You hang a light in front of your house using an elaborate system to keep the 12-kg object in static equilibrium (Figure 1). What are the magnitudes of the forces that the ropes must exert on the knot connecting the three ropes if θ2 = 63∘ and θ3 = 45∘? The image depicts a mechanical system involving three ropes and a mass labeled "m." Rope 1 is vertically oriented and connects to the mass, suspending it in equilibrium. Ropes 2 and 3 are attached to the ceiling and converge at a common point along with Rope 1.Key elements in the diagram include:1. **Rope 1**: This rope is vertical, holding the mass "m" in place through tension.2. **Rope 2** and **Rope 3**: These ropes form angles with the horizontal, indicated as \( \theta_2 \) and \( \theta_3 \), respectively. They are shown pulling the mass at the top where they join with Rope 1.3. **Angles**: The angle \( \theta_2 \) is between Rope 2 and the horizontal axis, while \( \theta_3 \) is between Rope 3 and the horizontal axis.The system demonstrates the principles of static equilibrium, where the sum of forces in both horizontal and vertical directions are zero, allowing the mass to remain stationary. The tensions in the ropes are balanced by the gravitational force acting on the mass.

Name: You hang a light in front of your house using an elaborate system to k
Uploaded: 2024-03-20T06:56:46.000Z
Channel: Bartleby
Description: You hang a light in front of your house using an elaborate system to keep the 12-kg object in static equilibrium (Figure 1). What are the magnitudes of the forces that the ropes must exert on the knot connecting the three ropes if θ2 = 63∘ and θ3 = 4

Lami's Theorem : When three forces acting at a point are in equilibrium, then each force is proportional to the sine of the angle between the other two forces. i.e. Psin(a)=Qsin(b) = Rsin(c)The mass of the object is 12 k-g so its weight = m*g =12*9.8 P = 117.6N this force is in the down word direction From figure ∠c = 45° +90° = 135°∠b = 63° + 90° =153°∠a = 360° -(153°+135°)∠a = 72° now apply the lami's theorem 117.6sin(72) = Qsin(135) = Rsin(153)equating it we get Q = sin(135)*117.6sin(72)Q = 87.43N and R = sin(153)*117.6sin(72)R = 56.14N Hence the magnitudes of the forces that the ropes must exert on the knot connecting the three ropes 87.43N, 56.14N & 117.6N.