A 175 kg horizontal beam is supported at each end. A 300 kg piano rests a third of the way from one end.
What is the vertical force on the support closest to the piano?
N
What is the vertical force on the support furthest from the piano?
N
Sum forces in the vertical
Fc= force closest
Ff=force furtherest
Fc+Ff-W-mg=0 where mg is the weight of the beam, W is the weight of the piano
Then sum moments about any point. I will sum about the closest end.
W*L/3+mgL/2-Ff*L=0
solve for Ff, then use the first equation for Fc
To find the vertical forces on the supports, we need to consider the weight of the beam and the weight of the piano. Let's break down the problem step by step:
Step 1: Calculate the weight of the beam
The weight of an object can be calculated using the formula: weight = mass × gravity, where gravity is approximately 9.8 m/s² on Earth.
Given that the mass of the beam is 175 kg, we can calculate its weight as follows:
Weight of the beam = 175 kg × 9.8 m/s² = 1715 N
Step 2: Calculate the weight of the piano
Since the piano is resting a third of the way from one end, it is located closer to that end than the other.
Let's assume the total length of the horizontal beam is L. Since the piano is resting a third of the way from one end, it is located at a distance of L/3 from that end, and 2L/3 from the other end.
Given that the weight of the piano is 300 kg, we can calculate the weight on each support as follows:
Weight on the closer support = (2L/3) / L × 300 kg × 9.8 m/s²
Weight on the farther support = (L/3) / L × 300 kg × 9.8 m/s²
Step 3: Simplify the equations
Weight on the closer support = (2/3) × 300 kg × 9.8 m/s²
Weight on the farther support = (1/3) × 300 kg × 9.8 m/s²
Step 4: Calculate the weights on each support
Using the simplified equations, we can calculate the weights as follows:
Weight on the closer support = (2/3) × 300 kg × 9.8 m/s² = 5880 N
Weight on the farther support = (1/3) × 300 kg × 9.8 m/s² = 2940 N
Therefore, the vertical force on the support closest to the piano is 5880 N, and the vertical force on the support furthest from the piano is 2940 N.