law of parallelogram of vectors experiment class 11

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Physics practicals class xi, the objective of the experiment :, our objective is to find the weight of a given body using the parallelogram law of vectors..

What does the Parallelogram Law of Vectors state?

If  two vectors acting simultaneously on a particle are represented in magnitude and direction by the two adjacent sides of a parallelogram drawn from a point, then their resultant is completely represented in magnitude and direction by the diagonal of that parallelogram drawn from that point.

Parallelogram Law of Vectors explained

On a Gravesand's apparatus, if the body of unknown weight (say S) is suspended from the middle hanger and balancing weights P and Q are suspended from othe two hangers then,

The unknown weight can be calculated from the equation (1).

On a Gravesand's apparatus, if the body of unknown weight (say S) is suspended from the middle hanger and balancing weights

P and Q are suspended from the other two hangers then,

Now construct a parallelogram OACB by assuming a scale (say 1cm=50 gwt) corresponding to the weights P and Q. The diagonal of the parallelogram OC will give the resultant vector. The weight of the unknown body,

If W is the actual weight of the body, then the percentage error in the experiment can be calculated using the equation,

Learning Outcomes

• Students learn what is parallelogram law of vectors.
• They become familiar with the Gravesands apparatus.
• Students are able to find the unknown weight of an object using the parallelogram law of vectors.

Materials Required :

• Parallelogram Law of Forces apparatus (Gravesand's apparatus)
• Two hangers with slotted weights
• A body (a wooden block) whose weight is to be determined
• Thin strong thread
• White drawing paper sheet
• Drawing pins
• Mirror strip
• Sharp pencil
• Half meter scale
• Set squares

Lab Procedure :

• Set up the Gravesand's apparatus and ensure its board is vertical.  This can be tested using the plumb line. Test if the pulleys (let us name them - P1 and Q1) are frictionless. If you feel any friction, oil them.
• Fix the white drawing paper sheet to the board using the drawing pins.
• Take three pieces of strong threads and tie one end of all three together to make a knot. (Let us name this knotted end - O). This knot becomes the junction of the three threads.
• From the other ends of the two threads, tie a weight hanger with the same slotted weights in each; we will name these weights as P and Q.
• From the end of third thread tie the given body, which is the wooden block, which we will name as S.
• Pass the threads with weights P and Q over the pulleys and let the third thread with the block S, stay vertical in the middle of the board.
• The weights P, Q and the wooden block S acts as the three forces along the three threads. At the junction O, the forces are in equilibrium.
• Now adjust the weights P and Q (forces) such that the junction O stays in equilibrium slightly below the middle of the paper.
• See that all the weights hang freely and that none of them touch the board or the table.
• Mark the position of junction O on the paper using a sharp pencil.
• Slightly disturb the weights P and Q and then leave them.
• Once settled, note the position of junction O.  Make sure that this point is very close to the earlier position.
•  Take the mirror strip and keeping it lengthwise under each thread, mark the position of the ends of the image of the thread in the mirror, covering the image by the thread. These new positions are P1, P2 for the thread with the weight P, and Q1 and Q2 for the thread with the weight Q and S1, S2 for the thread with the weight S.
• Remove the paper from the board and with the help of the half metre scale draw lines through the points P1 and P2 to represent P, through points Q1 and Q2 to represent Q and through points S1 and S2 to represent S. These lines must meet at point O.
• Assuming a scale of 1cm = 50 g, mark OA = 3 cm and OB =3 cm to represent P=150g and Q= 150g.
• Complete parallelogram OACB using the set squares and join OC. This represents the resultant vector R which corresponds to the weight S.
• Measure OC and multiply it by the scale (50 g) to get the value of the unknown weight (S).
• For different sets of observation, change P and Q suitably.
• We can find the weight of the wooden block (R) using the equation (1).
• Take the mean of the two values to get the actual weight of the body.
• To find the percentage error in the experiment, measure the actual weight of the body using a spring balance.
• Calculate the percentage error using equation (3).

Observations 

To find the actual weight of the unknown mass, w.

Least count of spring balance = _________g

Zero error of spring balance = ________g

Weight of unknown body by spring balance = ________g

To find the weight of the unknown mass using parallelogram law of vectors

Scale . Let  50 g=1 cm

Calculation

Mean value of unknown weight S = ---------- gwt.

Mean value of unknown weight, R =---------gwt

Unknown weight = (S+R)/2 = ------------gwt= ---------------kgwt

Percentage error = ---------

The unknown weight of given body = ------------------ kgwt .

The result shows the error is within limits of the experiment error.

Precautions

• 1. The board should be stable and ver
• 2. The pulleys should be friction
• 3. The hangers should not touch the board or
• 4. Junction O should be in the middle of the paper
• 5. Points should be marked only when weights are at r
• 6. Points should be marked with sharp
• 7. Arrows should be marked to show direction of for
• 8. A proper scale should be taken to make fairly big parallelogr

Sources of error :

• 1. Pulleys may have
• 2. Weights may not be accurate
• 3. Points may not be marked correctly.
• 4. Weight measured by spring balance may not be much accurate

Viva-Voce [ Parallelogram Law of Vectors ]

Q.1: What are scalar and vector quantities?

