Content Curator
- Vectors are geometrical entities that possess magnitude and direction.
- A vector can be represented as a line with an arrow pointing in its direction.
- The length of the line represents the magnitude of the vector.
- Therefore, vectors are represented by arrows, which have both starting and terminal points.
- The concept of vectors evolved over 200 years.
- Vectors indicate physical quantities such as displacement, velocity, and acceleration.
- Two vectors can not be added simply by algebraic methods.
- There are different methods to add vectors such as the Triangle law of vector addition and the Parallelogram law of vector addition.
- One of the main objectives of this experiment is to find the weight of the given object or body by using the parallelogram law of vectors.
- This is the basic law that is followed by basic mechanics, and its applications are used for lifting loads of cranes bracket stay wires, and so on.
| Table of Content |
Key Terms: Parallelogram law, Gravesand’s apparatus, Least count, Zero error, Spring balance, Pulleys, Weight, Parallelogram law of vector addition, Vectors, Resultant vector
Aim
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To find the weight of a given body using the parallelogram law of vectors.
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Apparatus Required
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The apparatus required to perform the experiment is
- Gravesand’s apparatus which is a parallelogram law of forces apparatus
- Plumb line
- Two hangers with slotted weights
- Thin strong thread
- A body whose weight is to be determined
- White drawing paper
- Half-meter scale
- Drawing pins
- Mirror strip
- Protractor
- Sharp pencil
- Set squares
Theory
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If an unknown-weight body S is hanging from the center of the hanger, and P and Q are the two balancing weights dangling from the other two ends, the unknown weight is determined as follows:
| \(S= \sqrt {P^{2} + Q^{2 }+ 2PQcos \theta}\) |
Where
- S stands for unknown weight.
- The balance weights are P and Q.
Diagram
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Procedure
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The following are the procedures to perform the experiment
- Place Gravesand's apparatus on a table vertically and firmly.
- Make that the pulleys are in good working order.
- Using thumb pins, secure a white piece of paper between the pulleys on the board.
- Tie the hooks of the hanger weights to the two ends of a thread that is draped over the pulley.
- Suspend the unknown weight from this string by tying another string in the middle of the string traveling over the pulleys.
- Place a slotted weight on the hangers and adjust until the knot (of the two strings) is in the center of the sheet.
- To identify the direction of forces, lay the plane mirror strip under each string one at a time.
- Mark the ends of the mirror strip by putting your eye in a position where the image of the string is covered by the string itself.
- Take the paper away and link each pair of points so that they meet at O.
- Using a handy scale, mark the OA and OB sides (for example, if P = 200 gm wt. and Q = 150 gm wt.).
- OA equals 4 units, while OB equals 3 units.
- Complete the OACB parallelogram.
- Join the diagonal OCS, measure them, and then convert them to comparable gm wt. using the scale you've chosen.
- The weight hung at S will be equal to this. For extra assurance, use a spring balance to weigh the item.
- Change the weight in hangers P and Q two more times to repeat the experiment.
Observation
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Spring balance Least Count = …… g
Spring balance zero error = ……. g
Weight of unknown body measured by spring balance = ……. g
Scale used: Let 1 cm = 50 g
| Sl. no | Forces | Sides | Resultant force R (g wt) | Unknown weight S (g wt) | Weight by spring balance (g wt) | Error (g wt) | |||
|---|---|---|---|---|---|---|---|---|---|
| P (g wt) | Q (g wt) | OA (cm) | OB (cm) | OC (cm) | |||||
| 1. | 145 | 145 | 3.5 | 3.5 | 3.6 | 180 | 200 | 205 | 5 |
| 2. | |||||||||
| 3. | |||||||||
Calculation
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OC = 3.6 cm, R = 50 × 3.6 = 180 g
Unknown weight, S = 200 g
Mean unknown weight, S = (S1 + S2 + S3)/S = 200 g
Weight by spring balance = 205 g
Difference = 5 g
Result
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200 g is the unknown weight of the supplied body.
The error is within the experimental error range.
Precautions
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The following are the precautions that should be taken while performing the experiment
- The board position should be vertical and stable.
- The pulleys should have a low friction coefficient.
- The hangers must not come into contact with the board or table.
- The middle of the paper page should have Junction O.
- Only mark points while the weights are at rest.
- A sharp pencil should be used to mark points.
- To represent the direction of forces, arrows should be used.
- To build a pretty large parallelogram, an appropriate scale should be used.
Source of Error
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The following are the sources of error
- There may be friction between the pulleys.
- It's possible that the weights aren't accurate.
- It's possible that certain points aren't indicated appropriately.
- The accuracy of a spring balance weight measurement is unknown.
Things to Remember
- According to the parallelogram law of vector addition, if two vectors can be represented in magnitude and direction by the adjacent sides of a parallelogram, then the results can be represented in magnitude and direction by the diagonal of the parallelogram.
- For two vectors P and Q at an angle θ with each other, the resultant vector is given by R = √(P2 + Q2 + 2PQ cosθ).
- This law can be calculated by the use of Gravesand's apparatus.
- The concept is that the vector sum of the forces that are experienced by the two masses hanging on the pulley is equivalent to the force of the object hanging in the middle.
- The same force is experienced by the mass in the middle.
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Sample Questions
Ques. Define the term "scalar quantity"? (1 Mark)
Ans. It is defined as a physical quantity with only one magnitude, such as length or mass.
Ques. Define the term “vector quantity”? (1 Mark)
Ans. It is defined as a physical quantity that has both a magnitude and a direction, such as force or velocity.
Ques. Define the term “vector addition”? (1 Mark)
Ans. The method of adding two or more vectors together to form a resultant vector sum is known as vector addition.
Ques. What makes vector addition different from scalar addition? (2 Marks)
Ans. The addition of a vector has both direction and magnitude, whereas adding a scalar simply has magnitude. As a result, adding a vector is not the same as adding a scalar.
Ques. Why is vector addition referred to as vector composition? (2 Marks)
Ans. The term "composition" refers to a collection of items. In addition, we combine (convert) a large number of vectors into a single vector.
Ques. What does the term "vector resolution" mean? (2 Marks)
Ans. It's the inverse of vector composition. The resolution of vectors is the process of breaking down a single vector into its constituents.
Ques. What are the main sources of inaccuracy in Gravesand's apparatus experiment? (2 Marks)
Ans. (1) Friction in the pulleys is one of its sources of inaccuracy.
(2) The threads' weights.
Ques. What can be done to lessen the friction? (1 Mark)
Ans. It is accomplished by lubricating the pulleys.
Ques. Why are the dangling weights not attached to the board or table? (1 Mark)
Ans. So that their effective weight does not change as a result of the board or table's reaction.
Ques. Rectangular components are what they sound like. (2 Marks)
Ans. Rectangular (or right angular) components are the two components of a vector that are perpendicular to each other.
Ques. What is the parallelogram law of two-vector addition? (2 Marks)
Ans. The parallelogram rule states that if two vectors are placed in a parallelogram with the same initial point and then completed, the total of the vectors is the directed diagonal that starts at the same position as the vectors.
Ques. State triangle law of addition of two vectors. (2 Marks)
Ans. According to triangle law of vector addition, when two vectors are represented as the magnitude of two sides of the triangle taken in the same order, then the third side of the triangle represents the magnitude of resultant vector taken in opposite order.
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