To Find the Weight of a Given Body Using Parallelogram Law of Vectors
1. Aim
To determine the weight of a given body using the parallelogram law of vectors.
2. Apparatus Used
- A solid body of unknown weight
- Two spring balances
- A rigid support (stand)
- Thread or string
- Protractor
- Ruler
- Graph paper
- Drawing board
- Drawing pins
- Pencil
3. Diagram
Fig 1. Experimental setup showing the body suspended by two spring balances at an angle
4. Theory
The parallelogram law of vectors states that if two vectors acting simultaneously at a point are represented in magnitude and direction by the two adjacent sides of a parallelogram drawn from a point, then their resultant is represented in magnitude and direction by the diagonal of the parallelogram passing through that point.
Mathematically, if two vectors \(\vec{A}\) and \(\vec{B}\) act at a point O, then their resultant \(\vec{R}\) is given by:
$$\vec{R} = \vec{A} + \vec{B}$$
The magnitude of the resultant can be calculated using the formula:
$$|\vec{R}| = \sqrt{|\vec{A}|^2 + |\vec{B}|^2 + 2|\vec{A}||\vec{B}|\cos\theta}$$
where \(\theta\) is the angle between the two vectors.
In this experiment, we use two spring balances to support a body. The tensions in the threads attached to the spring balances represent two forces acting on the body. By the principle of equilibrium, the resultant of these two forces must be equal and opposite to the weight of the body.
$$\vec{T_1} + \vec{T_2} + \vec{W} = 0$$
$$\vec{W} = -(\vec{T_1} + \vec{T_2})$$
$$|\vec{W}| = |\vec{T_1} + \vec{T_2}|$$
Where \(\vec{T_1}\) and \(\vec{T_2}\) are the tension forces in the threads connected to the spring balances, and \(\vec{W}\) is the weight of the body.
5. Formula
The weight of the body can be calculated using the following formula:
$$W = \sqrt{T_1^2 + T_2^2 + 2T_1T_2\cos\theta}$$
where:
\(W\) = Weight of the body (N)
\(T_1\) = Reading of the first spring balance (N)
\(T_2\) = Reading of the second spring balance (N)
\(\theta\) = Angle between the two threads
6. Procedure
- Set up the apparatus as shown in the diagram. Ensure that the rigid support is firmly fixed.
- Suspend the two spring balances from the support.
- Attach threads to the spring balances and tie them together at a point.
- Hang the body whose weight is to be determined from the point where the threads meet.
- Adjust the positions of the spring balances so that the system comes to equilibrium with a suitable angle between the threads.
- Measure the angle \(\theta\) between the two threads using a protractor.
- Record the readings of both spring balances \(T_1\) and \(T_2\).
- Repeat steps 5-7 for different angles between the threads.
- For each set of readings, calculate the weight of the body using the formula.
- Take the average of the calculated weights to determine the actual weight of the body.
- Alternatively, plot the diagram to scale on a graph paper and determine the weight graphically.
7. Observation Table
| S.No. | First Spring Balance Reading \(T_1\) (N) | Second Spring Balance Reading \(T_2\) (N) | Angle between threads \(\theta\) (degrees) | Calculated Weight \(W\) (N) |
|---|---|---|---|---|
| 1 | ||||
| 2 | ||||
| 3 | ||||
| 4 | ||||
| 5 | ||||
| Mean Value of Weight | ||||
8. Calculations
Sample calculation for the first set of readings:
Given:
\(T_1\) = _______ N
\(T_2\) = _______ N
\(\theta\) = _______ degrees
Using the formula:
$$W = \sqrt{T_1^2 + T_2^2 + 2T_1T_2\cos\theta}$$
Substituting the values:
$$W = \sqrt{(\_\_\_)^2 + (\_\_\_)^2 + 2(\_\_\_)(\_\_\_)\cos(\_\_\_)}$$
$$W = \_\_\_\_ \text{ N}$$
For graphical method:
- Draw a scale diagram of the forces \(T_1\) and \(T_2\) on graph paper.
- Construct a parallelogram with \(T_1\) and \(T_2\) as adjacent sides.
