Box 1, Folder 9: Notebook

Manuscripts Library State Historical Society

Increase A. Lapham Papers

Notebook containing tables and rules, used by Lapham at Milwaukee in 1840.

Wis Mss DB Vol. 15

Left Page

Engineering Note Book used by Dr. I.A. Lapham at Milwaukee, 1840. H5769

Right Page

Wis Mss DB

Autumn in Parro

[Left Page:] Blank

Right Page: Mechanic Powers - 1 1st. The lever. The power is to the weight as the length of its two ends, respectively, are from the fulcrum, or center of motion.

2nd. Wheel and axle. This depends on the principles of the lever, and like it, the power being always to the weight, as the radius or lever, at which the weight acts, to that at which the power acts: so that they are always in the receiprocol ratio of their velocities.

3rd. The Pulley. If a power sustain a weight by means of one moveable pulley; the power will be equal to only one half of the weight; or the effect will be to the power in the ratio of 2 to 1.

The effect of a combination of any number of fixed and moveable pulleys, is, that every cord going over a moveable pulley, always adds 2. to the powers; since each moveable pulleys rope bears an equal share of the weight; while each rope that is fixed, to a pulley, only increases the power by unity.

4th. Inclined Plane. The power gained by the inclined plane, is in proportion, at the length of the plane, is to its height, ie. when a weight is sustained on an inclined plane, by a power acting parallel to the plane, then the weight is in proportion to the power, at the length of the plane is to its height.

5th. The Wedge. When a wedge is in equilibrious the power acting against the back, is to the force acting perpendicular against each side; as the breadth of the back, is to the length of the sides.

6th. The Screw. The force or power applied to turn a screw around, is to the force with which it presses, upward or downward; at the

[left page] Mechanic Power s- Descent of Bodies distance between the apex, or centers of two threads, is to the circumference where the power is applied

Descent of Bodies A body will descend 16 1/12 feet in one seccond [second] of time, and then has a velocity of 32 1/6 feet per seccond [second]: hence this general Rule.

The Velocities are as the times, and the spa- ces, as the squares of the times, or as the squares of the velocities. Application.

1st. To find the spaces descended by a body in any given time: square the number of seconds, and multiply the product by 16 1/12 or 16.0833: the answer will be the feet descended

2nd. To find the time of descending any given distance: divide the given distance by 16 1/12 or 16.0833: and extract the square root of the quotient, which will be the time required in seconds.

3rd. To ascertain the velocity acquired by a falling body at any period of time during its descent: multiply the time, or number of seconds, by 32 1/6 or 32.1666: which product will be the velocity per sec. generated at the end of that time.

4th. To find the time of generating any given velocity; divide the velocity given by 32 1/6 or 32.1666: the quotient will be the time required in seconds.

[right page] 3

Table of Descent of Bodies

Space time of Velocity quotient Space Time Velocity descended descent in at the end of 2.00 0.3526 11.34 in feet seconds that time per sec .20 0.3698 11.89 .01 0.0249 0.80 .40 0.3863 12.42 .02 0.0352 1.13 .60 0.4020 12.93 .03 0.0432 1.39 .80 0.4172 13.42 .04 0.0499 1.50 3.00 0.4318 13.89 .05 0.0557 1.79 .25 0.4495 14.45 .10 0.0788 2.53 .50 0.4665 15.00 .15 0.0965 3.10 .75 0.4821 15.51 .20 0.1115 3.58 4.00 0.4985 16.02 .25 0.1246 4.00 .50 0.5289 17.01 .30 0.1365 4.38 5.00 0.5574 17.95 .35 0.1475 4.73 6.00 0.6107 19.64 .40 0.1577 5.07 7.00 0.6597 21.22 .45 0.1673 5.38 8.00 0.7052 22.68 .50 0.1763 5.67 9.00 0.7480 24.06 .55 0.1849 5.94 10.00 0.7885 25.34 .60 0.1931 6.11 11.00 0.8270 26.26 .65 0.2010 6.46 12.00 0.8637 27.72 .70 0.2086 6.70 13.00 0.8990 28.92 .75 0.2159 6.94 14.00 0.9330 30.01 .80 0.2230 7.18 15.00 0.9657 31.06 .85 0.2299 7.39 16.00 0.9974 32.08 .90 0.2365 7.60 17.00 1.0281 33.06 .95 0.2430 7.81 18.00 1.0579 34.02 1.00 0.2493 8.01 19.00 1.0869 34.95 .10 0.2615 8.41 20.00 1.1151 35.86 .20 0.2731 8.64 30.00 1.3657 43.92 .30 0.2843 9.14 40.00 1.5770 50.72 .40 0.2950 9.58 50.00 1.7632 56.71 .50 0.3053 9.82 60.00 1.9315 62.12 .60 0.3153 10.14 70.00 2.0862 67.10 .70 0.3251 10.45 80.00 2.2302 71.73 .80 0.3345 10.76 90.00 2.3655 76.09 .90 0.3437 11.04 100.00 2.4934 80.19

4 Centrifugal Force.

A body moving in a circle 16 feet in diameter with a velocity of 16 feet per second, its centrifugal force is just equal to gravity, that is its centrifugal force is equal to its own weight. Also,

If any pendulum oscillates in an arch of a circle, whose cord divides radius into two equal parts, its centrifugal force at the lowest point is just equal to gravity; or the tension of the chord by which it is suspended, at the lower point of oscillation, will be just twice as great, as when the pendulum hangs at rest.

