A 18kg box slides 4.0 in down the frictionless ramp shown in the figure(Figure 1

Question

A 18kg box slides 4.0 in down the frictionless ramp shown in the figure(Figure 1), then collides with a spring whose spring constant is 150N/m What is the maximum compression of the spring? Express your answer to two significant figures and include the appropriate units. At what compression of the spring does the box have its maximum speed? Express your answer to two significant figures and include the appropriate units.

Explanation / Answer

let maximum compression be h. then at this point, the block will have speed zero. let the horizontal level at the normal (unstretched) length of spring be the zero potential energy point. then at the initial condition configuration, total energy=m*g*4*sin(30)=352.8 J when it compresses the spring by h, then total energy=-m*g*h*sin(30)+0.5*k*h^2 comparing both the energy, we get h=2.8 m part B. let at compression h1, the velocity is maximum. applying conservation of energy, 352.8=-m*g*h1*sin(30)+0.5*k*h1^2+0.5*m*v^2 ==> 9*v^2=352.8+88.2*h1-75*h1^2 taking derivative, 18*v*dv=(88.2-150*h1)*dh1 v to be maximum, dv/dh1=0 88.2=150*h1 so h1=0.588 m

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