Ans. (a) Scalar is a physical quantity which is completely represented only by magnitude with a suitable unit, having no direction, e.g. time, mass, speed, density, work, energy, etc.

(b) Vector is a physical quantity has both ‘magnitude and a specified direction’ e.g. displacement, velocity, acceleration, force, weight, torque, momentum, magnetic field intensity, electric field intensity.

  Q.2: Define resolution of vectors?

Ans. The splitting up of a single vector into two or more vectors is called resolution of vector.

  Q.3: What do you mean by components of a vector?

Ans. Two or more such vectors which are at right angle to each other are called rectangular components.

  Q.4: What are rectangular component?

Ans. The components of a vector which are at right angle to each other are called rectangular components.

  Q.5: Define the following: (a) unit vector (b) null vector (c) position vector (d) negative vector

Ans. (a) Unit vector is that vector whose magnitude is unity ‘1’, and it simply indicates the direction.

(b) Null vector is that vector whose magnitude is zero and it may have any arbitrary direction or no direction. It is also called zero vector.

(c) Position vector is that vector which specifies the position of a point with respect to origin of reference axes.

(d) The negative of a vector is that vector which is equal in magnitude but opposite in direction of that vector.

  Q.6: State head to tail rule of addition of vectors?

Ans. It states that ‘When the representative lines of all the given are drawn, arrange them in such a way that head of first vector line joins with the tail of second vector, then head of second vector joins with the tail of third vector and so on. The line joining the tail of first vector with head of last vector will represent the resultant vector in magnitude and direction’.

  Q.7: State parallelogram law of vector addition?

Ans. It states that ‘If two vectors are completely represented by two adjacent sides of a parallelogram, then the diagonal of the parallelogram from the tails of two vectors gives their resultant vector’.

Q.8: What is a scalar product?

Ans. When the product of two vectors is a scalar quantity it is called scalar product or dot product, e.g. work is a dot product of force and displacement.

Q.9: What is a vector product?

Ans. When the product of two vectors is a vector quantity it is called vector product or cross product, e.g. torque is a vector product of force and force arm.

Q.10: How a vector is multiply by a number?

Ans. When a vector is multiplied by a positive number its magnitude changes but direction remains the same. But when a vector is multiplied by a negative number not only its magnitude changes but its direction is also reversed.

  Question. 11. Can the law of vectors be used to add forces and velocities ?

Answer. Yes, they can be used, because forces and velocities are also vectors.

Question. 12. Why is addition  of vectors different from addition  of scalars ?

Answer. Because vectors have direction also, which makes all the difference

Question. 13. Why is addition  of vectors called composition of vectors ?

Answer. Composition means collection. By addition we collect (convert) many vectors into one single vector.

Question. 14. What is meant by resolution of vectors ?

Answer. It is reverse of composition of vectors. The breaking of a single vector into its components is called resolution of vectors.

Question. 15. What are rectangular components ?

Answer. The two components of a vector, which are perpendicular to each  other, are called rectangular (or right angular)  components.

  Question. 16. What are the main sources of error in the experiment using Gravesand’s apparatus ?

Answer. Its sources of error are

(1) Friction in the pulleys. (2) Weights in the threads.

Question. 17. Why the thread junction  does not come at rest  at same position always  ?

Answer. It is due to friction on the pulleys.

Question. 18. How can this friction be reduced ?

Answer. It is done by oiling the pulleys.

Question. 19. Why the suspended weights are kept free from board  or table ?

Answer. So that their effective weight may not become different due to reaction of board or table.

Question 20.: What is the unit of force?

Ans. The force is measured in Newton (MKS system) or in dyne (CGS system) or in pound (BE system).

Physics is one of the most important subjects in Class 12. As the CBSE exam approaches, students get busy preparing for different subjects. But an essential part of the  CBSE exam  is the practical exams which consist of 30 marks.

Students must know all the experiments along with theorems, laws, and numerical to understand all the concepts of 12th standard physics in a detailed way. Two experiments (8 + 8 marks) are asked from each section in the practical exam. The experiment records and activities consist of 6 marks, the project has 3 marks and viva on the experiment consist of 5 marks.

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Parallelogram triangle law of vector addition experiment class 11 | gravesand’s apparatus.

To find the weight of a given body (Wooden Block) using parallelogram law of vectors.

• Parallelogram Apparatus (Gravesand’s Apparatus)
• Two Slotted Weights  With Hanger
• Wooden Block With Hook
• Spring Balance
• Mirror Strip
• Cotton Thread (roll)
• Drawing Pin

In Fig. 5.1 we see the Gravesand’s apparatus or Parallelogram apparatus.  It consists of a wooden board A fixed vertically on two pillars. There are two pulleys P and Q fitted at the same level at the top of the board. Three set of slotted weights are supplied with the apparatus which can be used to verify the parallelogram law of vectors.  A thread carrying hangers for addition of slotted weights is made to pass over the pulleys so that two forces P and Q can be applied by adding weights in the hangers. By suspending the given object, whose weight is to be determined, in the middle of the thread, a third force W is applied.