The centrifugal force is directly, as the square of the velocity, and inversely as the diameter of the circle in which the body moves. From the principles, is derived this Rule.

Multiply the weight of the body in pounds, by the square of the number of feet passed over in a second of time, then divided the product, thus obtained by 16 times the diameter of the circle, in feet, the quotient will be the absolute centrifugal force in pounds. For example,

Let a ball weighing 30 pounds, revolve in a circle, the diameter of which is 3 feet in 1 second of time; the number of feet passed over in a sec, or the circumference of the circle is 9.42 feet; the square of which is 88.54. This square being mul -tiplied by 30. The weight of the ball, gives 2656.20 Divide this quantity by 16 times the diameter, 16 x 3 = 48, the quotient will be 55.33 pounds; the centrifugal force.

In the following table look at the back of it for the velocity of the revolving body, under which and opposite to the diameter of the circle in the

[right page] Centrifugal Force.

left hand column will be found the centrifugal force. for a body, weighing one pound, which must be multiplied by the weight in pounds of any required body for its true centrifugal force in pounds.

Table of Centrifugal Force. [NOTE - table is not COMPLETE] diam of Distance in feet passed over per sec. by revolving body circle in feet 4 6 8 10 12 14 16 18 20 25 30 [columns 35 and 40] [also table not checked] 1 1.00 2.25 4.00 6.25 9.00 12.25 16.00 20.25 25.00 39.00 56.25 2 0.50 1.12 2.00 3.12 4.50 6.12 8.00 10.12 12.50 19.53 28.12 3 0.33 0.75 1.33 2.08 3.00 4.08 5.33 6.75 8.33 13.02 18.75 4 0.25 0.56 1.00 1.56 2.25 3.06 4.00 5.06 6.25 9.76 14.06 5 0.20 0.45 0.80 1.25 1.80 2.45 3.20 4.05 5.00 7.81 11.25 6 0.16 0.38 0.66 1.04 1.50 2.04 2.66 3.38 4.16 6.15 9.38 7 0.14 0.32 0.57 0.89 1.28 1.75 2.28 2.89 3.57 5.58 8.04 8 0.12 0.28 0.50 0.78 1.12 1.53 2.00 2.53 3.12 4.88 7.03 9 0.11 0.25 0.44 0.70 1.00 1.38 1.77 2.25 2.77 4.43 6.25 10 0.10 0.22 0.40 0.62 0.90 1.22 1.60 2.02 2.50 3.90 5.62 11 0.09 0.20 0.36 0.57 0.82 1.11 1.45 1.84 2.27 3.55 5.11 12 0.08 0.19 0.33 0.52 0.75 1.02 1.33 1.64 2.08 3.26 4.64 13 0.08 0.17 0.31 0.48 0.69 0.94 1.23 1.56 1.92 3.00 4.23 14 0.07 0.16 0.28 0.44 0.64 0.88 1.14 1.44 1.78 2.79 4.02 15 0.07 0.15 0.27 0.41 0.60 0.82 1.06 1.35 1.66 2.60 3.75 16 0.06 0.14 0.25 0.39 0.56 0.76 1.00 1.26 2.44 3.52 3.52 17 0.06 0.13 0.23 0.37 0.53 0.72 0.94 1.18 1.46 2.30 3.31 18 0.05 0.12 0.22 0.35 0.50 0.69 0.89 1.12 1.39 2.17 3.12 19 0.05 0.11 0.21 0.33 0.47 0.65 0.84 1.06 1.32 2.05 2.96 20 0.05 0.11 0.20 0.31 0.45 0.61 0.80 1.01 1.25 1.05 2.81 22 0.04 0.10 0.18 0.28 0.41 0.55 0.72 0.92 1.14 1.63 2.34 24 0.04 0.10 0.16 0.26 0.38 0.51 0.66 0.84 1.04 1.63 2.34 26 0.04 0.09 0.15 0.24 0.35 0.47 0.61 0.78 0.96 1.50 2.16 28 0.03 0.08 0.14 0.22 0.32 0.44 0.57 0.72 0.89 1.40 2.01 30 0.03 0.07 0.13 0.20 0.30 0.41 0.53 0.67 0.83 1.30 1.88 35 0.03 0.06 0.11 0.17 0.26 0.35 0.46 0.58 0.72 1.13 1.63 40 0.03 0.05 0.10 0.15 0.22 0.30 0.40 0.50 0.62 0.98 1.40