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THE PARALLELOGRAM LAW OF VECTOR ADDITION.
References: Stewart, Physics, Sect. 2-6; Kimball, College Physics, Sect. 18-20, 43; Duff, College Physics, Sect. 20-28; Spinney, Text-Book of Physics, Sect. 8-12.
The parallelogram law may be used in adding any two nonparallel vectors. The student must remember that forces are not the only vector quantities and that the force-table method, here described, is only one way of testing the parallelogram law for forces. One should try to recognize the many examples of this principle seen in daily life.
The apparatus consists of a circular disk, divided into degrees at the edge and having three adjustable pulleys clamped at the edge. Three weight-pans, attached by cords to a ring at the center of the disk, hang from the pulleys. It is best to use fairly large forces, say of more than 500 grams-weight, and to avoid using angles of 120° more than once.
Put weights on the pans until a balance is secured with the ring steady at the center of the disk. Displace it and see if it comes back to the center. Record the setting of each pulley and the corresponding weights, taking into account the weights of the scale-pans, but neglecting those of the cords.
These three forces are vector quantities, and the resultant of any two of them is equal in magnitude and opposite in direction to the third force when there is equilibrium. Using the protractor, lay off carefully on paper one of the angles between the forces, and measure on its sides distances proportional to the
forces at that angle, according to some such scale as 1 cm. for 100 gm. Complete the parallelogram by geometrical construction, measure the diagonal and compare with the third force. There should be quite close agreement. The angles between the extension of the diagonal and the two sides of the parallelogram should also agree with the angles on the disk.
If force-tables are not available, satisfactory arrangements may be made up of three spring balances or two balances and a weight. Spring balances work best in a vertical position.
Repeat for four different combinations of forces and angles, and tabulate data as follows.
Parallelogram of Forces
Trial Setting Setting Setting Force Force Force Calculated Pulley Pulley Pulley
MOMENTS OF FORCE ON A BALANCED BAR.
References: Stewart, Physics, Sect. 9-13; Kimball, College Physics, Sect. 51-59; Duff, College Physics, Sect. 34-39, Spinney, Text-Book of Physics, Sect. 33, 36, 40.
The second condition of equilibrium of a rigid body is, that the sum of all the moments of force acting on the body about
any axis is equal to zero, or, in the special case, where the forces are in one plane, the sum of the counter-clockwise moments must be equal to the sum of the clockwise moments. The present experiment shows how this rule is applied in three simple cases.
(a) Slide a meter bar in a knife-edge support until it balances horizontally and fasten it in that position. Suspend an object of known weight in a scale-pan from a knife-edge support near one end of the bar and suspend, in similar manner, an object of unknown weight near the other end. positions of these weight-supports till the system balances horizontally. Each scale-pan and its knife-edge must be weighed and taken into account. If desired, the objects may be suspended by threads from the meter stick and no scale-pans used. Using the condition of equilibrium stated above, calculate the unknown weight. All forces in this experiment may be measured in grams-weight.
(b) Suspend masses weighing 200 and 500 grams, respectively, near the two ends of the bar, and shift the bar in its support until the moments of force are again in equilibrium. Now, remembering that the total weight of the whole bar may be considered as acting at its center of gravity, calculate the weight of the bar. The same condition for equilibrium that was used in (a) must also be used here.
Why was the weight of the bar neglected in (a)?
(c) Substitute the object of unknown weight for one of the knowns, and shift the bar in its support until equilibrium is secured. Use the same condition for equilibrium as above to calculate the unknown weight. The weight of the bar must be taken into account.
In reporting data, draw a diagram, as indicated above, for each part and enter on the sketch the weights used, their positions, the positions of the point of support and the center of gravity of the bar. Later tabulate data for each part of the experiment under such headings as:
Check results by weighing the meter bar and the unknown weight on a balance. Calculate the per cent of error in each
EXPERIMENT NO. 104
THE CONDITIONS OF EQUILIBRIUM OF A RIGID BODY.
References: Stewart, Physics, Sect. 9-13; Kimball, College Physics, Sect. 51-59; Duff, College Physics, Sect. 34-39; Spinney, Text-Book of Physics, Sect. 33, 35, 36, 40.
The two conditions of equilibrium of a rigid body are very broad and apply to forces and corresponding moments of force acting in any direction on the body, but experimental difficulties here limit us to the special case where the forces are all in one plane. In this limited case, the conditions may be stated as follows:
(1) The sum of all the forces acting in one direction must be equal to the sum of all the forces acting in the opposite direction.
(2) The sum of all the moments of force in a clockwise direction about any arbitrary axis must be equal to the sum of all the moments of force in a counter-clockwise direction about the same axis.
Balance the bar on a knife-edge to find its center of gravity. Weigh the bar and suspend it in a horizontal position by means of two spring balances fastened, one near each end of the bar by loops of thread. The balances must be hung in a vertical position from a cross-piece above the table. Keep the rod horizontal by adjusting the lengths of the balance suspensions. At different points along the bar hang two scale-pans and place in each pan from 500 to 1000 grams. The five forces the two spring balances, the two suspended weights, and the weight of the bar are in equilibrium. Weights of scale-pans and their supports must be taken into account.
Make four trials, shifting the positions of balances and suspended weights and changing the values of the latter each time. For original records, make sketches of the arrangement, record on each sketch the centimeter marks on the bar where
the various forces are applied, placing the weight of the bar at the center of gravity, and indicate the magnitude of each force in grams-weight. In every case, mark an arbitrarily chosen axis, using a different point each time.
The form below is suggested for the final tabulation. Check condition 1 by comparing the sums of the two force columns and condition 2 by similar comparison of the sums of the moments of force.
If desired, three or more weights can be suspended from the bar.
Upward Downward Moment Counter- Clockwise Axis Forces Forces Arms clockwise
EXPERIMENT NO. 105
References: Stewart, Physics, Sect. 10, 12; Kimball, College Physics, Sect. 51, 52, 54; Duff, College Physics, Sect. 34-39; Spinney, Text-Book of Physics, Sect. 33, 35, 36.
In the previous experiments on forces and moments of force in cases of equilibrium, arrangements have been used in which the points of application of all the forces lay on one straight line. This condition is not always present. In the whiffletree or evener of a wagon we have a case of equilibrium between two parallel forces in one direction and a third force acting in the opposite direction. The points of application of these three forces need