18-05-2015
Sensitivity of Common Balance / Beam bar weighing scale Dr. Muhammed Arif M Associate Prof. Govt. Arts College Thiruvananthapuram Kerala, India
[email protected] 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. 1
18-05-2015 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
2
18-05-2015
Fig 3 From theFig 3 Total mass of the system = 40kg Taking moments about point of suspension p
3
18-05-2015
40π¦ = 38 Γ 5 β 2 Γ 100 π¦ = β0.25ππ Now the system can be simplified as follows
Taking moments about P 40 Γ 0.25π πππ = 0.2 Γ 50πππ π π‘πππ =
0.2 Γ 50 =1 40 Γ 0.25
π =45 o What are the factors upon which sensitivity of a common balance depends? With reference to above problem π‘πππ =
πΓπ πΓπ·
Sensitivity is the angle turned by the beam per unit mass π‘πππ π
π
= πΓπ·
The sensitivity of the balance is directly proportional to the arm length βdβ and inversely proportional to the mass of the balance βMβ β and the distance between the centre of mass and point of suspension βDβ. Frictional force is also a considerable factor when the angle βΞΈβ 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 βLβ shaped weight less frame as shown and suspended using a flexible string as shown find the angle βΞΈβ from the horizontal
4
18-05-2015
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 ΞΈ can be calculated using the equation π‘πππ =
πΓπ πΓπ·
5
18-05-2015 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..
6
