# Conditions Required to Achieve Static Equilibrium

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### The Idea

A scaffold is built from a board placed across a base without anything holding it down. How far from the edge of the board can a painter stand without tipping the board? This experiment investigates the condition for stability called static equilibrium.

### What You Need

• section of a 2" × 4" block about 6" long
• meterstick
• 20 g mass

### Method

1. Set the block on the table. This can be either with the 2" edge parallel to the table or the 2" edge perpendicular to the table. Each case gives a different result.
2. Measure the mass of the meterstick.
3. Lay the meterstick over the block, as shown in Figure 33-1, with the 50-centimeter mark of the meterstick centered over the middle of the block.
4. Predict how far the 20-gram mass (the "painter") can be placed from the center without tipping the meterstick, as is the case shown in Figure 33-2.
5. The principle to use is that the torque trying to tip the "scaffold" must not be greater than the torque that holds it in place. Here are the formulas:
6. Tipping torque

< supporting torque d1w1

< d2w2

d1 = distance from edge of block to center of 20g mass

w1 = weight of block

d2 = distance from edge of block to the 50 cm mark of the meterstick

w2 = weight of the meterstick

A 2" × 4" block has actual measurements of 1½" × 3½" (or 3.8 cm × 8.9 cm). (The 3.8 cm side is the height and the 8.9 cm side is the width of the block.) The distance, d2, is one-half the supporting edge. This would be 4.45 cm (with the width of the block along the table) or 1.6 cm (with the height of the block on the table).

7. Try it with other masses.

### Expected Results

For a 90-gram mass meterstick balanced on top of a nominally 2" × 4" block, the following table shows the maximum distance the painter, m2, can go without toppling the meterstick.

### Why It Works

The amount of mass carried at a point of support is the result of a torque generated around the pivot point. In this case, the spring scales form a pivot point. The greater the mass supported, and the further from the pivot point, the greater the torque.

### How weight is distributed

Place two bathroom scales on the floor separated by the length of the board. Set a stiff board about 8 feet long over each scale. Adjust the scales to read zero, to eliminate the effect of the weight of the board. Predict and measure the reading directly over the scales, in the middle, and at arbitrary positions in between.

### Vertical static equilibrium

Assemble the apparatus shown in Figure 33-3. Based on balancing clockwise and counter-clockwise torque, develop other combinations that establish equilibrium. This is based on a demonstration found on the U.C. Berkley Physics Lecture Demonstration website http://www.mip.berkeley.edu/physics/noteindex.html (item: A+60+0).

### The Point

Static equilibrium reflects a balance of forces that results in a collection of objects remaining stable and stationary. The condition for static equilibrium is that the sum of the force and the sum of the torques on an object is zero.