College Physics I: BIIG problem-solving method

Equilibrium

• Statics is the study of forces in equilibrium.
• Newton’s second law states that

F_net  =   m a

so the net external force is zero for all stationary objects and for all objects moving at constant velocity.

• There are forces acting, but they are balanced. That is, they are in equilibrium.

The First Condition for Equilibrium

• The first condition necessary to achieve equilibrium is:  the net external force on the system must be zero

            F_net    =   0

• Note that if F_net is zero, then the net external force in any direction is zero.
• For example, the net external forces along the typical x- and y-axes are zero.

            F_net-x    =    0      and      F_net-y   =   0

• F_net is zero for both Static equilibrium (motionless) and Dynamic equilibrium (constant velocity)

The Second Condition for Equilibrium

• The second condition necessary to achieve equilibrium is that the net external torque on a system must be zero.

            τ_net   =   0

• It involves avoiding accelerated rotation (maintaining a constant angular velocity.
• A rotating body or system can be in equilibrium if its rate of rotation is constant and remains unchanged by the forces acting on it.

Torque

• The general expression for the torque exerted by force F applied at a distance r from the pivot, at an angle θ to the radial line is

                  τ   =   r F sin θ 

• The SI unit of torque is  N · m
• Is calculated or measured about a particular point.
• A torque that tends to rotate the object in a counterclockwise direction is positive, while a

clockwise direction is a negative torque.

• Problem (E9.1):  The two children are balanced on a seesaw of negligible mass. The first child has a mass of 26.0 kg and sits 1.60 m from the pivot.  If the second child has a mass of 32.0 kg, how far is she from the pivot?                                                                                                                                 ( 1.30 m ) 

Center of Gravity

• Center of gravity is the point where the total weight of the body is assumed to be concentrated.

Base of Support

• Base of support is the area between the points it would pivot on if tilted slightly in either direction.

Stable Equilibrium

• A system is said to be in stable equilibrium if, when displaced from equilibrium, it experiences a net force or torque in a direction opposite to the direction of the displacement.

Unstable Equilibrium

• A system is in unstable equilibrium if, when displaced, it experiences a net force or torque in the same direction as the displacement from equilibrium.

Neutral Equilibrium

• A system is in neutral equilibrium if its equilibrium is independent of displacements from its original position.

Static Equilibrium Situations

• The first step is to determine whether or not the system is in static equilibrium.

             F_net   =   0       and      τ_net   =   0

• It is particularly important to draw a free body diagram for the system of interest.

Carefully label all forces, and note their relative magnitudes, directions, and points of application whenever these are known.

• Problem (E9.2):  Suppose a pole vaulter is holding a pole which is uniform and has a mass of 5.00 kg. The pole’s cg lies halfway between the vaulter’s hands. The hands are 0.900 m apart, and the cg of the pole is 0.600 m from the left hand.  Calculate the forces exerted by each hand.         ( 32.7 N  ;  16.3 N ) 

Simple Machines

• Simple machines are devices that can be used to multiply or augment a force that we apply –

often at the expense of a distance through which we apply the force.

• Levers, gears, pulleys, wedges, and screws are some examples of machines.
• Energy is still conserved for these devices because a machine cannot do more work than the energy put into it.  However, machines can reduce the input force that is needed to perform the job.

Mechanical advantage

• The ratio of output to input force magnitudes for any simple machine is called its mechanical advantage (A_M).

            A_M   =    F_o  /  F_i  

Falcrum

• One of the simplest machines is the lever, which is a rigid bar pivoted at a fixed place called the fulcrum.

Lever

• The mechanical advantage A_M  for levers is given by

             A_M   =    F_o  /  F_i    =    l_i  /  l_o

• Where F_i  and F_o are the input force and output force, l_i and l_o are the distances from where the input and output forces are applied to the pivot.
• Rearranging

             l_i  F_i    =    l_o  F_o

• The longer the handle on the lever, the greater is the force exerted with it.
• Problem (E9.3): In the wheelbarrow, the load has a perpendicular lever arm of 7.50 cm, while the hands have a perpendicular lever arm of 1.02 m.  What upward force must you exert to support the wheelbarrow and its load if their combined mass is 45.0 kg?                                                     ( 32.4 N ) 

Combination of Pulleys

• The combination of pulleys is used to multiply force.
• The force is an integral multiple of tension if the pulleys are frictionless.

Forces and Torques in Muscles and Joints

• Muscles, bones, and joints are some of the most interesting applications of statics.
• There are four forces acting on the forearm and its load: The magnitude of the force of the biceps is F_B ;  that of the elbow joint is F_E ;  that of the weights of the forearm is w_a, and its load is w_b.

BIIG: Problems & Solutions