How Much Sand Is Needed to Balance a Plank Horizontally?

[iwonde]

Homework Statement

A box of negligible mass rests at the left end of a 2.00-m, 25.0-kg plank (see image). The width of the box is 75.0 cm, and sand is to be distributed uniformly throughout it. The center of gravity of the nonuniform plank is 50.0 cm from the right end. What mass of sand should be put into the box so that the plank balances horizontally on a fulcrum placed just below its midpoint?

Homework Equations

x_cg = (sum of (the masses x position)) / sum of the masses

The Attempt at a Solution

m_p = mass of plank
m_s = mass of sand
x_cg= (m_p(0)+m_s(0.25))/(25+m_s)
I don't know what to set the equation of x-cg equal to. Please help.

 

[Feldoh]

It's equal to whatever position you called the fulcrum

 

[Moetasim]

I would first clarify the task at hand. It seems that the goal is to find the mass of sand that needs to be added to the box in order for the plank to balance horizontally on a fulcrum placed just below its midpoint.

To solve this problem, we can use the principle of moments, which states that the sum of the moments on one side of a fulcrum must be equal to the sum of the moments on the other side in order for the system to be in equilibrium.

In this case, the moments on the left side of the fulcrum are due to the weight of the plank and the box, while the moments on the right side are due to the weight of the sand. Since the plank and box are already in place, we can focus on the right side and find the mass of sand that would create an equal moment to balance the system.

Using the equation for the center of gravity, we can set the moments on either side of the fulcrum equal to each other:

(m_p + m_s)(0.5) = m_s(0.5 + 0.75)

Simplifying this equation, we get:

m_s = m_p(0.5)/(0.25)

Substituting in the values given in the problem, we get:

m_s = (25 kg)(0.5)/(0.25) = 50 kg

Therefore, 50 kg of sand should be added to the box in order for the plank to balance horizontally on a fulcrum placed just below its midpoint.