Torque and Statics Equilibrium

Torque and Statics
Equilibrium

Objectives
O Students should understand the concept of torque, so
they can:
O Calculate the magnitude and direction of the torque
associated with a given force.
O Calculate the torque on a rigid object due to gravity.
O Students should be able to analyze problems in statics,
so they can:
O State the conditions for translational and rotational
equilibrium of a rigid object.
O Apply these conditions in analyzing the equilibrium of a
rigid object under the combined influence of a number of
coplanar forces applied at different locations.

8-4: Torque
O To make an object start rotating, a force is needed; the
position and direction of the force matter as well.
O The perpendicular distance from the axis of rotation to
the line along which the force acts is called the lever
arm.
O The torque is defined as:
τ = rF sin θ

O The most efficient direction of force is perpendicular to
a radius from the axis of rotation, and the least efficient
direction is parallel to a radius form the axis of rotation.
O Torque is additive when force is applied in the same
angular direction, and it is subtracted when applied in
the opposite angular direction.
O Counterclockwise torque is, by convention, a positive
quantity, whereas clockwise torque is considered a
negative quantity

Example 8-8: Biceps torque
O The biceps muscle exerts a
vertical force on the lower
arm as shown in figure (a)
and (b). For each case,
calculate the torque about the
axis of rotation through the
elbow joint, assuming the
muscle is attached 5 cm from
the elbow as shown.

Exercise B
O Two forces (F B = 20 N and F A = 30 N) are applied to a
meter stick which can rotate about its left end. Force F B
is applied perpendicularly at the midpoint. Which force
exerts the greater torque?

Example 8-9: Torque on a compound wheel
O Two thin disk-shaped wheels, of radii r A = 30 cm
and r B = 50 cm, are attached to each other on an
axle that passes through the center of each, as
shown. Calculate the net torque on this compound
wheel due to the two forces shown, each of
magnitude 50 N.

9-1: The Conditions for Equilibrium
O An object with forces acting on it, but that is not
moving, is said to be in equilibrium.
O The first condition for equilibrium is that the forces
along each coordinate axis add to zero.
O The second condition of equilibrium is that there be
no torque around any axis; the choice of axis is
arbitrary.

Example 9-1: Straightening teeth
O The wire band shown has a tension F T of 2 N along it. It
therefore exerts forces of 2.0 N on the highlighted tooth
(to which it is attached) in the two directions shown.
Calculate the resultant force on the tooth due to the
wire, F R .

Example 9-2: Chandelier cord tension
O Calculate the tensions F A and F B in the two cords
that are connected to the vertical cord supporting
the 200-kg chandelier.

Exercise A
O In Example 9-2, F A has to be greater than the
chandelier’s weight, mg. Why?

Example 9-3: A lever
O The bar shown is being used as a lever to pry up a large
rock. The small rock acts as a fulcrum (pivot point). The
force F P required at the long end of the bar can be quite
a bit smaller than the rock’s weight mg, since it is the
torques that balance in the rotation about the fulcrum.
If, however, the leverage isn’t sufficient, and the large
rock isn’t budged, what are two ways to increase the
leverage?

Exercise B
O For simplicity, we wrote the equation in Example 9-3 as
if the lever were perpendicular to the forces. Would the
equation be valid even for a lever at an angle as shown
in the figure.

9-2: Solving Statics Problems
1. Choose one object at a time, and make a free-body
diagram showing all the forces on it and where they
act.
2. Choose a coordinate system and resolve forces into
components.
3. Write equilibrium equations for the forces.
4. Choose any axis perpendicular to the plane of the
forces and write the torque equilibrium equation. A
clever choice here can simplify the problem
enormously.
5. Solve.

Example 9-4: Balancing a seesaw
O A board of mass M = 2.0 kg serves as a seesaw for two
children. Child A has a mass of 30 kg and sits 2.5 m
from the pivot point, P (his center of gravity is 2.5 m
from the pivot). At what distance x from the pivot must
child B, of mass 25 kg, place herself to balance the
seesaw? Assume the board is uniform and centered
over the pivot.

Exercise C
O We did not need to use the force equation to solve
Example 9-4 because of our choice of the axis. Use the
force equation to find the force exerted by the pivot.

Example 9-5: Forces on a beam and supports
O A uniform 1500-kg beam, 20.0 m long, supports a
15,000-kg printing press 5.0 m from the right support
column. Calculate the force on each of the vertical
support columns.

Example 9-6: Hinged beam and cable
O A uniform beam, 2.20 m long with mass m = 25.0 kg, is
mounted by a hinge on a wall as shown. The beam is
held in a horizontal position by a cable that makes an
angle θ = 30° as shown. The beam supports a sign of
mass M = 28.0 kg suspended from its end. Determine
the components of the force F H that the hinge exerts on
the beam, and the tension F T in the supporting cable.