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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.
Part A
Solution One
System:
Interactions:
Model:
Approach:
Diagrammatic Representation
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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
Mathematical Representation
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. .
\begin
[\ M_
x_
+ \sum M_
x_
= M_
x_
+ \sum M_
x_
]\end
Here xi is the position of the center of the *i*th cannonball and xBoxcar 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.
\begin
[ N = F_
= \mbox
]\end
Part B
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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?
Solution
System:
Interactions:
Model:
Approach:
Diagrammatic Representation
We begin with a free body diagram for the box:
Unable to render embedded object: File (thatfbd2.jpg) not found.
Mathematical Representation
From the free body diagram, we can write the equations of Newton's 2nd Law.
\begin
[\sum F_
= F_
\cos\theta - N = ma_
]
[ \sum F_
= F_
\sin\theta - mg = ma_
]\end
Because Because the box is held against the wall, it has no movement (and no acceleration) in the x direction (ax = 0). Setting ax = 0 in the x direction equation gives:
\begin
[ N = F_
\cos\theta = \mbox
]\end
Part C
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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?
Solution
System:
Interactions:
Model:
Approach:
Diagrammatic Representation
We begin with a free body diagram for the box:
Unable to render embedded object: File (thatfbd3.jpg) not found.
The ceiling must push down to prevent objects from moving up through it.
Mathematical Representation
From the free body diagram, we can write the equations of Newton's 2nd Law.
\begin
[\sum F_
= F_
\cos\theta = ma_
]
[ \sum F_
= F_
\sin\theta - mg - N = ma_
]\end
Because Because the box is held against the ceiling, it has no movement (and no acceleration) in the y direction (ay = 0). Setting ay = 0 in the y direction equation gives:
\begin
[ F_
\sin\theta - mg - N = 0 ]\end
which we solve to find:
\begin
[ N = F_
\sin\theta - mg = \mbox
]\end
We can check that the y direction is in balance. We have N (52 N) and mg (98 N) on one side, and FA,y on the other (150 N).