A body of mass 2.5 kg is suspended by two strings XP and YP. If the angle between XP and the vertical is 40 degrees and the angle between YP and the vertical is 50 degrees, find the tensions in the st

The mass of the body is $m = 2.5 \text{ kg}$.
The acceleration due to gravity is $g = 10 \text{ ms}^{-2}$.
The weight of the body, $W$, is given by $W = mg$.

$$W = 2.5 \text{ kg} \times 10 \text{ ms}^{-2} = 25 \text{ N}$$

Let $T_1$ be the tension in string $XP$ and $T_2$ be the tension in string $YP$.
The body is in equilibrium under the action of three forces: $T_1$, $T_2$, and its weight $W$.
The angles are given with respect to the downward vertical.

• The angle between $T_1$ and the downward vertical is $40^\circ$.
• The angle between $T_2$ and the downward vertical is $50^\circ$.

For Lami's Theorem, we need the angles between the forces.

• The angle between $T_1$ and $W$ is $40^\circ$.
• The angle between $T_2$ and $W$ is $50^\circ$.
• The angle between $T_1$ and $T_2$ is the sum of their angles from the vertical, which is $40^\circ + 50^\circ = 90^\circ$.

Lami's Theorem states that for three forces in equilibrium, each force is proportional to the sine of the angle between the other two forces.

$$\frac{T_1}{\sin(\text{angle between } T_2 \text{ and } W)} = \frac{T_2}{\sin(\text{angle between } T_1 \text{ and } W)} = \frac{W}{\sin(\text{angle between } T_1 \text{ and } T_2)}$$

Substituting the values:

$$\frac{T_1}{\sin(50^\circ)} = \frac{T_2}{\sin(40^\circ)} = \frac{25 \text{ N}}{\sin(90^\circ)}$$

Since $\sin(90^\circ) = 1$:

$$\frac{T_1}{\sin(50^\circ)} = \frac{T_2}{\sin(40^\circ)} = 25 \text{ N}$$

From the equations:

$$T_1 = 25 \sin(50^\circ)$$ $$T_2 = 25 \sin(40^\circ)$$

Using approximate values for sine: $\sin(50^\circ) \approx 0.7660$ and $\sin(40^\circ) \approx 0.6428$.

$$T_1 = 25 \times 0.7660 = 19.15 \text{ N}$$ $$T_2 = 25 \times 0.6428 = 16.07 \text{ N}$$

Rounding to three significant figures:

$$T_1 \approx 19.2 \text{ N}$$ $$T_2 \approx 16.1 \text{ N}$$

The tensions in the strings are:

$$\boxed{\mathbf{T_1 \approx 19.2 \text{ N}, T_2 \approx 16.1 \text{ N}}}$$