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{
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Composition Setup
Deck of Cards
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Image Added

}{composition-setup} {deck:id=bigdeck} !Pushing a Box^pushingbox.png|width=400! {card:label=Part A} h3. Part A A person pushes a box of mass 15 kg along a floor by applying a perfectly horizontal force _F_. The box accelerates horizontally at a rate of 2.0 m/s{color:black}^2^{color}. {excerpt}Assuming the coefficient of kinetic friction between the box and the ground is 0.45, what is the magnitude of _F_?{excerpt} h4. Solution {toggle-cloak:id=sysa} *System:* {cloak:id=sysa}Box as [point particle].{cloak} {toggle-cloak:id=inta} *Interactions:* {cloak:id=inta} External influences from the person (applied force) the earth (gravity) and the floor (normal force and friction).{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} The free body diagram for this situation is: !pushingboxfrfbd1.png! {cloak:diaga} {toggle-cloak:id=matha} {color:red} *Mathematical Representation* {color} {cloak:id=matha} With this free body diagram, [Newton's 2nd Law|Newton's Second Law] can be written: {latex}
Card
labelPart A

Part A

A person pushes a box of mass 15 kg along a floor by applying a perfectly horizontal force F. The box accelerates horizontally at a rate of 2.0 m/s2.

Excerpt

Assuming the coefficient of kinetic friction between the box and the ground is 0.45, what is the magnitude of F?

Solution

Toggle Cloak
idsysa
System:
Cloak
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 and friction).

Toggle Cloak
idmoda
Model:
Cloak
idmoda

.

Toggle Cloak
idappa
Approach:

Cloak
idappa

Toggle Cloak
iddiaga
Diagrammatic Representation

Cloak
iddiaga

The free body diagram for this situation is:

Image Added

Cloak
diaga
diaga

Toggle Cloak
idmatha
Mathematical Representation

Cloak
idmatha

With this free body diagram, Newton's 2nd Law can be written:

Latex
\begin{large}\[ \sum F_{x} = F - F_{f} = ma_{x} \]
\[ \sum F_{y} = N - mg = ma_{y} = 0 \]\end{large}
{latex}

where

we

have

assumed

that

the

_

y

_

acceleration

is

zero

because

the

box

is

sliding

along

a

horizontal

floor,

not

moving

upward

or

downward.

This

realization

is

important,

because

we

know

_

F

_~f~ = μ_N_. Thus, because the _y_ acceleration is zero, we can solve Newton's 2nd Law in the _y_ direction to yield: {latex}

f = μN. Thus, because the y acceleration is zero, we can solve Newton's 2nd Law in the y direction to yield:

Latex
\begin{large}\[ N = mg\]\end{large}
{latex}

so

that:

{

Latex
}
\begin{large}\[ F = ma_{x}+F_{f} = ma_{x} + \mu_{k}N = ma_{x} + \mu_{k}mg = \mbox{96 N} \] \end{large}
{latex} {cloak:matha} {cloak:appa} {card} {card:label=Part B} h3. Part B A person pushes a box of mass 15 kg along a 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. Assuming the coefficient of kinetic friction between the box and the ground is 0.45, what is the magnitude of _F_? {toggle-cloak:id=sysb} *System, Interactions and Model:* {cloak:id=sysb} As in Part A.{cloak} {toggle-cloak:id=appb} *Approach:* {cloak:id=appb} Just as above, [Newton's 2nd Law|Newton's Second Law] can be written: {latex}\begin{large}\[ \sum F_{x} = F - F_{f}= ma_{x} \] \[ \sum F_{y} = N - mg = 0\] \end{large}{latex} This time, however, the _x_ acceleration is also zero, since the box maintains a constant speed. This implies: {latex}\begin{large}\[ F = ma_{x} + F_{f} = F_{f} = \mu_{k}mg = \mbox{66 N} \] \end{large}{latex} {cloak} {card} {deck}
Cloak
matha
matha

Cloak
appa
appa

Card
labelPart B

Part B

A person pushes a box of mass 15 kg along a 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. Assuming the coefficient of kinetic friction between the box and the ground is 0.45, what is the magnitude of F?

Toggle Cloak
idsysb
System, Interactions and Model:
Cloak
idsysb

As in Part A.

Toggle Cloak
idappb
Approach:

Cloak
idappb