Problem-Solving Techniques

When solving static problems, you need to identify all forces and torques, confirm directions, solve equations, and check the results.

Learning Objective

Formulate and apply six steps to solve static problems

Key Points

First, ensure that the problem you're solving is in fact a static problem—i.e., that no acceleration (including angular acceleration) is involved.
Choose a pivot point -- use the location at which you have the most unknowns.
Write equations for the sums of torques and forces in the x and y directions.
Solve the equations for your unknowns algebraically, and insert numbers to find final answers.

Terms

moment of inertia
A measure of a body's resistance to a change in its angular rotation velocity
torque
A rotational or twisting effect of a force; (SI unit newton-meter or Nm; imperial unit foot-pound or ft-lb)

Example

For the situation shown in, calculate: (a) F_R, the force exerted by the right hand, and (b) F_L, the force exerted by the left hand. The hands are 0.900 m apart, and the cg of the pole is 0.600 m from the left hand. The strategy includes a free-body diagram for the pole, the system of interest. There is not enough information to use the first condition for equilibrium (F_net = 0), since two of the three forces are unknown, and the hand forces cannot be assumed to be equal in this case. There is enough information to use the second condition for equilibrium (τ_net= 0) if the pivot point is chosen to be at either hand, making the torque at that hand zero. We choose to locate the pivot at the left hand in this part of the problem to eliminate the torque from the left hand.Solution for (a)There are now only two nonzero torques: that from the gravitational force (τ_w) and that from the push or pull of the right hand (τ_R). Stating the second condition in terms of clockwise and counterclockwise torques,τ_cwnet = –τ_ccwnetThat is to say, the algebraic sum of the torques is zero. Here this means:τ_R= –τ_wsince the weight of the pole creates a counterclockwise torque and the right hand counters with a clockwise torque. Using the definition of torque (τ = rFsinθ), noting that θ = 90º, and substituting known values, we obtain:(0.900 m)(F_R) = (0.600 m)(mg)Therefore:F_R= (0.667)(5.00 kg)(9.80m/s²) = 32.7 NSolution for (b)The first condition for equilibrium is based on the free-body diagram in the figure. This implies, by Newton's second law, that:F_L+ F_R– mg = 0From this we can conclude:F_L+ F_R= w = mgSolving for F_L, we obtain:F_L= mg − F_R = mg − 32.7 N = (5.00 kg)(9.80m/s²) − 32.7 N = 16.3 N

Full Text

Statics is the study of forces in equilibrium. Recall that Newton's second law states:

$\sum F=ma$

Therefore, for all objects moving at constant velocity (including a velocity of 0 -- stationary objects), the net external force is zero. There are forces acting, but they are balanced -- that is to say, they are "in equilibrium. "

When solving equilibrium problems, it might help to use the following steps:

First, ensure that the problem you're solving is in fact a static problem—i.e., that no acceleration (including angular acceleration) is involvedRemember:$\sum F=ma=0$ for these situations. If rotational motion is involved, the condition $\sum \tau = I\alpha=0$ must also be satisfied, where is torque, is the moment of inertia, and is the angular acceleration.
Choose a pivot point. Often this is obvious because the problem involves a hinge or a fixed point. If the choice is not obvious, pick the pivot point as the location at which you have the most unknowns . This simplifies things because forces at the pivot point create no torque because of the cross product: $\tau = rF$
Write an equation for the sum of torques, and then write equations for the sums of forces in the x and y directions. Set these sums equal to 0. Be careful with your signs.
Solve for your unknowns.
Insert numbers to find the final answer.
Check if the solution is reasonable by examining the magnitude, direction, and units of the answer. The importance of this last step cannot be overstated, although in unfamiliar applications, it can be more difficult to judge reasonableness. However, these judgments become progressively easier with experience.

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