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Composition Setup
 Deck of Cards
idbigdeck
 Wiki Markup
h4. Part A

!pushingbox.png|width=40%!

A person pushes a box of mass 15 kg along a smooth floor by applying a perfectly horizontal force _F_.  The box accelerates horizontally at a rate of 2.0 m/s{color:black}^2^{color}.  What is the magnitude of _F_?

System:  Box as [point particle] subject to external influences from the person (applied force) the earth (gravity) and the floor (normal force).

Model: [Point Particle Dynamics].

Approach:  The word *smooth* in the problem statement is a keyword, telling us that the floor exerts no horizontal force on the box.  Thus, [Newton's 2nd Law|Newton's Second Law] for the _x_ direction can be written:

{latex}
 Card
labelPart A
Part A
Image Added
 Excerpt
A person pushes a box of mass 15 kg along a smooth floor by applying a force F at an angle of 30° below the horizontal.
 The box accelerates horizontally at a rate of 2.0 m/s2. What is the magnitude of F?
Solution
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idsysa
 System: 
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idsysa
Box as .
 Toggle Cloak
idinta
 Interactions: 
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idinta
External influences from the person (applied force) the earth (gravity) and the floor (normal force).
 Toggle Cloak
idmoda
 Model: 
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idmoda
.
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idappa
 Approach: 
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idappa
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iddiaga
  Diagrammatic Representation 
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iddiaga
Before writing Newton's 2nd Law for the x direction, we choose coordinates and break the applied force F into x- and y-components:
Image Added
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diaga
diaga
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idmatha
  Mathematical Representation 
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idmatha
Using the free body diagram, we can write the relevant x-component of Newton's 2nd Law:
 Latex
\begin{large}\[ \sum F_{x} = F\cos\theta = ma_{x}\] \end{large}
Solving for F:
 Latex
\begin{large}\[ F = \frac{ma_{x}}{\cos\theta} = \mbox{
30
34.6 N}\]
 
\end{large}
{latex}

h4. Part B

A person pushes a box of mass 15 kg along a smooth floor by applying a perfectly horizontal force _F_.  The box moves horizontally at a constant speed of 2.0 m/s in the direction of the person's applied force.  What is the magnitude of _F_?

System and Model:  As in Part A.

Approach:  Just as above, [Newton's 2nd Law|Newton's Second Law] for the _x_ direction can be written:

{latex}
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matha
matha
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appa
appa
 Card
labelPart B
Part B
Image Added
A person pulls a box of mass 15 kg along a smooth floor by applying a force F at an angle of 30° above the horizontal.. The box accelerates horizontally at a rate of 2.0 m/s2. What is the magnitude of F?
Solution
 Toggle Cloak
idsysb
 System: 
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idsysb
 Box as .
 Toggle Cloak
idintb
 Interactions: 
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idintb
 External influences from the person (applied force) the earth (gravity) and the floor (normal force).
 Toggle Cloak
idmodb
 Model: 
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idmodb
.
 Toggle Cloak
idappb
 Approach: 
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idappb
 Toggle Cloak
iddiagb
  Diagrammatic Representation 
 Cloak
iddiagb
Before writing Newton's 2nd Law for the x direction, we choose coordinates and break the applied force F into x- and y-components:
Image Added
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diagb
diagb
 Toggle Cloak
idmathb
  Mathematical Representation 
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idmathb
The free body diagram implies:
 Latex
\begin{large}\[ \sum F_{x} = F\cos\theta = ma_{x}\] \end{large}
{latex}

This time, however, the acceleration requires some thought.  The speed of the box and its direction of motion are constant.  Thus, by definition, the acceleration is zero.  This implies:

{latex}
Solving for F:
 Latex
\begin{large}\[ F = \frac{ma_{x}
 = \mbox{(15 kg)(0 m/s
}
^
{
2
\cos\theta}
)
 = \mbox{
0
34.6 N}
 
\]
 
\end{large}
{latex}

{info}This result is probably not consistent with your everyday experience.  The reason for this is that it is very difficult to find a box and floor combination with zero friction.  Instead, consider the effort that would be required to keep an air-hockey puck moving at constant speed on the air-table (friction is very small) or to keep a soccer ball rolling at constant speed on a smooth, level floor (friction is unimportant since the ball is rolling).{info}




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