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Imagine that you have an indestructible boxcar sitting on frictionless railroad track. The boxcar has length *L*, height *H*, and width *W*. It has *N* cannonballs of radius *R* and mass *M* stacked up against one end. If I move the cannonballs in any fashion -- slowly carrying them, rolling them, firing them out of a cannon -- what is the furthest I can move the boxcar along the rails? Which method should I use to move the boxcar the furthest? Assume that the inside walls are perfectly absorbing, so that collisions are perfectly inelastic.

h2. Part A


h4. Solution One

{toggle-cloak:id=sysa} *System:*  {cloak:id=sysa}Boxcar and cannonballs as [point particles|point particle].{cloak}

{toggle-cloak:id=inta} *Interactions:* {cloak:id=inta} Not Important in this part.{cloak}

{toggle-cloak:id=moda} *Model:*  {cloak:id=moda}[Point Particle Dynamics].{cloak}

{toggle-cloak:id=appa} *Approach:*  

{cloak:id=appa}

{toggle-cloak:id=diaga} {color:red} *Diagrammatic Representation* {color}

{cloak:id=diaga}


!thatfbd1.jpg!

the system consists of the Boxcar on rails and the Cannonballs, plus whatever devices we use for propulsion inside.There are thus no external influences

{cloak:diaga}

{toggle-cloak:id=matha} {color:red} *Mathematical Representation* {color}
{cloak:id=matha}

Since there are no external influences, which includes forces, the [Center of Mass] of the system is not affected, and by the Law of Conservation of [Momentum] must remain fixed. . 

{latex}\begin{large}\[\ M_{Boxcar} x_{Boxcar, initial} + \sum M_{i} x_{i, initial} = M_{Boxcar} x_{Boxcar, final} + \sum M_{i} x_{i, final}  \]\end{large}{latex}

Here {*}x{~}i{~}{*}  is the position of the center of the *i*th cannonball and {*}x{~}Boxcar{~}{*} is the position of the center of the boxcar. The subscripts *initial* and *final* indicate the positions at the start and the end of our operation.  

{latex}\begin{large}\[ N = F_{A} = \mbox{300 N} \]\end{large}{latex}

{cloak:matha}
{cloak:appa}

h2. Part B

!thatnormal2.jpg|width=500!

A person moves a 10 kg box up a smooth wall by applying a force of 300 N. The force is applied at an angle of 60° above the horizontal.  What is the magnitude of the normal force exerted on the box by the wall?

h4. Solution

{toggle-cloak:id=sysb} *System:* {cloak:id=sysb}Box as [point particle].{cloak}

{toggle-cloak:id=intb} *Interactions:* {cloak:id=intb}External influences from the earth ([gravity|gravity (near-earth)]), the wall ([normal force]) and the person ([applied force]).{cloak}

{toggle-cloak:id=modb} *Model:*  {cloak:id=modb}[Point Particle Dynamics].{cloak}

{toggle-cloak:id=appb} *Approach:* 

{cloak:id=appb}

{toggle-cloak:id=diagb} {color:red} *Diagrammatic Representation* {color}

{cloak:id=diagb}

 We begin with a free body diagram for the box:

!thatfbd2.jpg!

{cloak:diagb}

{toggle-cloak:id=mathb} {color:red} *Mathematical Representation* {color}

{cloak:id=mathb}

From the free body diagram, we can write the equations of [Newton's 2nd Law|Newton's Second Law]. 

{latex}\begin{large}\[\sum F_{x} = F_{A}\cos\theta - N = ma_{x}\]
\[ \sum F_{y} = F_{A}\sin\theta - mg = ma_{y}\]\end{large}{latex}

Because Because the box is held against the wall, it has no movement (and no acceleration) in the _x_ direction (_a_~x~ = 0).  Setting _a_~x~ = 0 in the _x_ direction equation gives:

{latex}\begin{large}\[ N = F_{A}\cos\theta = \mbox{150 N} \]\end{large}{latex}

{cloak:mathb}

{cloak:appb}

h2. Part C

!thatnormal3.jpg|width=500!

A person scrapes a 10 kg box along a low, smooth ceiling by applying a force of 300 N at an angle of 30° above the horizontal.  What is the magnitude of the normal force exerted on the box by the ceiling?

h4. Solution

{toggle-cloak:id=sysc} *System:*  {cloak:id=sysc}Box as [point particle].{cloak}

{toggle-cloak:id=intc} *Interactions:* {cloak:id=intc}External influences from the earth ([gravity|gravity (near-earth)]), the ceiling ([normal force]) and the person ([applied force]).{cloak}

{toggle-cloak:id=modc} *Model:*  {cloak:id=modc}[Point Particle Dynamics].{cloak}

{toggle-cloak:id=appc} *Approach:*  

{cloak:id=appc}

{toggle-cloak:id=diagc} {color:red} *Diagrammatic Representation* {color}

{cloak:id=diagc}

We begin with a free body diagram for the box:

!thatfbd3.jpg!

{note}The ceiling must push down to prevent objects from moving up through it.{note}

{cloak:diagc}

{toggle-cloak:id=mathc} {color:red} *Mathematical Representation* {color}

{cloak:id=mathc}

From the free body diagram, we can write the equations of [Newton's 2nd Law|Newton's Second Law]. 

{latex}\begin{large}\[\sum F_{x} = F_{A}\cos\theta = ma_{x}\]
\[ \sum F_{y} = F_{A}\sin\theta - mg - N = ma_{y}\]\end{large}{latex}

Because Because the box is held against the ceiling, it has no movement (and no acceleration) in the _y_ direction (_a_~y~ = 0).  Setting _a_~y~ = 0 in the _y_ direction equation gives:

{latex}\begin{large}\[ F_{A}\sin\theta  - mg - N = 0 \]\end{large}{latex}

which we solve to find:

{latex}\begin{large}\[ N = F_{A}\sin\theta - mg = \mbox{52 N}\]\end{large}{latex}

{tip}We can check that the _y_ direction is in balance.  We have N (52 N) and mg (98 N) on one side, and _F_~A,y~ on the other (150 N).{tip}

{cloak:mathc}
{cloak:appc}