Please show all steps. Thank you Students can identify and apply the relevant la

Question

Please show all steps. Thank you Students can identify and apply the relevant laws of physics along with the necessary mathematics to successfully solve a mechanics problem. There are two problems. Use the space provided. A block with mass M rests on a frictionless surface and is connected to a horizontal spring of force constant k. The other end of the spring is attached to a wall. A second block with mass m rests on top of the first block. The coefficient of static friction between the blocks is mu_s. Find the maximum amplitude of oscillation such that the top block will not slip on the bottom block.

Explanation / Answer

The friction force of the large object on the small object is
Ff = mg
the accelerating force on the small object must not be larger than the friction force. Otherwise the small object will start to slip.
The maximum acceleration by this maximum force is
amax = Ff/m = mg/m = g

the equation for the movement of the spring is
y = A*sin(t)

where = (k/(m+M)) and A = amplitude.

the acceleration a is the second derivative
v = A(k/(m*M))*cos((k/(m+M))
a = - A*k/(m+M)*sin((k/(m+M)) where the maximum acceleration is
amax = - A*k/(m+M)

set both equations for amax equal and solve for A:
g = - A*k/(m+M)
Amax = (m+M)*g* / k = maximum amplitude

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