a particle, A of weight W, is suspended by two strings AB and AC. AB is inclined at 30 degrees to the vertical and AC at an angle theta to the vertical.
The tensions in AB and AC are 40N and 60N respectively.
Calculate the values of w and the angle.
W = -(40*Cos30+60*Cos(theta)).
-40*sin30 + 60*sin(theta) = 0.
60*sin(theta) = 40*sln30,
sin(theta) = 2/3 * sin30 = 0.3333,
theta = 19.5o
W = -(40*Cos30+60*Cos19.5) = -91.2 N.
To calculate the values of weight (W) and angle (θ), we need to use the principles of equilibrium.
Let's start by analyzing the forces acting on particle A:
1. Weight (W): It acts vertically downwards and can be represented by a vector pointing downwards.
2. Tension in AB (TAB): It acts along AB and can be represented by a vector pointing upwards at an angle of 30 degrees to the vertical.
3. Tension in AC (TAC): It acts along AC and can be represented by a vector pointing upwards at an angle of θ degrees to the vertical.
Since the particle is in equilibrium, the vector sum of the forces acting on it must be zero.
Vertically: We can write the equation by considering vertical components of forces:
TAB * sin(30°) + TAC * sin(θ) - W = 0
Horizontally: Since there are no horizontal forces acting on the particle, the horizontal components of forces will be zero:
- TAB * cos(30°) - TAC * cos(θ) = 0
Now, we can substitute the given values into these equations:
From the horizontal equation, we have:
TAB * cos(30°) = TAC * cos(θ)
From the vertical equation, we have:
TAB * sin(30°) + TAC * sin(θ) = W
Substituting the known values:
40N * cos(30°) = 60N * cos(θ)
40N * sin(30°) + 60N * sin(θ) = W
Simplifying these equations, we get:
40N * 0.866 = 60N * cos(θ)
40N * 0.5 + 60N * sin(θ) = W
Solving these simultaneous equations will give us the values of W and θ.