- Draw the diagonal from the origin and measure its length.
- Convert the length to force using the scale factor.
- This gives the magnitude of the resultant force, which is equal to the weight of the body.
9. Result
The weight of the given body as determined by the parallelogram law of vectors is _______ N.
The percentage error in the measurement is:
$$\text{Percentage Error} = \frac{|W_{\text{measured}} - W_{\text{actual}}|}{W_{\text{actual}}} \times 100\%$$
Where \(W_{\text{actual}}\) can be determined by directly measuring the weight using a single spring balance or a standard weight.
10. Precautions
- Ensure that the spring balances are calibrated and checked for zero error before use.
- The body should be allowed to come to complete rest before taking readings.
- The spring balances should be positioned such that they are not too close to each other to ensure a measurable angle.
- Parallax error should be avoided while taking readings from the spring balances and while measuring the angle.
- The threads should be light and inextensible.
- The support should be rigid and not move during the experiment.
- The weight of the threads should be negligible compared to the weight of the body.
- The angle should be measured accurately using a good quality protractor.
11. Sources of Error
- Zero error in the spring balances.
- Friction at the supports and in the spring balances may affect the readings.
- The weight of the threads may not be negligible.
- Parallax error while reading the spring balances and measuring the angle.
- The threads may not be perfectly inextensible.
- Air currents may cause the system to oscillate, making it difficult to take accurate readings.
- Human error in measuring the angle between the threads.
- The spring balances may not be perfectly calibrated.
- The support may not be perfectly rigid.
12. Viva Voice Questions
Q1. What is the parallelogram law of vectors?
A: The parallelogram law of vectors states that if two vectors acting simultaneously at a point are represented in magnitude and direction by the two adjacent sides of a parallelogram drawn from a point, then their resultant is represented in magnitude and direction by the diagonal of the parallelogram passing through that point.
Q2. Why do we use the parallelogram law of vectors to find the weight of a body?
A: The parallelogram law of vectors is used to find the weight of a body because it allows us to determine the resultant of two forces acting at a point. In this experiment, the two forces are the tensions in the threads attached to the spring balances, and their resultant is equal and opposite to the weight of the body.
Q3. How does changing the angle between the threads affect the readings of the spring balances?
A: As the angle between the threads increases, the readings of the spring balances also increase for the same body weight. This is because the vertical component of each tension force must together balance the weight of the body. When the angle increases, the vertical components decrease, so the tensions must increase to maintain equilibrium.
Q4. What would happen if the angle between the threads is 180°?
A: If the angle between the threads is 180°, the system would be in a straight line. In this case, the tensions in both threads would be equal to each other and equal to half the weight of the body (assuming the threads are horizontal). However, practically, this arrangement is unstable and difficult to achieve.
Q5. What is the relationship between the tension in the threads and the weight of the body when the threads are at right angles to each other?
A: When the threads are at right angles to each other (θ = 90°), the cosine term in the formula becomes zero. The weight of the body is given by: $$W = \sqrt{T_1^2 + T_2^2}$$ This is the Pythagorean theorem, where the weight is the hypotenuse of a right-angled triangle with the tensions as the other two sides.
Q6. Why is it important to check the spring balances for zero error?
A: It is important to check the spring balances for zero error because any deviation from zero when no load is applied will lead to incorrect readings. This error will propagate through the calculations and affect the final result of the weight determination.
Q7. How can we minimize experimental errors in this experiment?
A: We can minimize experimental errors by:
- Calibrating the spring balances before use
- Ensuring the system is in static equilibrium before taking readings
- Taking multiple readings at different angles and averaging the results
- Using precision instruments to measure the angle
- Minimizing friction in the system
- Using lightweight and inextensible threads
- Avoiding parallax error while taking readings
Q8. What are the applications of the parallelogram law of vectors in real life?
A: The parallelogram law of vectors has numerous applications in real life, such as:
- Engineering design of structures like bridges and buildings
- Navigation in ships and aircraft
- Determination of resultant forces in mechanical systems
- Analysis of forces in robotics and mechanics
- Study of fluid dynamics and aerodynamics
- Solving problems in electricity and magnetism