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A large block P executes horizontal simple harmonic motion as it slides across a frictionless surface with a frequency f = 1.00 Hz. Block B rests on it, as shown in Figure P15.53, and the coefficient of static friction between the two is µs = 0.450. What maximum amplitude of oscillation can the system have if block B is not to slip?
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Friction force between block P & B prevents blocks from sliding,
and maximum frictional force on B = ?Mbg
where Mb = mass of block B
This frictional force will be maximum when P is at its maximum displacement i.e. Amplitude of oscillation;
lets assume it A;
so,
Frictional force = Maximum spring force on Block P

Maximum force on block P = kA ; A is amplitude and k is spring constant

so,

kA = ?Mbg
A=  ?Mbg/k...................................(i)

also we know that in simple harmonic oscillation
f = 1/2??k/m......................... where f is frequency of oscillation
so k = (2?f)2m
putting value of k in eq. i

A = ?Mbg / (2?f)2Mb
A = ?g / (2?f)2.............................(ii)

Putting values in eq. (ii) as f=1; ?=0.45; g=9.8

A = 0.45*9.8/(2?)2
A = 0.119 m or 11.9 cm (approx)
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