marlon f. sacedon dept. of mathematics, physics, statistics (dmps) college of arts and sciences,...

Marlon F. SacedonDept. of Mathematics, Physics, Statistics (DMPS)

College of Arts and Sciences, Visayas State UniversityVisca, Baybay City, Leyte, Philippines

GENERALIZED MATHEMATICAL FORMULA FOR DYNAMICS ON ONE-

DIMENSIONAL MOTION OF TWO-OBJECT SYSTEM

Formerly ViSCA

Baybay City, Leyte

What is the most important part in the solutions of solving problems in dynamics of motion? Free-body diagram or FBD

A vector diagram showing all external forces acting on a body

Baybay City, Leyte

What is the acceleration of the ball if air friction is negligible?

w = mg

ΣFy = ΣFΣF a

mg = ma

FBD

a = g = 9.8 m/s2

from FBD from Second Law

+y

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θ

μk L

What is the acceleration of the ball if air friction is negligible?

ΣFx = ΣF

+mgsinθ -f = ma

a = g(sinθ-μkcosθ)

from FBD from Second Law

mg

+y

+x

+N-f

+mgsinθ

-mgcosθ

+mgsinθ -μkN= ma+mgsinθ –μkmgcosθ = ma

aΣF

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What about this?

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mA=

mB=

+x

+y

+mAg

-T

+y

+x

+mBg

+mBgcosø+T

-mBg.sinθ

a ΣFA

aΣFB

-μmBg.cosθ

What about this?

Shows complicated FBD!

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Student’s response on FBD.

Wrong!Wrong!

Wrong!

Wrong!

Wrong!Correct!

Misconceptions on Free-Body diagram

Problem given to students…..

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This paper presents a generalized mathematical formula that can be used in solving problems of dynamics on one-dimensional motion of two-object system which was derived from the principles of Newton’s Laws of Motion.

The formula helps the difficulty of students in solving the problems because the solutions process escapes FBD.

Baybay City, Leyte

BA

AAB

mm

m

cossincossinBmga

Generalized formula for dynamics on two-object system

θ β

μAmA

mB

μBmBmA

Atwood-machine

mB

mA

β

mA

mB

mB

mA

mB

mA

mB

mA

mBmAmBmA

mB

mA

mB

mA

Baybay City, Leyte

BA

AAB

mm

m

cossincossinBmga

Generalized formula for dynamics on two-object system

θ β

μAmA

mB

μB

Assumptions:• maximum of one pulley and it’s a frictionless.• Air friction is negligible.• motion is due to gravity only.• each object moves on a straight line.• object B accelerates down the plane. If calculated a

is negative, then it moves up the plane.• Neglect the effects of rotation of pulley & masses.• Negligible weight of cord, and no elongations.

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BA

AAB

mm

m

cossincossinBmga

Generalized formula for dynamics on two-object system

θ β

μAmA

mB

μB

Assumptions:• maximum of one pulley and it’s a frictionless.• Air friction is negligible.• motion is due to gravity only.• each object moves on a straight line.• object B accelerates down the plane. If calculated a

is negative, then it moves up the plane.• Neglect the effects on rotation of pulley & masses.• Negligible weight of cord, and no elongations.

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θ β

μA

mA

μB

+y

+x

mBg

N -T

mBg.sinβ

aΣFB

-μBmBg.cosβ

+y

+x

+mAg

-μAmAgcosθ

+T

-mAg.sinθ

aΣFB N

-mBg.cosβ-mAg.cosθ

Derivation of Formula

mB

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+y

+x

mBg

N -T

mBg.sinβ

aΣFB

-μBmBg.cosβ

-mBg.cosβ

+y

+x

+mAg

-μAmAgcosθ

+T

-mAg.sinθ

aΣFA N

-mAg.cosθ

FBD of mA FBD of mB

ΣFx =

+T–mAgsinθ-μAmAgcosθ=mAa (eq.1)

ΣFx =

-T+mBgsinβ-μBmBgcosβ=mBa (eq.2)

Adding equations 1& 2

+mBgsinβ-μBmBgcosβ –mAgsinθ-μAmAgcosθ =mBa +mAa g[mB(sinβ-μBcosβ –mA(sinθ+μAcosθ )]=a(mB +mA)

