Mr. Miyamoto\'s Moving Company has two stellar moving engineers, Mario and Luigi

Question

Mr. Miyamoto's Moving Company has two stellar moving engineers, Mario and Luigi. Mario has invented a new ramp which he thinks will make the company's most talented movers much more productive. Mario's new additions to the company's standard ramp include two hard nylon rails which run the length of the ramp; the crates which have to be moved will slide along the hard nylon rails with almost no friction. In the space on the ramp which is in between the nylon rails, Mario has installed a special super-grip carpet, which will give the mover great traction as he or she pushes the crates along the rails and up the ramp.

Mario and Luigi plan to test the new ramp design at the company's testing lab. They begin with the ramp in a North-South orientation flat on the lab floor, and they put two crates of equal mass side-by-side at the south end of the ramp. They then use two hydraulic jacks to raise the north end of the ramp to a desired height; one crate is then at the bottom of the ramp (where the ramp intersects with the lab floor) and the other crate is adjacent to the first and just a little higher up along the ramp. We will now refer to the first crate as the "lower crate" and to the second crate as the "upper crate".

Mario knows that most of the company movers will use the ramp to push crates at constant speed, but he wishes to test extreme possibilities, so he plans to push the crates up the ramp with increasing speed. He selects two crates, each of mass 47.0  kg . The start of his push will be at the bottom of the ramp, and he puts a piece of tape at a point 4.90  m along the ramp (not along the floor) which will mark the endpoint of his hard push; upon arriving at this endpoint, he will "ease up" so that the crates will move at constant speed for the rest of their journey the top of the ramp. Luigi measures the height from the lab floor to the marked endpoint as being 176.40  cm . Luigi also designs a system of two electric eyes connected to a digital timer which will time Mario during his trip from the starting point to the marked endpoint. The size of the selected crates is just right so that Mario will push HORIZONTALLY (parallel to Earth's surface) on the lower crate.

Part A) If Mario is moving the crates up the ramp at constant speed, what will be the force on the upper crate by the rails attached to the ramp?

Give your answer as an ordered pair, with magnitude first, followed by a comma, then followed by the direction. Give the direction in terms of an angle measured CCW from North when looking West.

Part B) If Mario is moving the crates up the ramp at constant speed, what will be the magnitude of the force on the lower crate by the rails attached to the ramp?

Part C) Mario puts on his track shoes and manages to push with a force stronger than the force calculated in Part (A), so that the crates move up the ramp, starting from rest, with a constantly increasing speed. Luigi's timing system measures the time for Mario's trip from start to endpoint (where Mario then "eases up") as being 3.5354  seconds .What was the magnitude of Mario's pushing force during Mario's extreme pushing experiment?

Part D) What was the magnitude of the force on the upper crate by the lower crate during Mario's extreme pushing experiment?

Part E) What was the magnitude of the force on the upper crate by the rails of the ramp during Mario's extreme pushing experiment?

Part F) What was the magnitude of the force on the lower crate by the rails of the ramp during Mario's extreme pushing experiment?

Give your answer as an ordered pair, with magnitude first, followed by a comma, then followed by the direction. Give the direction in terms of an angle measured CCW from North when looking West.

Part G)Answer the following questions about the forces on Mario during Mario's extreme pushing experiment. Mario's mass is 85.0  kg .

What was the force on Mario by the lower crate during Mario's extreme pushing experiment?

Part H) On average, what was the magnitude of the normal component of the force on Mario's feet by the carpet during Mario's extreme pushing experiment?

Part I) On average, what was the "parallel-to-ramp" component of the force on Mario's feet by the carpet during Mario's extreme pushing experiment? Let Up the Ramp be the positive direction.

Explanation / Answer

given two crates of mass m = 47 kg each

length along the ramp upto which the bvlock will be pushed harder, l = 4.9 m

height of this point from the ground, h = 1.764 m

Applied force is pasrallel to earth surface

A. if he is pushing at constant velocity

then the upper crate faces just horizontal force from the lower crate

hence force from rails is the normal reaciton force

N = mg*cos(theta)

but sin(theta) = 1.764/4.9

m = 47 kg

g = 9.81

so N = 430.156 N

B. let force applied by mario be F

then for lower crate, from force balance

Fcos(theta) = 2mgsin(theta)

N = mgcos(theta) + Fsin(theta) = mgcos(theta) + 2mg*tan(theta)sin(theta)

so force on lower crate due to rails

N = 47*9.81[cos^2(theta) + 2sin^2(theta)]/cos(theta) = 47*9.81(1 + sin^2(theta))/cos(theta)

also theta = 21.1 deg

hence N = 558.2536 N

C. time taken to move till length l, t1 = 3.5354 s

let acceleration of system be a

then from force balance

Fcos(theta) - 2mgsin(thjeta) = 2ma

also, l = 0.5*a^t^2

so a = 2l/t^2 = 2*4.9/3.5354^2 = 0.784 m/s/s

hence F = 435.115 N

D. force on upper crate F' = 47*a = 36.848 N

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