fisica1-youn-e-freedman-exercicios-resolvidos

6.100: Use the Work–Energy Theorem, W = ∆KE,
and integrate to find the work.
x
1 2
∆KE
= 0 − mv0 andW
= ∫(
−mg sin α − µ mg cos α)
dx.
2
0
Then,
W
= −mg
x
∫
0
(sin α + Ax
cos α)
dx,
W
⎡ Ax
= −mg⎢sin
αx
+
⎣ 2
2
⎤
cos α ⎥.
⎦
Set W = ∆KE.
2
1 2 ⎡ Ax
⎤
− mv
0
= −mg
sin cos .
2
⎢ αx
+ α
2
⎥
⎣
⎦
To eliminate x, note that the box comes to a rest when the force of static friction balances
the component of the weight directed down the plane. So, mg sin α = Ax mg cos α;
solve
this for x and substitute into the previous equation.
sin α
x = .
A cos α
Then,
1
v
2
2
0
2
⎡
⎛ sin α ⎞ ⎤
⎢
A⎜
⎟ ⎥
⎢ sin α A cos
sin
⎝ α
= + g
cos
⎥
⎢
α +
⎠
α ,
A cos α 2 ⎥
⎢
⎥
⎢⎣
⎥⎦
2
2 3g
sin α
and upon canceling factors and collecting terms, v
0
= . Or the box will remain
A cos α
2
2 3g
sin α
stationary whenever v0
≥ .
A cos α