A simply supported beam is 6 meters long and has a mass of 15 kg/m of length. It carries loads = 180 N and 270 N at points 1.2 meters and 3.6 meters from the left end respectively. Find the reactions.

I guess it is supported at the two ends?

mass of beam = 6 * 15 = 90 kg at 3 m from left

moments about left
180 * 1.2 + 90 * 3 + 270 * 3.6 - right reaction * 6 = 0

solve for right reaction

then right reaction + left reaction = 180 + 90 + 270
solve for left reaction

right is 243N ?

forgot to convert so it will be 648.45N?

Sorry, I also did not notice the kg/Newton problem

doing it all in Newtons:
WEIGHT of beam = 9.81*90 = 882.9 N
180*1.2+882.9*3+270*3.6 = 6*right
right = I get 639 N

To find the reactions of a simply supported beam, we need to consider the equilibrium of forces and moments acting on the beam.

1. Start by drawing a diagram of the beam and labeling all the given information. The beam is simply supported, 6 meters long, and has loads of 180 N and 270 N acting at distances of 1.2 meters and 3.6 meters from the left end, respectively.

1.2 m 3.6 m
<--------------|--|--|--|--|---------------> (Beam)
Load1 Load2

2. Calculate the total mass of the beam. Since the mass is given as 15 kg/m of length, the total mass of the beam can be calculated by multiplying the length of the beam by the mass per unit length:
Mass of beam = 15 kg/m * 6 m = 90 kg

3. Convert the masses into forces. Since acceleration due to gravity is 9.8 m/s², the weight of the beam is:
Weight of beam = Mass of beam * g = 90 kg * 9.8 m/s² = 882 N

4. Consider the equilibrium of forces in the vertical direction (y-axis). The reactions at the supports will provide the necessary upward forces to balance the downward forces.

The sum of the forces in the y-direction is:
R₁ + R₂ - Load1 - Load2 - Weight of beam = 0

5. Consider the equilibrium of moments around either support. Since the beam is symmetrically loaded, taking moments around any point will give the same result. Let's take moments around support R₁.

The sum of the moments around R₁ is:
(Load1 * distance from R₁ to Load1) + (Load2 * distance from R₁ to Load2) + (Weight of beam * distance from R₁ to center of beam) + (R₂ * distance from R₁ to R₂) = 0

We can assign directions for moments. In this case, we assume counterclockwise (+) as the positive direction and clockwise (-) as the negative direction.

6. Now we can substitute the given values into the equations and solve them simultaneously to calculate the reactions R₁ and R₂.

From the force equilibrium equation:
R₁ + R₂ - 180 N - 270 N - 882 N = 0

From the moment equilibrium equation:
(180 N * 1.2 m) + (270 N * 2.4 m) + (882 N * 3 m) + (R₂ * 4.8 m) = 0

Solve these equations to find the reactions R₁ and R₂.

The solution to this problem will provide the reactions R₁ and R₂, which represent the upward forces at the supports.