A mass m = 10 kg rests on a frictionless table and accelerated by a spring with

Question

A mass m = 10 kg rests on a frictionless table and accelerated by a spring with spring constant k = 4461 N/m. The floor is frictionless except for a rough patch. For this rough path, the coefficient of friction is mu_k = 0.49. The mass leaves the spring at a speed v = 2.8 m/s. How much work is done by the spring as it accelerates the mass? How far was the spring stretched from its unstreched length? The mass is measured to leave the rough spot with a final speed v_f = 1.3 m/s. How much work is done by friction as the mass crosses the rough spot? What is the length of the rough spot? In a new scenario, the block only makes it (exactly) half-way through the rough spot. How far was the spring compressed from its unstretched length?

Explanation / Answer

2)Using conservation of energy

Energy lost in spring = kinetic energy gained by mass

1/2*k*x^2 = 1/2*m*v^2

1/2*4461*x^2=1/2*10*2.8^2

gives

x=0.1325 m

5) Kinetic energy of spring = work done by friction

So

umg*0.64/2 = 1/2*m*v^2

0.49*10*9.8*0.64/2 = 1/2*10*v^2

gives v=1.753m/s

Now we know that work done by spring == kinetic energy of mass

so

1.2*k*x^2=1/2*m*v^2

gives

1/2*4461*x^2 = 1/2*10*1.753^2

gives x = 0.083m

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