Question

A mass m = 13 kg is connected to two different springs, one on the left and another on the right. It is displaced a distance x to the right from its equilibrium position. (Take k1 = 115 N/m and k2 = 198 N/m.)

(a) What is the net force acting on the mass? (Use any variable or symbol stated above along with the following as necessary: k_eff for the effective spring constant.)
What is the effective spring constant?

(b) What will the frequency of the oscillation be when the mass is released?

(c) If x = 0.1 m, what is the total energy of the mass-spring system after the mass is released, and what is the maximum velocity of the mass?


Im not sure how to solve these, or where to begin. Any help or advice is appreciated, thank you!

Answer 1

(a)
F1= - k1•x
F2 =- k2•x
F=F1+F2= - (k1+k2) •x
k(eff) = k1+k2 =115+198=313 N/m.

(b)
ω =sqrt{k(eff)/m}=sqrt(313/13) = 4.9 rad/s.
f = ω/2•π =4.9/2•π = 0.78 Hz.
(c)
x=A=0.1 m.
E=kA²/2 =313•0.1²/2 =1.565 J.
v(max) = A• ω =0.1•4.9=0.49 m/s.

Answer 2

Thank you so much for your help!