A horizontal board weighing 227.0 N is supported at points A and B. The plank is 3.8 m in length, and points A and B are sawhorse supports a distance of D in from each end of the plank. The plank serves as a platform for a painter, who weighs 553.0 N. Find the maximum distance D, such that the plank will not tip, no matter where the painter stands. (Let L = 3.80 m.)

Question

A horizontal board weighing 227.0 N is supported at points A and B. The plank is 3.8 m in length, and points A and B are sawhorse supports a distance of D in from each end of the plank. The plank serves as a platform for a painter, who weighs 553.0 N. Find the maximum distance D, such that the plank will not tip, no matter where the painter stands. (Let L = 3.80 m.)

Answer 1

To solve this problem, we need to find the maximum distance D such that the plank will not tip, no matter where the painter stands. To do this, we need to find the point on the plank where the torque is maximum and equal to zero.

To calculate the torque, we need to know the weight of the plank and the painter, as well as the distances from the supports to the center of the plank. The torque can be calculated using the formula:

Torque = Force * Distance

In this case, the force is the weight of the plank and the painter, and the distances are the distances from the supports to the center of the plank.

Let's calculate the torque for the plank and the painter separately.

Torque for the plank:
The weight of the plank is 227.0 N, and the distance from each support to the center of the plank is D/2. Therefore, the torque for the plank is:

Torque_plank = 227.0 N * (D/2)

Torque for the painter:
The weight of the painter is 553.0 N, and the distance from each support to the center of the plank is L/2 - D/2. Therefore, the torque for the painter is:

Torque_painter = 553.0 N * (L/2 - D/2)

Now, to find the maximum value of D, we need to find the point where the torque is maximum and equal to zero. This occurs when the torques for both the plank and the painter are equal and in opposite directions:

Torque_plank = -Torque_painter

227.0 N * (D/2) = -553.0 N * (L/2 - D/2)

Now, let's solve this equation for D.

227.0 N * D/2 = -553.0 N * L/2 + 553.0 N * D/2

Multiplying through by 2 to eliminate fractions:

227.0 N * D = -553.0 N * L + 553.0 N * D

Rearranging the equation:

227.0 N * D - 553.0 N * D = -553.0 N * L

Combining like terms:

-326.0 N * D = -553.0 N * L

Dividing both sides of the equation by -326.0 N:

D = (-553.0 N * L) / -326.0 N

Simplifying:

D = (553.0/326.0) * L

D ≈ 0.01 * 3.80 m

D ≈ 0.038 m

Therefore, the maximum distance D such that the plank will not tip, no matter where the painter stands, is approximately 0.038 m.