If  two vectors acting simultaneously on a particle are represented in magnitude and direction by the two adjacent sides of a parallelogram drawn from a point, then their resultant is completely represented in magnitude and direction by the diagonal of that parallelogram drawn from that point.

Let two vectors  (P) ⃗  and (Q) ⃗  act simultaneously on a particle O at an angle θ as shown in figure 5.2. They are represented in magnitude and direction by the adjacent sides OA and OB of a parallelogram OACB drawn from a point O. Then the diagonal OC, will represent the resultant R in magnitude and direction. Mathematically we can say, 

(R) ⃗= (P) ⃗ + (Q) ⃗

If a body of unknown weight W is suspended from the middle hanger and balancing weights P and Q are suspended from other two hangers then, (R) ⃗ and the three vectors (P) ⃗,(Q) ⃗ ,(W) ⃗  are in equilibrium. Under equilibrium  |W| ⃗= |R| ⃗. Weight of a body is a force.

Hence, |W| ⃗  = |P| ⃗+|Q| ⃗.

If  S  is the actual weight of the body, then the percentage error in the experiment can be calculated using 

Procedure :

• Set the Gravesand’s apparatus with its board in vertical position with the help of plumb line.
• Ensure that the pulleys are moving smoothly. Oil them if necessary.
• Fix a drawing paper sheet on the board with the help of drawing pins.
• Take a long thread and tie two hangers H1 and H2 to the both ends of the thread.
• Tie the given body B, whose weight is to be found, with one end of another shorter thread.
• Tie the other end of the shorter thread in the middle of the longer thread to form a small knot, O. This knot becomes the junction of the three threads.
• Pass the longer threads over the two pulley P1 and P2 of the apparatus and place two equal slotted weights P and Q the hangers H1 and H2 .
• The body B is made to hang vertically with the knot O somewhere in the drawing paper.
• Adjust the weights P and Q to keep the knot O at a position slightly below the middle of the paper. This is the equilibrium position of O. Ensure that all the weights hang freely and do not touch the board.
• Mark the position of junction O on the white paper by a pencil.
• Ensure that there is no friction at the two pulleys P1 and P2. For that purpose, disturb the positions of P and Q a little and leave them. The knot should go back to its initial equilibrium position, If it does not, oil the pulleys to remove friction.
• Mark the position of the equilibrium of the knot (O) by a dot of a pencil on the paper sheet.
• Take the thin mirror strip m and place it breadthwise on the paper under one of the threads.  See the image of the thread in the mirror. Adjust your line of sight such that it is at right angle to strip and the thread. At this point cannot see the image of the thread (meaning, parallax is removed). Mark two points P1 and P2 on the paper, on either side of the strip exactly behind the thread.
• Similarly, mark two points Q1, Q2 and W1, W2 for the other two threads.
• Remove the threads and hangers from the apparatus and note the weights of the two hanger H1 and H2 along with the slotted weights in table 5.1. Let these two weights be P and Q.
• Remove the paper from the board and join P1 and P2, Q1, Q2 are W1, W2 with the help of ruler to get three lines which should meet O. These three lines represent the directions of three forces.
• To represent the vector, chose a suitable scale, say 10 gm (0.1N) = 1 cm.
• Cut the lines OA and OB to represent the forces  and  respectively. For example, if P = 60 gm and Q = 60 gm, then OA = 6 cm and OB = 6 cm.
• With OA and OB as adjacent sides, complete the parallelogram OACB as shown in fig 5.2.
• Join O and C. The length OC represents the resultant vector  which corresponds to the unknown weight W.
• Measure OC ( l ) and multiply it by the scale (10 g) to get the value of R.
• Measure the angle  () using a protractor and calculate W using equation (i).
• Repeat the experiment with fresh paper and using different weights (un-equal) P and Q in the hangers.
• Find the calculated value of weight R.
• Measure the actual weight (S) of the unknown weight (W) by the spring balance and calculate the percentage error using equation (ii).

Observation:

Least count of spring balance, L.C. = _______________ gm

Weight of B by spring balance, S = ________________gm

Scale factor: Let ______ cm = ___________gm

Table 5.1 Measurement of weight of given body

Calculations:

For equal weights

Weight of unknown body by observation,  W:   ……………………….

Weight of unknown body by calculation,  W ‘ : ……………………….

For un-equal weights

Weight of unknown body by observation,  W: ……………………….

Weight of unknown body by observation,  W ‘ : ……………………….

Precautions :

• The pulleys should be frictionless.
• The boards should be stable and vertical.
• The hangers should not touch the board.
• There should be one central knot and it should be small.
• When the lines of action of the forces are marked, the hangers should at rest.
• The scale should be so chosen that a fairly large parallelogram is obtained.
• Threads should be strong and thin, so that they may be regarded as massless.
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