Amplitude and static friction?

Question

Hi,

There is a 2.0 kg mass riding on top of a 3.8 kg mass as it oscillates on a frictionless surface with a period of 3.0 s. The upper block just begins to slip when the amplitude is increased to 50 cm. What is the coefficient of static friction between the blocks? The spring constant is 70 N/m.

I am having a hard time incorporating friction into my equations. I believe that I am supposed to use F=uN, w = (k/m)^.5 as well as w = 2(pi)f and V = wA. Any ideas on how to start this one?


Thanks!

Answer 1

The acceleration (a) of the system (m1+m2) is the key

The force on the 2 kg mass(m1) is
F1=m1 a
but what is a?
Consider the system equation for
The harmonic oscillator is
x(t)=A cos(wt +Ph)
A - amplitude
w- angular frequency=2 pi f
t – time
Ph – phase angle
The acceleration the second derivative is
a=x’’(t)= -A w^2 cos(wt +Ph)
also
w=sqrt(k/m)

I guess now we have all the ingredients.
Let phase =0 we have
a= - A (k/m)cos (sqrt(k/m)t)=
a=-.5(70/(2+3.8)) cos(sqrt(70/(2.0 +3.8)) (0) ) note t=0 for max acceleration
a= 5.7m/s^2
also you are correct
F=uN = m1 n and since N=m1g
u g m1 = m1 a
u=a/g
u=5.7m/9.81=0.581
This is it!