Prof. I. Biaggio
Solutions HW 14
Physics 90, Fall 2006
17
Solutions Homework 14
73
••
In this problem the mass m will accelerate downwards while pulling on the sphere. The pull on the
sphere, through the string, will cause it to rotate faster and faster. The problem can be solved by
relating the rate at which the angular velocity of the speed increases (via its moment of inertia) to
the torque acting on it (which comes from the pull from the string), and realizing that the
acceleration of the hanging object must be the same as the tangential acceleration of a point at the
surface of the sphere where the string is wrapped.
Picture the Problem
The force diagram shows the
forces acting on the sphere and the hanging object. The
tension in the string is responsible for the angular
acceleration of the sphere and the difference between
the weight of the object and the tension is the net force
acting on the hanging object. We can use Newton’s 2
nd
law to obtain two equations in
a
and
T
that we can
solve simultaneously.
(
a
)Apply Newton’s 2
nd
law to the sphere and the
hanging object:
!
=
=
"
#
sphere
0
I
TR
(1)
and
!
=
"
=
ma
T
mg
F
x
(2)
Substitute for
I
sphere
and
α
in equation (1) to
obtain:
(
)
R
a
MR
TR
2
5
2
=
(3)
Eliminate
T
between equations (2) and (3) and
solve for
a
to obtain:
m
M
g
a
5
2
1
+
=
(
b
) Substitute for
a
in equation (2) and solve for
T
to obtain:
M
m
mMg
T
2
5
2
+
=
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Prof. I. Biaggio
Solutions HW 14
Physics 90, Fall 2006
27
74
••
This is an extension of the problem above. Again one can look at the rotational acceleration of the
pulley (caused by the tensions in the stings), and again at the accelerations of the two hanging
objects. All accelerations are related to each other because the objects are connected with a string.
In this problem, note how the tensions in the left and right strings need to be different if the pulley
has mass, but would tend to be equal for when its moment of inertia becomes negligible.
Picture the Problem
The diagram shows the forces
acting on both objects and the pulley. By applying
Newton’s 2
nd
law of motion, we can obtain a system of
three equations in the unknowns
T
1
,
T
2
, and
a
that we
can solve simultaneously.
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 Physics, Force, Mass, Work, Prof. I. Biaggio, Solutions HW

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