Question

(Differential Equations Problem) A Mass of weight 1N stretches a spring a length of 0.75m. Assume that there is a negligible drag force. The mass will also be acted upon by an external force F(t) = cos(ωt)N. At time t = 0, the mass is stretched 0.5m past equilibrium and is released with zero velocity.

(a) Write a differential equation for the displacement u(t) of the mass from equilibrium.

(b) What value of ω causes resonance?

(c) Find the motion of the mass for this value of ω.

Answer 1

spring constant k

where x is the equilibrium distance from natural length of spring.

k = 4/3 N/m

This is the case of forced oscillation where F(t) = cos(ωt)

Net force at any distance x from equilibrium position

k=4/3 ,m=1N/9.8 Kg= 0.102kg

putting values in above differential equation we get

b

:

condition for resonance for above equation is

by comapring we find

w=3.615 rad/sec

c

steady state solution for any general value of w is

f = 9.8m/s^2