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13.2 The Equation of Rotational Motion 405

The components of the forces along the ramp are f for the friction force and
Mg sin for the weight. Hence

Ma Mg sin f

or

f Mg sin Ma

Substituting this into Eq. (13.25), we find

a 2g sin 2a

which we can immediately solve for a:
2
a g sin
3
COMMENT: Note that the force Mg sin along the ramp here produces an accel-
eration that is two-thirds of the acceleration that the cylinder would have if it were
to slip down a frictionless ramp without rolling. This is consistent with the last
3
example, where we saw that a force as large was required to produce a given accel-
2
eration.The same factor occurs in both cases, because both the disk and the cylin-
1
der have the same moment of inertia, MR 2 .
2

PROBLEM-SOLVING TECHNIQUES TORQUES AND ROTATIONAL MOTION

The general techniques for the solution of problems of rota- coordinate axes for the translational motion, preferably
tional motion are similar to the techniques we learned in placing one of the axes along the direction of motion.
Chapters 5 and 6 for translational motion.
4 Select an axis for the rotation of the rigid body, either an
axis through the center of mass, or else a fixed axis (such
1 The first step is always a careful enumeration of all the
as an axle or a pivot mounted on a support) about which
forces. Make a complete list of these forces, and label each
the body is constrained to rotate. Calculate the torque of
with a vector symbol.
each force acting on the body about this center. Remember
2 Identify the body whose motion or whose equilibrium is that the sign of the torque is positive or negative depend-
to be investigated and draw the “free-body” diagram show- ing on whether it produces an angular acceleration in the
ing the forces acting on this body. If there are several dis- positive or the negative direction of rotation.
tinct bodies in the problem (as in Example 5), then you
5 Then apply the equation of rotational motion, I , to
need to draw a separate “free-body” diagram for each.
each rotating body, where is the net torque on a given
When drawing the arrows for the forces acting on a rotat-
body.
ing body, be sure to draw the head or the tail of the arrow
at the actual point of the body where the force acts, since 6 If the rigid body has a translational motion besides the
this will be important for the calculation of the torque. rotational motion, apply Newton’s Second Law, F ma,
Note that the weight acts at the center of mass (we will for the translational motion (see Examples 5 and 6). For
establish this in the next chapter). rolling without slipping, the translational and the rota-
tional motions are related by v R and a R.
3 Select which direction of rotation will be regarded as pos-
itive (for instance, in Example 5, we selected the counter- 7 If there are several distinct bodies in the problem, you need
clockwise direction of rotation as positive). If the problem to apply the equation of rotational motion or Newton’s
involves joint rotational and translational motions, select Second Law separately for each (see Example 5).