eloquent logic - equilibrium

Legend

Force

A balance we often use in school applies the principles of equilibrium. The small weight, the poise, is moved downed the arm until a balance is reached with the object being weighed.

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Balance Stand

Problem:
	Calculate where to put the ruler so that the whole system stays in equilibrium.
Material:	masses (200 g, 100 g)
	1 m stick (0.19 kg)
	pivot

Setup:	Apparatus setup as below diagram.

m₁= 0.200 kg
m₂= 0.100 kg
X = ?

Background info & calculations:

F = 0

1) F_x = 0 2) F_y = 0 3) = 0	1) all horizontal forces add up to zero 2) all vertical forces add up to zero 3) all twisting, rotating forces add up to zero (called torque, )
= dF	= Torque d = distance perpendicular from pivot point to point where force is applied F = force applied

Experiment:

Calculation:

Using

= 0 for calculation.

picking the rotation point at the pivot point, we get:
= m₁gx - m₂g(L-x) - m_balanceg(1/2 L-x) = 0
Simplifies to:
m₁x - m₂L + m₂x - 1/2Lm_balance + m_balancex = 0
Rearrange the equation:
m₁x + m₂x + m_balancex = m₂L + 1/2Lm_balance
Factor out the x on the left side and L on the right:
x (m₁ + m₂ + m_balance) = L (m₂ + 1/2m_balance)
Finally, the mathematical expression for x:
x = L (m₂ + 1/2m_balance) / (m₁ + m₂ + m_balance)

We plug in the numbers:

x = 1 [0.1 + 1/2(0.19)] / (0.200 + 0.100 + 0.19)
= 0.398 m

Conclusion:
	Theoretical result was 39.8 cm and experimental result was 40.5 cm. These results are extremely close. The results were different due to friction between balance and fulcrum. The meter stick was most likely not uniform.

Rotational

Formulas