How Much Sand Is Needed to Balance a Plank Horizontally?

In summary, a box of negligible mass is resting on a 2.00-m, 25.0-kg plank with a width of 75.0 cm. To balance the plank horizontally on a fulcrum placed just below its midpoint, sand needs to be distributed uniformly throughout the box. The center of gravity of the plank is 50.0 cm from the right end. Using the equation x_cg = (sum of (the masses x position)) / sum of the masses, the mass of sand should be added to the box so that the center of gravity is at the desired position of the fulcrum.
  • #1
iwonde
31
0

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?

centerofgravity.jpg


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.
 
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  • #2
It's equal to whatever position you called the fulcrum
 
  • #3


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.
 

Related to How Much Sand Is Needed to Balance a Plank Horizontally?

1. What is the "Center of Gravity Problem"?

The "Center of Gravity Problem" is a physics problem that involves finding the point on an object where the total weight is evenly distributed in all directions. This point is known as the center of gravity or center of mass.

2. Why is the center of gravity important?

The center of gravity is important because it helps determine the stability and balance of an object. It is also crucial in understanding the motion of objects and how they behave under the influence of external forces.

3. How is the center of gravity calculated?

The center of gravity can be calculated by finding the weighted average of all the individual points of an object's mass. This can be done using the equation:
Center of Gravity = (Sum of (mass x distance))/Total mass

4. What factors affect the center of gravity?

The center of gravity is influenced by the shape, size, and distribution of mass of an object. It can also be affected by external forces such as gravity, friction, and air resistance.

5. How does the center of gravity affect an object's stability?

The lower the center of gravity is, the more stable an object will be. This is because a lower center of gravity means there is less chance of the object tipping over when disturbed by external forces. On the other hand, a higher center of gravity can make an object more prone to tipping or falling.

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