৪৯তম বিসিএস ⎯ ফলিত গণিত [৫৬১]

    W = Fs cos θ
⇒ W = 30 × 17 × cos 60° 
⇒ W = 255 Joules

The principle of virtual work states, for bodies in equilibrium, for a small arbitrary displacement, the total work done by the system is zero.

Females usually have a lower centre of gravity than males. This is because they are usually shorter and have smaller bone frameworks than males. This makes their movement more stable.The centre of gravity of females is typically located somewhere below their belly button.

Angle Between force and displacement= 60-30=30

 W = Fs cos θ
⇒ W = 20 × 17 × cos 30° 
⇒ W = 170√3 Joules

The primary force acting on the conductor is its weight, which causes it to sag. This weight acts uniformly along the length of the conductor.
The tension in the conductor varies along its length, being higher at the higher support.
The sag of the conductor is directly related to the tension. Higher tension results in less sag, and lower tension results in more sag.Since the tension is lower near the lower support, the conductor sags more in that region, causing the lowest point to shift closer to the lower supportand lower at the lower support.

Given Data:
Constant of the catenary, c = 200 m
Span between the two towers, x= 200 m
Weight of the conductor per unit length, w = 1000 kg/km = 1 kg/m
S = c sinh(x/c)
Where: S = Length of the conductor
Calculate the value of x /c = 200 /200  = 1
s=200sinh(1)=200(e-1/e)

Where,S is the sag of the conductor
W is the weight of the conductor per unit
l is the span length of the conductor
T is the working tension on the conductor

A common catenary is best described as the curve formed by a flexible cable or chain suspended between two points and acted upon by gravity. It's not a parabola, though it may appear similar, and is characterized by the hyperbolic cosine function.

Virtual work is the work done by a real force acting through a virtual displacement.

A common catenary is best described as the curve formed by a flexible cable or chain suspended between two points and acted upon by gravity. It's not a parabola, though it may appear similar, and is characterized by the hyperbolic cosine function.

The cable’s weight acts vertically downward.

At the lowest point (the vertex), the tension is purely horizontal, no vertical component yet. So it is the minimum tension in the cable.
As you move upward toward a support, the cable must also carry the vertical weight of the segment below that point. This adds a vertical component to the horizontal tension, making the total tension larger.

At the supports, both the full horizontal tension and the maximum vertical component (weight of half the cable) combine, giving the maximum resultant tension.

h=4, r=4, R=10
1/h=1/4, 1/r -1/R=1/4-1/10
So, 1/h>1/r-1/R
Hence the equilibrium is stable

The rods are weightless.
W∂(AC) + T∂(BD) = 0

h = √(5³ - 3²) = 4
The centre of gravity located at from the base = h/4 = 1cm
Then from top point = 4-1 = 3 cm 

-
-4W∂(GA) - T∂(BD) = 0
The weight located upward related to the line. So The virtual work done by weight is negative.

Virtual work is the work done by a real force acting through a virtual displacement i.e.Virtual work done by Actual forces. 
Virtual work done by real forces.

The distance of centre of gravity of uniform semicircular lamina of radius a, from the centre is 4a/3Π.
So ,The distance of centre of gravity of uniform semicircular lamina of radius a, from the centre is 4*3/3Π= 4/Π.

Circle, rhombus, and rectangle are symmetrical plane lamina. Therefore the center of the gravity of these lamina lies in the geometrical center. But the right angle triangle is unsymmetrical therefore it's center of gravity does not lie on its geometrical center.

The centroid or Centre of Gravity of a triangle is the intersection of the three medians of the triangle (each median connecting a vertex with the midpoint of the opposite side).

In the context of a catenary, sag refers to the vertical distance between the lowest point of the suspended cable or chain and a horizontal line connecting the two points of support.

The Unit Load Method, alos known as Virtual work load method, was developed by Jhon Bernoulli in 1717.

A lower center of gravity generally increases stability, making an object less prone to tipping over, while a higher center of gravity makes it more unstable and likely to topple. 

The CG of a rhombus located at the mid point of its diagonals.
x = (0+3)/2 = 3/2
y = (0+4)/2 = 2 

h = 2, r = 4, R = 4
1/h = 1/2,         1/r +1/R = 0 + 1/4
So, 1/h > 1/r+1/R
​
Hence the equilibrium is stable.

h = 4/2√3
​= 2/√3 cm.

Virtual work load method:
The virtual work load method is also known as unit-load method or virtual force method used to find the deflection and slope at a point of a structure with the help of law of conservation.
When considering forces applied to a body in static equilibrium, the principle of least action requires the virtual work of these forces to be zero.

The centre of gravity of the body coincides with the centre of mass in uniform gravity or gravity-free space. If the body is so extended that gravitational acceleration varies from part to part of the body, then the centre of gravity and centre of mass will not coincide.