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Rotational Dynamics
Translational and
Rotational Motion
Which one of the above is the easiest to open a door?
Definition of Torque
Torque is a vector quantity.
Direction: The torque is positive when the force tends to
produce a counterclockwise rotation about the axis, and
negative when the force tends to produce a clockwise
SI Unit of Torque: newton · meter (N · m)
Example 1
Different Lever Arms,
Different Torques
In Figure 9.3 a force whose magnitude is 55 N is applied to a
door. However, the lever arms are different in the three parts of
the drawing. Find the magnitude of the torque in each case.
The Achilles Tendon
9.2 Rigid Objects in
Equilibrium Of A Rigid
A rigid body is in equilibrium if it has zero translational
acceleration and zero angular acceleration. In equilibrium, the
sum of the externally applied forces is zero, and the sum of
the externally applied torques is zero:
Applying the Conditions of Equilibrium to a Rigid Body
1.Select the object to which the equations for equilibrium are to be
2.Draw a free-body diagram that shows all the external forces acting on
the object.
3.Choose a convenient set of x, y axes and resolve all forces into
components that lie along these axes.
4.Apply the equations that specify the balance of forces at equilibrium:
SFx = 0 and SFy = 0.
5.Select a convenient axis of rotation. Identify the point where each
external force acts on the object, and calculate the torque produced by
each force about the axis of rotation. Set the sum of the torques about
this axis equal to zero: St = 0.
6.Solve the equations for the desired unknown quantities.
Example 4 Fighting a Fire
In Figure 9.7a an 8.00-m ladder of weight WL = 355 N leans
against a smooth vertical wall. The term “smooth” means that
the wall can exert only a normal force directed perpendicular
to the wall and cannot exert a frictional force parallel to it. A
firefighter, whose weight is WF = 875 N, stands 6.30 m from
the bottom of the ladder. Assume that the ladder’s weight acts
at the ladder’s center and neglect the hose’s weight. Find the
forces that the wall and the ground exert on the ladder.
9.3. Center of Gravity
The center of gravity of a rigid body is the point at which its
weight can be considered to act when the torque due to the
weight is being calculated.
Uniform Thin Rod
The center of gravity of the rod is at its
geometrical center.
Center of Gravity of a
Group of Objects
Example 6 The Center
of Gravity of an Arm
The horizontal arm in Figure 9.11 is composed of three parts: the
upper arm (weight W1 = 17 N), the lower arm (W2 = 11 N), and
the hand (W3 = 4.2 N). The drawing shows the center of gravity
of each part, measured with respect to the shoulder joint. Find the
center of gravity of the entire arm, relative to the shoulder joint.
Overloading a Cargo
(b) Correctly loaded (c) Incorrectly loaded