A large block P attached to a light spring executes horizontal, simple harmonic

Question

A large block P attached to a light spring executes horizontal, simple harmonic motion as it slides across a frictionless surface with a frequency f = 1.60 Hz. Block B rests on it as shown in the following figure, and the coefficient of static friction between the two is s = 0.560. What maximum amplitude of oscillation can the system have if block B is not to slip?
cm

A large block P attached to a light spring executes horizontal, simple harmonic motion as it slides across a frictionless surface with a frequency f = 1.60 Hz. Block B rests on it as shown in the following figure, and the coefficient of static friction between the two is mu s = 0.560. What maximum amplitude of oscillation can the system have if block B is not to slip? cm

Explanation / Answer

Even you dont mention it, but i assume that block P was attached to a spring to have a simple harmonic motion. Lets M be the mass of both block P&B ----> f = 1/(2pi) * sqrt(k/M) ----> k = 4*pi^2*f^2*M Now, first consider the block B: Newton's 2nd law: f(static) = m(B)* a f(static) m(B)*g*u = m(B)*a(max) , m(B) will be canceled out ----> a(max) = g*u 2nd, consider system of block P and block B with total mass of M: F(spring) = M*a k*x = M*a k*x(max) = M*a(max) 4*pi^2*f^2*M*x(max) = M*g*u , M will be canceled out ----> x(max) = g*u/(4*pi^2*f^2) = 9.8*0.6/(4*3.14^2*1.5^2) = 0.0662 m

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