Parallelogram Law of Vectors

This law can be explained as, “If two forces acting simultaneously on a particle are represented in magnitude and direction by the two adjacent sides of the parallelogram, the diagonal of that parallelogram will be expressed as the resultant of these two forces represented in direction and magnitude.”

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Aim

The main objective of this experiment is to find the weight of the given object (body) applying the law of forces.

This is the basic law followed by basic mechanics. Its applications are used as lifting loads of cranes and bracket stay wires, etc.

Apparatus

To conduct this experiment, we need some essential apparatus such as;

• Gravesand's apparatus which is an ideal apparatus for parallelogram law of forces

• An object with an unknown weight (used for identifying its weight)

• Plumb line

• Slotted weights are hung with two hangers

• The thread which is thin as well as durable

• White colour drawing sheet

• Pins to hook up drawing sheet

• Pointed pencil (2HB)

• Mirror strip

• Set squares

• Half-meter scale

• Protractor

To Find the Weight of a Given Body Using Parallelogram

It can be calculated by the use of Gravesand's apparatus. The concept is that the vector sum of the forces experienced by the two masses hanging on the pulley is equal 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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If an anonymous weight body (S) is suspended from the centre of the hanger, and P and Q are the two symmetric weights from the other end of the hanger, then that unknown weight can be calculated by using the equation below;

\[S=\sqrt{P^2+Q^2+2PQcos\theta }\]

Where,

\[\vec{P},\vec{Q}\] are two identical forces

The unknown weight can be termed S

P and Q are the balance weights used in the experiment

θ is the angle between two forces

Procedure

To Find the Weight of a Given Body Using Parallelogram Law of Vector. We need to follow certain steps to do so:

Gravesand's apparatus is set up with a board vertically with the help of a plumb line.

P1 and P2 pulleys should be oiled properly to make them frictionless

Fix the white sheet on the board with the help of drawing pins.

“O” is the knot shaped 

P and Q are the weights that are tied up at both the ends of the hanger and S be the third body tied at the third end.

Junction O should be sustained at equilibrium by maintaining weights P and Q.

P, Q, and S these three weights act as three forces \[\vec{P},\vec{Q}\]and\[\vec{S}\]

These weights should be hung freely without making any contact with the board.

Mark the position of the junction of O with the help of a dark pencil.

Disturb the weights at P and Q and leave them free.

The position of junction O will be closed as compared to the earlier position.

Let the position of P be P1 and P2, Q1 and Q2 will be the position of Q and S1 and S2 will be the position of S. All these positions are being written down with the use of a mirror.

By taking a scale, 1cm =50gm

OA = 3cm and

OB= 3cm

These parameters are taken to represent 

P = 150 gm and Q = 150 gm

Where R is represented by finishing the parallelogram OACB and by drawing OC line with the use of set squares

When measuring OC, the result shows 3.9cm.

P and Q can be altered for different sets.

By utilizing spring balance, calculate the weight of the wooden box.

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To Find the Weight of a Given Body Using Parallelogram Law of Vectors Observation Table

Least count of spring balance = …… g

Zero error of spring balance = …….. G

Weight of unknown body by spring balance = …….g

Scale used: Let 1 cm = 50 g

Calculations

After all the measurements

OC = 3.9cm, R = 50 * 3.9 = 195 g

Unknown weight is calculated as, S =195 g

Mean unknown weight will be S = \[\frac{(S_1+S_2+S_3)}{3}\] = 195 g

Weight measured from spring balance = 200 g

So the error is calculated through the difference between weight measured and mean unknown weight such as;

200g-195g = 5g

The error in this experiment is under its limits as per experimental error.

Results

The unknown weight of the given body = 195g

The error is within the limits of experimental error.

Learn about Precautions

• The board used for the experiment should be placed vertically and stable.

• Try to make these pulleys friction-free.

• The table and board should not make any contact with the hangers.

• The junction O should lie in the middle of the paper

• The points should be marked when weights are stationary.

• A sharp pencil (2HB) should be useful to mark all the points.

• Arrows should be indicated to show the direction of forces.

• A proper scale should be used for making a fairly big parallelogram.

To Evaluate Sources of Error

• Friction in the pulleys might cause an error.

• The accuracy of weights might vary.

• The marked point may be correct.

• The accuracy of weight obtained from spring balance may not be accurate.

Learning Outcomes

• Students learn exactly what the parallelogram law of vectors is.

• Gravesand's apparatus is familiarized with them.

• Using the parallelogram law of vectors, students can find the unknown weight of an object.