Experiment 4.3: Verification of the Principle of Moments

Principle of Moments

• The principle of moments states that for a body to be balanced, the sum of clockwise moments must equal the sum of anti-clockwise moments.
• When a rod is balanced, weights attached to it can cause rotation. This rotation is also known as torque.
• A weight on one side causes a clockwise moment (rotation), while weights on the opposite side cause an anti-clockwise moment.

Definition

• The principle of moments definition: The sum of clockwise moments and anti-clockwise moments acting on a body must equal zero for rotational equilibrium.
• Rotational equilibrium means the body does not rotate.
• Mathematically, this can be expressed as:
  $\sum \text{Clockwise Moments} + \sum \text{Anti-Clockwise Moments} = 0$

Sign Conventions

• Clockwise moments are considered negative, while anti-clockwise moments are considered positive.
• If the clockwise moment is $-10$ and the anti-clockwise moment is $+10$, their sum is zero, indicating a balanced state.

Balancing Condition

• For a balanced state, the clockwise moments equal the anti-clockwise moments.
• Therefore, the body does not rotate in either direction.

Experiment Setup

• A meter rule is balanced on a weight.
• The initial reading on the meter rule, where it balances, is recorded as 51.5. This point represents the center of gravity.
• Record the center of gravity in a table.

Preparing Weights

• Three weights are attached to the meter rule using threads.
• The threads should allow the weights to move freely.
• Tie the thread with a loosely knot to allow movement.

Determining Mass

• The mass of each weight can be found either by looking at the value written on it (e.g., 10 grams) or by using an electronic balance.
• The total mass is the sum of individual masses (e.g., $10 + 40 + 50 = 100$ grams).
• Using an electronic balance provides a more accurate mass measurement.
• Example mass reading from electronic balance: 29.05 grams.

Weight Calculation

• To find weight (in Newtons) from mass (in grams), divide the mass value by 100.
• If mass is in kg, the formula to find weight ($W$) is:
  $W = mg$
  where $m$ is mass and $g$ is the acceleration due to gravity ($\approx 10 m/s^2$).
• However, since mass is in grams, the formula becomes:
  $\text{Weight} = \frac{\text{Mass in grams}}{100}$
• For example, if the mass is 29.05 grams:
  $W = \frac{29.05}{100} = 0.2905 \text{ Newtons}$

Experimental Procedure

1. Record the weight $W_1$ in the table.
2. Place $W<em>1$ on one side of the meter rule and the other two weights ($W</em>2$ and $W_3$) on the opposite side.
3. Adjust the weights to balance the meter rule.
4. Record the masses of the other weights: $W<em>2 = 39.2$ grams, $W</em>3 = 30$ grams.
5. Convert these masses to weights in Newtons by dividing by 100: $W<em>2 = 0.39 N$, $W</em>3 = 0.30 N$.
6. Fill these values in the table.

Balancing the Meter Rule

• Adjust the positions of the weights to achieve balance.
• The meter rule should be flat, and the graduate side should be facing up.
• Ensure there is enough space for adjustments and that weights are properly aligned.
• Achieving balance requires fine adjustments; move the threads slightly to find the equilibrium.
• When the meter rule is close to balance, it may sway slightly. The goal is to find the point where it remains stable.

Recording Positions

• Record the positions of $W<em>1, W</em>2$, and $W_3$ on the meter rule.
• Example positions: $W<em>1$ at 89.8, $W</em>2$ at 28.6, and $W_3$ at 8.5.
• The position of center of gravity is 51.5.

Moment Arm Calculation

• The moment arm is the distance between the point where the meter rule is balanced (center of gravity) and the position of each weight.
• To find the moment arm for $W<em>1$, subtract the center of gravity (51.5) from the position of $W</em>1$ (89.8):
  $89.8 - 51.5 = 38.3 \text{ cm}$
• The moment arm is 38.3 cm.
• For $W_2$, subtract its position (28.6) from the center of gravity (51.5):
  $51.5 - 28.6 = 22.9 \text{ cm}$
• For $W_3$, subtract its position (8.5) from the center of gravity (51.5):
  $51.5 - 8.5 = 43 \text{ cm}$
• These moment arms are essential for calculating torques.

Torque Calculation

• Torque (or moment) is the product of the weight and its moment arm.
• $W<em>1$ tends to rotate the meter rule clockwise, while $W</em>2$ and $W_3$ tend to rotate it anti-clockwise.
• Calculate the torque for each weight.

Clockwise Torque

• For $W_1$: Weight = 0.49 N, Moment Arm = 38.3 cm.
• Torque $T_1$ = Weight × Moment Arm = $0.49 × 38.3 = 18.767 \text{ N·cm}$.

Anti-Clockwise Torque

• For $W_2$: Weight = 0.39 N, Moment Arm = 22.9 cm.
• For $W_3$: Weight = 0.30 N, Moment Arm = 43 cm.
• Torque $T<em>2$ (for $W</em>2$) = $0.39 × 22.9 = 8.931 \text{ N·cm}$.
• Torque $T<em>3$ (for $W</em>3$) = $0.30 × 43 = 12.9 \text{ N·cm}$.
• Total anti-clockwise torque = $T<em>2 + T</em>3 = 8.931 + 12.9 = 21.831 \text{ N·cm}$.

Verification

• Total clockwise moment = 18.767 N·cm.
• Total anti-clockwise moment = 21.831 N·cm.
• Difference = $|18.767 - 21.831| = 3.064 \text{ N·cm}$.
• Ideally, the difference should be zero. A small difference is expected due to experimental errors.

Conclusion

• The experiment verifies the principle of moments. For the meter rule to be balanced, the clockwise and anti-clockwise moments should be equal.
• Repeat the experiment with different weight positions and record the values. Recalculate moment arms, torques and find the difference between the clockwise and anticlockwise moments.
• Also, perform the experiment by keeping the values the same from the start. However, when balancing, values will be different.
• Ideally, the difference should be zero. A small difference is expected due to experimental errors.