Lab 3:  Vectors and the Force Table

The force table is a clever device that, in effect, allows you to redirect the force of gravity to produce any variety of two-dimensional force vectors.  The sum, or resultant, F_R, of these vectors can be measured directly on the force table.  This is done by hanging a few strings over the circular table via pulleys.  Weights are hung from the strings, providing the forces.  Typically, two or more forces are specified, and then one other force is added so that it balances out the effect of the other forces, bringing the strings into equilibrium.  This force that is added to balance the others is called the equilibrant, F_E.  When the strings are in equilibrium, the total force is equal to zero.  Therefore, the equilibrant equals the negative of the resultant.  For two given forces and their equilibrant,

– F_E  = F_R  = F_A + F_B.

For three given forces and their equilibrant,

– F_E  = F_R  = F_A + F_B + F_C.

These measured resultants can be compared with the sums of the vectors obtained graphically or by components.

            You will set up the given forces by hanging masses over the pulleys at specified directions.  Technically, the magnitude of the force is actually the amount of mass hanging times the acceleration due to gravity, g.  But since g multiplies all the forces (including the equilibrant), we can deal directly with the hanging masses themselves and not worry about the common factor of g.

            Here are some points to apply in setting up and using your force table.  (1) To reduce parallax error, set the pulleys so that the strings are as close to the table as possible without dragging the strings or ring on the table.  (2)  Screw up the center post far enough so that it will keep the ring from sliding off the table when out of equilibrium.  (3) When checking for equilibrium, give the masses a slight jiggle and see if the ring is centered.  This prevents the appearance of equilibrium when one of the pulleys sticks.  (4) When checking for equilibrium, view the ring from directly over the center of the table to reduce parallax error.  (5) When checking for equilibrium, make sure the strings are in line with the center of the table and their pulleys, and not twisted over to one side of center.

Procedure

Sum of Two Vectors

·        Obtain your force table with 4 strings on the white ring, 4 super pulleys, one or two sets of hanging masses, and a few 1-, 2-, and 5-g slotted masses.

·        Position two pulleys with masses to give you these two vectors:

§         F_A = 50.0 g @ 0.0°

§         F_B = 100.0 g @ 120.0°

·        Using a third pulley and string, hang masses off the string and position the pulley until the table is in equilibrium.  Record the mass and position of the third string as F_E.

·        Find F_R from F_E.

Sum of Three Vectors

·        Position three pulleys with masses to give you these three vectors:

§         F_A = 50.0 g @ 0.0°

§         F_B = 100.0 g @ 120.0°

§         F_C = 40.0 g @ 330°

·        Using a fourth pulley and string, hang masses off the string and position the pulley until the table is in equilibrium.  Record the mass and position of the fourth string as F_E.

·        Find F_R from F_E.

·        Return all of your equipment to the place from which you obtained it.

Analysis

·        For both trials, construct the sum of the vectors F_A + F_B and F_A + F_B + F_C graphically.

·        For both trials, calculate the sum of the vectors F_A + F_B and F_A + F_B + F_C using vector components.  Express these sums both in component form and in magnitude/direction form.

·        For both trials, calculate the % error between the measured resultant’s magnitude and the magnitude of the sum calculated from the components.  (The sum calculated from the components is the standard value.)

·        For both trials, compare the direction of the measured resultant with the direction of the calculated sum.