Class 11 Physics Notes Chapter 5 (Chapter 5) – Lab Manual (English) Book

Detailed Notes with MCQs of Chapter 5 from your Physics Lab Manual. This chapter deals with a fundamental concept: Vector Addition, specifically verifying the Parallelogram Law of Vector Addition experimentally. Understanding this is crucial, not just for your practical exams, but also as it forms the basis for many concepts tested in government exams.

Here are the detailed notes:

Experiment: To find the weight of a given body using the Parallelogram Law of Vector Addition.

1. Aim:
To determine the unknown weight (mass) of a given object using Gravesand's apparatus by applying the Parallelogram Law of Vector Addition and comparing the experimentally determined weight with the actual weight.

2. Apparatus:

• Gravesand's apparatus (Vector board)
• Two frictionless pulleys
• Plumb lines (2)
• Slotted weights (Weight box)
• Strong thread
• A body of unknown weight (e.g., a wooden or metal block)
• White drawing paper sheet
• Drawing pins or adhesive tape
• Mirror strip
• Sharp pencil
• Protractor
• Metre scale (ruler)

3. Theory:

• Scalar: A physical quantity having only magnitude but no direction (e.g., mass, distance, speed, temperature, work).
• Vector: A physical quantity having both magnitude and direction, and obeying laws of vector addition (e.g., displacement, velocity, acceleration, force, momentum). Vectors are represented by arrows, where the length represents magnitude and the arrowhead indicates direction.
• Parallelogram Law of Vector Addition: If two vectors acting simultaneously at a point can be represented both in magnitude and direction by the two adjacent sides of a parallelogram drawn from that point, then their resultant vector is represented both in magnitude and direction by the diagonal of the parallelogram passing through that same point.
• Resultant Vector (R): A single vector that produces the same effect as two or more vectors acting together. For two vectors P and Q acting at an angle θ, the magnitude of the resultant R is given by:
  R = √(P² + Q² + 2PQcosθ)
  The direction of the resultant (angle β it makes with vector P) is given by:
  tan β = (Qsinθ) / (P + Qcosθ)
• Equilibrant Vector (S): A single vector that, when applied at the same point as the original vectors, produces equilibrium (net force is zero). The equilibrant is equal in magnitude to the resultant but exactly opposite in direction. |S| = |R|.
• Equilibrium: A body is said to be in equilibrium if the net force acting on it is zero. In this experiment, the central knot 'O' where the three threads meet is in equilibrium under the action of three forces: Tension T<0xE1><0xB5><0x83> due to weight P, Tension T<0xE1><0xB5><0x84> due to weight Q, and Tension T<0xE2><0x82><0x9B> due to the unknown weight S. Since the system is at rest and pulleys are frictionless, T<0xE1><0xB5><0x83> = P, T<0xE1><0xB5><0x84> = Q, and T<0xE2><0x82><0x9B> = S. The force S acts as the equilibrant for forces P and Q.
• Lami's Theorem (Can be used for verification): If three concurrent forces acting on a body keep it in equilibrium, then each force is proportional to the sine of the angle between the other two forces.
  P / sin(∠QOS) = Q / sin(∠POS) = S / sin(∠POQ)

4. Diagram:
(Imagine a vertical board - Gravesand's apparatus. Two pulleys are clamped at the top corners. A thread passes over pulley 1, carrying weight P. Another thread passes over pulley 2, carrying weight Q. A third thread hangs vertically downwards from the meeting point (knot 'O') of the first two threads, carrying the unknown weight S. A drawing paper is fixed behind the threads.)

