A mass m = 16 kg rests on a fr
A mass m = 16 kg rests on a frictionless table and acceleratedby a spring with spring constant k = 4739 N/m. The floor isfrictionless except for a rough patch. For this rough path, thecoefficient of friction is μk = 0.45. The mass leavesthe spring at a speed v = 2.9 m/s.
1)How much work is done by the spring as it accelerates themass?
2)How far was the spring stretched from its unstrechedlength?
3)The mass is measured to leave the rough spot with a finalspeed vf = 1.7 m/s. How much work is done by friction asthe 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-waythrough the rough spot. How far was the spring compressed from itsunstretched length?
6)in this new scenario, what would the coefficient of frictionof the rough patch need to be changed to in order for the block tojust barely make it through the rough patch?
Answer:
1) The work done by the spring is converted into kinetic energyof the body as there is no friction. The kinetic energy of theblock when it leaves the spring is 1/2 m v2
W = 1/2 x 16 x 2.92 = 67.28 J
2) Intial elastic energy stored in the spring is converted intokinetic energy of the block, Hence
1/2 k x2 = 1/2 m v2 => x2 = m v2 / k = 16 x 2.92 /4739 = 0.0284; then x = 0.17 m
The compression in the spring is 0.17m
3) The speed of the body before rough spot is 2.9 m/s and itsspeed after rough spot is 1.7 m/s. The decrease in kinetic energyis due to workdone by frictional force.
Hence Wf = KEf – KEi = 1/2 m Vf2 – 1/2 m Vi2 = 1/2 m (Vf2 – Vi2)= 1/2 x 16 x (1.72 – 2.92)
The work done by friction is Wf = – 44.16 J
4) Wf = 
mg x S => 44.16 = 0.45 x 16 x 9.8 x S => S =0.626 m = 62.6 cm
The length of the rough patch is 0.626m
5) If the body travels only half of the rough patch then thework done by the spring is equal to work done against thefrictional force. Let y is new compression in spring.
1/2 k y2 = mg x S/2 => y2 = mg x S / k = 0.45 x 16 x 9.8 x 0.626 / 4739 = 0.0093
The new compression in the spring is y = 0.096 m = 9.6cm
6) The body crosses the rough patch in this new scenario if thecoefficient of kinetic friction is half of its original value. Thenew coefficient of kinetic friction is 0.225.
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