Mozaffari

Armeen Mozaffari

Honors Physics

Mr. Bostian

29 October 2013

6 November 2013

Finding the Equilibrant using Trigonometry

Abstract: For this experiment, we were trying to prove that trigonometry could be used in place

of a force table to find the equilibrant. The trigonometry way includes making two right triangles

with the given angles and the given forces. Then, we use trigonometry to figure out the massesx

and the massesy. Then, you add the massesx together and the massesy together and make a new

triangle with the sums. After that, you figure out the hypotenuse of that triangle using

Pythagorean Theorem and figure out the angle using tangent. Then using that angle, you must

figure out the resultant angle. Then, simply add 180 to that angle and that’s the final angle of the

equilibrant. After the experiment, we concluded that we could use the trigonometry way instead

using a force table every time.

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Introduction: In this experiment, we performed three trials to try figuring out the equilibrant

using trigonometry. A certain physics teacher who accidently drops bricks off of buildings to kill

students gave us two angles with the force applied to each angle. For the first two trials, we did

the trigonometry first, then we checked our calculations by testing it with a force table.

Fortunately, the calculations added up and the ring was balanced in the middle of the force table.

For the 3rd trial, we did not use the force table and only used trigonometry. There are three major

ideas that play part in this experiment: Newton’s 2nd Law, Newton’s 3rd Law, and vectors.

Newton’s 2nd Law helps us figure out the amount of force applied to the angle. The given mass

would be multiplied by 9.8 which gives you the force. Newton’s 3rd law states that for every

action, there is always an equal and opposite reaction. This applies to this lab because to

neutralize the resultant angle, there must be an equal amount of force pulling in the exact

opposite direction. That is why you add 180 to the resultant angle. Lastly, vectors show

magnitude and direction. In this lab, vectors were used multiple times to show in which way the

forces pulled the ring. For example, in trial 2, for one angle, it pulled one way and for the other

angle, it pulled the opposite way. Whichever angle had a stronger force, the ring would be pulled

more towards that direction. Also, the resultant angle pulls one way and its opposite angle pulls

the other way with the same force. It is very similar to a tug-of-war rope. When one teams pulls

one way and the other team pulls the other way with the same force, the rope won’t move; only

the tension of the rope will increase. This is why the ring in the center of the table stays exactly

where it was only raised up slightly. At the end of the experiment, we can conclude that

trigonometry can be used in place of a force table.

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Materials/Equipment: 1 force table

1 calculator

1 electronic balance

mass rings

1 pencil

Procedure: Trial 3

1. Be given two angles and two masses/forces

2. Make an x-y axis and make right triangles with given information

3. Figure out the massx and the massy of both right triangles by using sine/cosine

4. Add the massesy together and add the massesx together.

massy=132.62 + 182.19=314.81 g

massx=-364.37 + 233.19=-131.18 g

38°

295.92 g160°

20°

387.76 g

38°

387.76 g

20°132.62 g

-364.37 g (negative because it’s on left side of y-axis)

Plug in calculator:Sin(20)x387.76=132.62Cos(20)x387.76=364.37

295.92 g182.19 g

233.19 g g

Plug in calculator:Sin(38)x295.92=182.19Cos(38)x295.92=233.19

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5. Make another right triangle with the new massx and the new massy to figure out

the angle and the hypotenuse using tangent and Pythagorean theorem.

6. Figure out the actual angle where the force should be applied

Equilibrant of 2.9 N (295.92 g) @ 38°and 3.8 N (387.76 g) @ 160°:

341.05 g @ 292.62°

Data:Trial 1:

Angle Placed Angle made in right triangle

Mass applied at angle

Massx Massy

35° 35° 28 g 22.94 g 16.06 g

85° 85° 42 g 3.66 g 41.84 g

245.33° 65.33° 63.71 g 26.6 g 57.9 g

Boldface-given

0.31 g314.812 + 131.182=c2

116313.53=c2

341.05=c

Tan-1(314.81/131.18)= °67.38=°

67.38°

112.62°

112.62° is the angle in between the 38° and 160° given the forces, so to neutralize that angle with the force, there must be another force pulling back exactly opposite of 112.62° and with the same amount of force so you add 180

292.62°

0.13 g

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28 g (274.4 N)

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Trial 2: *Negative means mass was on left side of y-axis

Angle Placed Angle made in right triangle

Mass applied at angle

Massx Massy

48° 48° 31 g 20.74 g 23.04 g

152° 28° 45 g -39.73 g* 21.13 g

293.27° 66.73° 48.07 g -18.99 g* 44.16 g

63.71 (624.358 N)

295.92 g (2.9 N)

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Trial 3 (no force table used):

Angle Placed Angle made in right triangle

Force applied at angle

Massx Massy

38° 38° 2.9 N (295.92 g) 233.19 g 182.19 g

160° 20° 3.8 N (387.76 g) -364.37 g* 132.62 g

387.76 g (3.8 N)

341.05 g (471.09 N)

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292.62° 67.38° 3.34 N (341.05g) -131.18 g* 314.81 g

Data Analysis:

Trial 1-

42 g

35°

28 g16.06 g

22.94 g

85°

41.84 g

3.66 g

26.6 g

57.9 g

65.33°

63.71 g

245.33°

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Equilibrant: 63.71 g @ 245.33°

Trial 2-

Equilibrant: 48.07 g @293.27

48°28° 31 g

20.74 g

23.04 g45 g

39.73 g21.13 g

66.73°

18.99 g

44.16 g

293.27°

48.07 g

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Trial 3-

Equilibrant: 341.05 g @ 292.62°

Discussion/Conclusion: In this experiment, we were trying to prove that trigonometry is a more

effective way of figuring out the equilibrant rather than using a force table. The trigonometry

38°20°

295.92 g387.76 g

233.19 g

182.19 g132.62 g

364.37 g

67.38°

131.18 g

341.05 g

314.81 g

292.62 °

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way includes making two right triangles with the angles given and then using sine and cosine to

figure out the massesx and the massesy. Once you figure those out, you add them up and you

make a new right triangle with the new numbers. After that, you use tangent and the Pythagorean

Theorem to figure out the hypotenuse and the angle. The hypotenuse is the actual force pulling at

the angle. Then, you have to figure out the angle in relation to the reference angle and you just

add 180 to that and that gives you the equilibrant. You add 180 because you have to pull in the

exact opposite direction.

References: Force table

diagram-http://titan.bloomfield.edu/facstaff/dnicolai/Physics/Physics105/Experiments/force.htm

Research- http://csep10.phys.utk.edu/astr161/lect/history/newton3laws.html

Trial 1 force table pic Trial 2 force table pic

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