A mass m = 16 kg rests on a frictionless table and accelerated by a spring with

Question

A mass m = 16 kg rests on a frictionless table and accelerated by a spring with spring constant k = 4967 N/m. The floor is frictionless except for a rough patch. For this rough path, the coefficient of friction is ?k = 0.51. The mass leaves the spring at a speed v = 3.9 m/s.
3)The mass is measured to leave the rough spot with a final speed vf = 2.1 m/s.

How much work is done by friction as the mass crosses the rough spot?
4)What is the length of the rough spot?

5)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?

6)In this new scenario, what would the coefficient of friction of the rough patch need to be changed to in order for the block to just barely make it through the rough patch?

7)

Return to a scenario where the blcok makes it throgh the entire rough patch. If the rough patch is lengthened to a distance of three times longer, as the block slides through the entire distance of the rough patch, the magnitude of the work done by the force of friction is:

the same

three times greater

three times less

nine times greater

nine times less

Explanation / Answer

using energy conservation principle:
initial kinetic energy=work done against friction+final kinetic energy

==>0.5*16*3.9^2=work done+0.5*16*2.1^2

==>work done=86.4 J

4. work done=force*distance

force=friction coefficient*normal force

normal force=weight of the block

hence work done=0.51*16*9.8*distance

==>distance=1.0804 m

5. half way through the rough spot means half the work done .

so initial kinetic energy of the block=0.5*86.4 (as final veloicty=0)

==>potential eenrgy of the spring=0.5*86.4=43.2 J

==>0.5*4967*x^2=43.2

==>x=0.1319 m

6. to just make it barely through the rough patch , let coefficient of friction required is k

then same work (43.2 J) will be done for the whole length

hence k*16*9.8*1.0804=43.2

==>k=0.255

7. work done=force*distance

so if distance is increased three times, the work done will be increased by 3

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