m ? (Only consider the costs associated with purchasing the mass and the spring.)

Total Cost=

The completed configuration should cost a total of 37 dollars and the oscillations should have a period of (1/13) seconds. What are the values of k and m that will meet these design objectives?

m = ______ kg
k =_______ N

Solution

cost of spring = 6k $

cost of mass = 4 m$

so total cost = ( 6k + 4m)$

given that total cost = 37 $

so ( 6k+ 4m)$ = 37 $

6k + 4m = 37 -------------------------1

w( angular frequnecy) = 2 pi/ T

K( spring constant) = m w²

K = m( 2 pi/ T)²      ( T = time period= 1/13 sec)

after putting the value of T, K= (26 pi)² m

now put this value in eq 1:

6*(26 pi)² m + 4m = 37

6*(26*3.14)² m + 4m=37

on solving: m= 9.25*10^-4 kg

K= (26 pi)² m

k= ( 26*3.14)²*9.25*10^-4 = 6.615 N/m