Question #226074

With block C placed on top of B. the system undergoes simple harmonic motion with an amplitude of 0.12 m. Block B has a speed of 0.22 m/s at a displacement of 0.075 m from its equilibrium position. Determine the period of the motion. Express your answer with the appropriate units. What minimum value for the coefficient of static friction mu_s between B and C is needed if C is never to slip? Ignore any friction between B and the horizontal surface.


1
Expert's answer
2021-08-16T08:34:11-0400

We know the amplitude, so, we know the total energy of the system:


ET=12kA2.E_T=\frac12 kA^2.

This is equivalent to the sum of elastic potential and kinetic energy of the system at the moment when the speed is 0.22 m/s and displacement of 0.075 m.


ET=12kx2+12Mv2.E_T=\frac12 kx^2+\frac12Mv^2.

The period of motion is


T=2πMk.T=2\pi\sqrt{\frac{M}{k}}.

So, what we have is


12kA2=12kx2+12Mv2, k=Mv2A2+x2.\frac12 kA^2=\frac12 kx^2+\frac12Mv^2,\\\space\\ k=\frac{Mv^2}{A^2+x^2}.

Substitute this into the equation for period:


T=2πvA2+x2=4 s.T=\frac{2\pi}{v}\sqrt{A^2+x^2}=4\text{ s}.

Now, let's talk about the coefficient of friction. Write the equation for displacement:


x(t)=Asin(2πTt),x(t)=A\sin\bigg(\frac{2\pi}{T}t\bigg),

the acceleration is the second derivative of displacement by time:


a(t)=A(2πT)2sin(2πTt).-a(t)=A\bigg(\frac{2\pi}{T}\bigg)^2\sin\bigg(\frac{2\pi}{T}t\bigg).

So, the magnitude of acceleration is


a=A(2πT)2=0.3 m/s2.|a|=A\bigg(\frac{2\pi}{T}\bigg)^2=0.3\text{ m/s}^2.

The coefficient of friction is


μs=ag=0.03.\mu_s=\frac ag=0.03.


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Comments

Cpha
13.08.21, 23:20

Thanks

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