Sensitivity of Common Balance / Beam bar

18-05-2015
Sensitivity of Common Balance / Beam bar weighing scale
Dr. Muhammed Arif M
Associate Prof.
Govt. Arts College Thiruvananthapuram
Kerala, India
arifmmm@gmail.com
Key words: common balance, Beam bar weighing scale, equilibrium angle of common balance,
sensitivity of common balance, common balance sensitivity, static stability of ships
Abstract
The sensitivity of balance is defined as the tan of angle turned by the balance per unit mass. The
sensitivity of balance is proportional to the length of beam and inversely proportional to mass of the
balance and distance from the centre of mass of balance and point of suspension. It is important to
locate the position of centre of mass from the point of suspension to find the sensitivity.
The present problem
Consider a common balance having a beam weighing 38kg resting on a sharp edge as shown. The
ends of beam are attached to two pans weighing 1kg each connected by means of weight less string
as shown. The beam of the balance is perfectly horizontal with respect to gravity. Now a weight of
200 g is added to right pan of the balance find the equilibrium angle of the beam from the
horizontal.
Fig 1.
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Solution.
The balance is in a state of stable equilibrium only when the centre of mass of the system is below
the point of suspension
Fig 2
Let us calculate the position of centre of mass of the system
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Fig 3
From theFig 3
Total mass of the system = 40kg
Taking moments about point of suspension p
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40𝑦 = 38 × 5 − 2 × 100
𝑦 = −0.25𝑐𝑚
Now the system can be simplified as follows
Taking moments about P
40 × 0.25𝑠𝑖𝑛𝜃 = 0.2 × 50𝑐𝑜𝑠𝜃
𝑡𝑎𝑛𝜃 =
0.2 × 50
=1
40 × 0.25
𝜃 =45 o
What are the factors upon which sensitivity of a common balance depends?
With reference to above problem
𝑡𝑎𝑛𝜃 =
𝑚×𝑑
𝑀×𝐷
Sensitivity is the angle turned by the beam per unit mass
𝑡𝑎𝑛𝜃
𝑚
𝑑
= 𝑀×𝐷
The sensitivity of the balance is directly proportional to the arm length ‘d’ and inversely proportional
to the mass of the balance ‘M’ ’ and the distance between the centre of mass and point of
suspension ‘D’.
Frictional force is also a considerable factor when the angle ‘θ’ becomes smaller. So the reduction of
frictional force at the fulcrum is also important in judging the sensitivity.
Problems:
a). two mass are attached at the ends of an “L” shaped weight less frame as shown and suspended
using a flexible string as shown find the angle ‘θ’ from the horizontal
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b)
Static stability of ships
The common balance problem can be extended to stability of ships is eccentric loading. Here we
need to know the distance between centre of mass and centre of buoyancy (D), total mass of ship
(M), the eccentric load (m) and the perpendicular distance (d) from centre of mass and the load.
From this the angle of tilt θ can be calculated using the equation
𝑡𝑎𝑛𝜃 =
𝑚×𝑑
𝑀×𝐷
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Conclusion
The sensitivity of the balance is directly proportional to the arm length and inversely proportional to
the mass of the balance and the distance between the centre of mass and point of suspension..
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