To determine mass of a given body using a metre scale by principle of moments
Aim
To determine the mass of a given body using a metre scale by applying the principle of moments.
Apparatus Required
- A metre scale
- A knife edge (fulcrum)
- A standard mass set
- The body whose mass is to be determined
- Thread
- A stand with clamp
Theory
This experiment is based on the principle of moments, which states that for a body in equilibrium, the sum of clockwise moments equals the sum of anti-clockwise moments about any point.
Mathematically, the principle of moments can be expressed as:
$\sum \text{Clockwise moments} = \sum \text{Anti-clockwise moments}$
In this experiment, we balance a metre scale on a knife edge (fulcrum). When a standard mass ($m_1$) is suspended at a distance $d_1$ from the fulcrum and the unknown mass ($m_2$) at a distance $d_2$ from the fulcrum on the opposite side, the scale will be in equilibrium if:
$m_1 \times d_1 = m_2 \times d_2$
Therefore, $m_2 = \frac{m_1 \times d_1}{d_2}$
The principle of moments is a fundamental concept in physics that helps us understand rotational equilibrium.
The moment of a force about a point is defined as the product of the force and the perpendicular distance from the point to the line of action of the force:
$\text{Moment} = \text{Force} \times \text{Perpendicular distance}$
In our experiment, the forces are the weights of the masses (given by $m_1g$ and $m_2g$), and the perpendicular distances are $d_1$ and $d_2$ respectively.
Taking moments about the fulcrum:
$m_1g \times d_1 = m_2g \times d_2$
The acceleration due to gravity $g$ cancels out on both sides, giving us:
$m_1 \times d_1 = m_2 \times d_2$
Rearranging to find the unknown mass $m_2$:
$m_2 = \frac{m_1 \times d_1}{d_2}$
Experimental Setup
The setup consists of:
- A metre scale balanced on a knife edge (fulcrum)
- A standard mass ($m_1$) suspended at a distance $d_1$ from the fulcrum
- The unknown mass ($m_2$) suspended at a distance $d_2$ from the fulcrum on the opposite side
Procedure
- Find the center of gravity (CG) of the metre scale by balancing it on the knife edge. Mark this point as the origin (0 cm).
- Suspend the unknown mass on one side of the fulcrum at a known distance $d_2$.
- Suspend a suitable standard mass on the other side of the fulcrum and adjust its position until the metre scale is perfectly balanced (horizontal).
- Measure the distance $d_1$ of the standard mass from the fulcrum.
- Repeat the procedure for different positions of the unknown mass ($d_2$) and corresponding positions of the standard mass ($d_1$).
- For each observation, calculate the unknown mass using the formula $m_2 = \frac{m_1 \times d_1}{d_2}$.
- Take the average of all calculated values to determine the mass of the unknown body.
Observations
Position of the center of gravity of the metre scale: ________ cm mark
Value of standard mass ($m_1$): ________ g
| S.No. | Position of unknown mass from fulcrum $d_2$ (cm) | Position of standard mass from fulcrum $d_1$ (cm) | $m_2 = \frac{m_1 \times d_1}{d_2}$ (g) |
|---|---|---|---|
| 1 | |||
| 2 | |||
| 3 | |||
| 4 | |||
| 5 | |||
| Mean value of unknown mass ($m_2$) | |||
Calculations
For each observation, calculate the unknown mass using the formula:
$m_2 = \frac{m_1 \times d_1}{d_2}$
Sample calculation (for the first observation):
Given:
Standard mass ($m_1$) = ________ g
Distance of standard mass from fulcrum ($d_1$) = ________ cm
Distance of unknown mass from fulcrum ($d_2$) = ________ cm
Unknown mass ($m_2$) = $\frac{m_1 \times d_1}{d_2}$ = ________ g
The mean value of the unknown mass is calculated by:
$\text{Mean value of } m_2 = \frac{\sum m_2}{n}$
Where $n$ is the number of observations.
Sources of Error
- The metre scale may not be uniform throughout its length.
- The knife edge may not be sharp enough, resulting in increased friction.
- The centre of gravity of the metre scale may not be accurately determined.
- The masses may not be suspended exactly at the measured points.
- Air currents may affect the balance of the metre scale.
- Parallax error in reading the positions on the metre scale.
Precautions
- The knife edge should be sharp and positioned properly.
- The metre scale should be balanced properly before starting the experiment.
- The threads used to suspend the masses should be thin and of equal length.
- The scale should be allowed to come to rest completely before taking readings.
- Readings should be taken without parallax error.
- The experiment should be performed in a draft-free environment.
Result
The mass of the given body as determined by the principle of moments is ________ g.
Viva Questions
- What is the principle of moments?
- Why do we need to determine the center of gravity of the metre scale first?
- How would the results be affected if the metre scale is not uniform?
- What happens if we place the fulcrum at the 50 cm mark instead of at the center of gravity?
- How can we minimize errors in this experiment?
- What are some real-life applications of the principle of moments?
- What is the principle of moments?
The principle of moments states that for a body in equilibrium, the sum of clockwise moments equals the sum of anti-clockwise moments about any point. Mathematically, $\sum \text{Clockwise moments} = \sum \text{Anti-clockwise moments}$. - Why do we need to determine the center of gravity of the metre scale first?
We need to determine the center of gravity of the metre scale because it accounts for the weight of the scale itself. If we use the center of gravity as our fulcrum, the torque due to the weight of the scale becomes zero, simplifying our calculations. - How would the results be affected if the metre scale is not uniform?
If the metre scale is not uniform, its center of gravity will not coincide with its geometric center. This would introduce systematic errors in our measurements, as the weight distribution of the scale would create additional torques that are not accounted for in our calculations. - What happens if we place the fulcrum at the 50 cm mark instead of at the center of gravity?
If the fulcrum is placed at the 50 cm mark instead of at the center of gravity, the weight of the metre scale will create an additional torque that must be accounted for in the calculations. The equation would become: $m_1 \times d_1 + W_s \times d_s = m_2 \times d_2$, where $W_s$ is the weight of the scale and $d_s$ is the distance from the fulcrum to the center of gravity of the scale. - How can we minimize errors in this experiment?
Errors can be minimized by using a sharp knife edge, ensuring the metre scale is balanced properly, using thin threads of equal length, taking readings without parallax error, performing the experiment in a draft-free environment, and taking multiple readings at different positions to calculate the average. - What are some real-life applications of the principle of moments?
Real-life applications include: see-saws and lever systems, crane operations, wrenches and spanners, balance scales for weighing, door hinges and handles, and the human musculoskeletal system (levers formed by bones, joints, and muscles).