BA

AAB

mm

m

cossincossinmg

a B

mAa mBaƩFy =0 ƩFy =0

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Common problems on Two-Object system

mBmA

Atwood-machine

mB

mA

mA

mB

BA

AAB

mm

m

cossincossinBmga

mB

mA

mB

mA

OR

mBmAmBmA

OR

Next

mB

mAmB

mA

OR

Assump

Click buttonsto solve

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BA

AAB

mm

m

cossincossinBmga

mBmA

Atwood-machine

θ = 90o β= 90o

μA = 0

Generalized Formula:

μB = 0

0 01 1

BA

A

mm

m

BmgaBack to menu

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mBmA

Atwood-machine

BA

A

mm

m

Bmga If mB >mA

If mB <mA

,then a>0

,then a<0

If mA = 0 ,then a=g

If mB = mA ,then a=0

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mB

mA

β

BA

AAB

mm

m

cossincossinBmga

θ = 90o

μA = 0

Generalized Formula:

μB

01

BA

AB

mm

m

cossinBmga

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BA

AAB

mm

m

cossincossinBmga

θ = 0o μB = 0

Generalized Formula:

0 0

mA

mB

μA

β= 90o

1

BA

AA

mm

m

Bmga

( 1 )

Recall: sin(-A) = -sinA cos(-A) = cosA

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BA

AAB

mm

m

cossincossinBmga

θ = 0o

μB

Generalized Formula:

μA

β= 0o

0 0

BA

AAB

mm

m

Bm-ga

BA

AAB

mm

m

Bmg-a

mBmAmBmA

OR

Recall: sin(-A) = -sinA cos(-A) = cosA

1 1

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mBmA+y

+x

N

-R

-μAmAg

-mAg

FBD of mA

+y

+x

N

+R-μBmBg

-mBg

FBD of mB

ΣFx =

-R-μAmAg=mAa (eq.1)

ΣFx =+R-μBmBg=mBa (eq.2)

Adding equations 1& 2

-μBmBg -μAmAg =mBa +mAa

BA

AA

mm

m

BBmg-

a

mAa mBa

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mB

mA

BA

AAB

mm

m

cossincossinBmga

μB

Generalized Formula:

μA

β

BA

AAB

mm

m

cossincossinBmga

BA

AAB

mm

m

cossincossinBmga

-θ

BA

AAB

mm

m

cos)(sincossinmg

a B(- θ)

(- θ)

Recall: sin(-A) = -sinA cos(-A) = cosA

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mB

mA

mB

mA

OR

BA

AAB

mm

m

cossincossinBmga

Generalized Formula:

μB

μA -θ

BA

AAB

mm

m

cossincossinBmga(- θ)

(- θ)

BA

AAB

mm

m

cossincossinBmga

BA

AAB

mm

m

cossincossinBmga

β

BA

AAB

mm

m

cossincossinmg

a B

Recall: sin(-A) = -sinA cos(-A) = cosA

But : θ=β

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mB

mA

μB

μA

+y

+x

N

-R

mAg.sinβ

aΣFA

-μAmAg.cosβ

-mAg.cosβ

FBD of mA

+y

+x

N

mBg.sinβ

aΣFB

-μBmBg.cosβ

-mBg.cosβ+R

FBD of mB

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+y

+x

N

-R

mAg.sinβ

aΣFA

-μAmAg.cosβ

-mAg.cosβ

FBD of mA

+y

+x

N

mBg.sinβ

aΣFB

-μBmBg.cosβ

-mBg.cosβ+R

FBD of mB

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+y

+x

N

-R

mAg.sinβ

aΣFA

-μAmAg.cosβ

-mAg.cosβ

FBD of mA +y

+x

N

mBg.sinβ

aΣFB

-μBmBg.cosβ

-mBg.cosβR

FBD of mB

ΣFx =

-R+mAgsinβ-μAmAgcosβ=mAa (eq.1)

ΣFx =

+R+mBgsinβ-μBmBgcosβ=mBa (eq.2)

Adding equations 1& 2

+mBgsinβ-μBmBgcosβ +mAgsinβ-μAmAgcosβ =mBa +mAa

BA

AAB

mm

m

cossincossinmg

a B

mAa mBa

BA

AAB

mm

m

cossincossinmg

a B

But : θ=β

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mB

mA

mB

mA

mB

mA

Problems that cannot solve by the formula

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• The mathematical formula can solve many problems in dynamics on one-dimensional motion of two-object system.

• Solutions to the problems becomes shorter and simple.

• Escapes totally the constructions of FBD.

Conclusion

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Thanks

Email: mfsacedon@yahoo.com

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