The magnitude of tension in the muscle is obtained by solving
[link] :
The force at the elbow is obtained by solving
[link] :
The negative sign in the equation tells us that the actual force at the elbow is antiparallel to the working direction adopted for drawing the free-body diagram. In the final answer, we convert the forces into SI units of force. The answer is
Significance
Two important issues here are worth noting. The first concerns conversion into SI units, which can be done at the very end of the solution as long as we keep consistency in units. The second important issue concerns the hinge joints such as the elbow. In the initial analysis of a problem, hinge joints should always be assumed to exert a force in an
arbitrary direction , and then you must solve for all components of a hinge force independently. In this example, the elbow force happens to be vertical because the problem assumes the tension by the biceps to be vertical as well. Such a simplification, however, is not a general rule.
Solution
Suppose we adopt a reference frame with the direction of the
y -axis along the 50-lb weight and the pivot placed at the elbow. In this frame, all three forces have only
y -components, so we have only one equation for the first equilibrium condition (for forces). We draw the free-body diagram for the forearm as shown in
[link] , indicating the pivot, the acting forces and their lever arms with respect to the pivot, and the angles
and
that the forces
and
(respectively) make with their lever arms. In the definition of torque given by
[link] , the angle
is the direction angle of the vector
counted
counterclockwise from the radial direction of the lever arm that always points away from the pivot. By the same convention, the angle
is measured
counterclockwise from the radial direction of the lever arm to the vector
Done this way, the non-zero torques are most easily computed by directly substituting into
[link] as follows:
Free-body diagram for the forearm for the equivalent solution. The pivot is located at point E (elbow).
The second equilibrium condition,
can be now written as
From the free-body diagram, the first equilibrium condition (for forces) is
We see that these answers are identical to our previous answers, but the second choice for the frame of reference leads to an equivalent solution that is simpler and quicker because it does not require that the forces be resolved into their rectangular components.
Check Your Understanding Repeat
[link] assuming that the forearm is an object of uniform density that weighs 8.896 N.
A uniform ladder is
long and weighs 400.0 N. The ladder rests against a slippery vertical wall, as shown in
[link] . The inclination angle between the ladder and the rough floor is
Find the reaction forces from the floor and from the wall on the ladder and the coefficient of static friction
at the interface of the ladder with the floor that prevents the ladder from slipping.
