p13_007.pdf

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7.
Three forces act on the sphere: the
tension force
T
of the rope (acting
along the rope), the force of the wall
N
(acting horizontally away from the
wall), and the force of gravity
mg
(acting downward). Since the sphere
is in equilibrium they sum to zero.
Let
θ
be the angle between the rope
and the vertical. Then, the vertical
component of Newton’s second law
is
T
cos
θ
−
mg
= 0. The horizontal
component is
N
−
T
sin
θ
= 0.
↑
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r
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N
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mg
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√
(a) We solve the first equation for the tension:
T
=
mg/
cos
θ.
We substitute cos
θ
=
L/ L
2
+
r
2
to
√
obtain
T
=
mg L
2
+
r
2
/L.
√
(b) We solve the second equation for the normal force:
N
=
T
sin
θ.
Using sin
θ
=
r/ L
2
+
r
2
, we
obtain
√
Tr
mg L
2
+
r
2
mgr
r
√
N
=
√
=
=
.
L
L
L
2
+
r
2
L
2
+
r
2

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