5. Procedure:

• Set up the Gravesand's apparatus, ensuring the board is vertical using plumb lines. Check that the pulleys are frictionless (rotate easily).
• Fix the white drawing paper sheet onto the board using drawing pins or tape.
• Take a piece of thread and make loops at both ends. Take another piece of thread and tie it to the middle of the first thread, forming a knot 'O'. The unknown weight 'S' is attached to the lower end of this second thread.
• Pass the threads with loops over the pulleys and attach known weights P and Q (using hangers and slotted weights) to them.
• Adjust the weights P and Q such that the knot 'O' stays in equilibrium somewhere near the middle of the paper. Ensure the weights hang freely and do not touch the board or table.
• Mark the position of the knot 'O' on the paper.
• To mark the direction of the forces, place the mirror strip under each thread one by one. Mark points (like P₁, P₂ for force P; Q₁, Q₂ for force Q; S₁, S₂ for force S) on the paper such that the thread and its image in the mirror are collinear (coincide, eliminating parallax error).
• Remove the paper from the board. Join the marked points to 'O' to represent the directions of the three forces P, Q, and S acting at point O.
• Choose a suitable scale (e.g., 1 cm = 50 gwt or 0.5 N). Represent the forces P and Q as vectors OA and OB according to the chosen scale, starting from point O along the directions marked.
• Complete the parallelogram OACB using vectors OA and OB as adjacent sides.
• Draw the diagonal OC starting from point O. This diagonal represents the resultant R of forces P and Q in magnitude and direction.
• Measure the length of the diagonal OC and convert it back into force (weight) using the chosen scale. This gives the magnitude of the resultant R.
• Measure the angle (∠POQ = θ) between forces P and Q using a protractor. Also measure the angle the resultant OC makes with OA (∠AOC = β).
• Compare the magnitude of the calculated resultant R (from the parallelogram) with the magnitude of the unknown weight S (which acts as the equilibrant). They should be approximately equal.
• Compare the direction of the resultant R (vector OC) with the direction of the equilibrant S (vector OS', where OS' is the extension of OS backwards). The resultant R should be opposite in direction to S, meaning the angle between OC and OS should be 180°.
• Calculate the unknown weight analytically using the formula R = √(P² + Q² + 2PQcosθ). Compare this analytical value with the graphical value (from parallelogram length) and the actual weight S.
• Calculate the percentage error between the experimental resultant (graphical R) and the actual equilibrant weight S:
  Percentage Error = [|R - S| / S] × 100%
• Repeat the experiment two more times with different values of P and Q.

6. Observations:

• Least count of the metre scale = ... cm

• Scale chosen for vectors: 1 cm = ... gwt (or N)

• Weight P = ... gwt (or N)

• Weight Q = ... gwt (or N)

• Unknown Weight S = ... gwt (or N) (If measured using a spring balance for comparison)

• Observation Table:

7. Calculations:

• Graphical Method: Find R from the length of OC using the scale.
• Analytical Method: Calculate R using R = √(P² + Q² + 2PQcosθ).
• Calculate % Error = [|Graphical R - S| / S] × 100%

8. Result:
The unknown weight S determined experimentally using the parallelogram law (Graphical R) is found to be ... gwt.
The percentage error is ... %.
Within the limits of experimental error, the magnitude of the resultant R is equal to the magnitude of the equilibrant S, and its direction is opposite to S, thus verifying the Parallelogram Law of Vector Addition.

9. Precautions:

• The board should be stable and vertical.
• Pulleys should be frictionless (lubricate if necessary).
• Weights should hang freely and not touch the board or table.
• The knot 'O' should be small and ideally in the center of the paper.
• Use a sharp pencil for marking points.
• Mark directions accurately using the mirror strip to avoid parallax error.
• Choose a suitable and convenient scale for drawing vectors.
• Draw the parallelogram carefully.

10. Sources of Error:

• Friction in the pulleys.
• Weights may not be accurate.
• Error in marking directions due to parallax or thick thread.
• Error in measuring lengths and angles.
• Board may not be perfectly vertical.
• Knot 'O' might be thick.

11. Viva Voce (Potential Questions for Exams):

• Define scalar and vector quantities with examples.
• State the Parallelogram Law of Vector Addition.
• What is a resultant vector? What is an equilibrant vector?
• What is the relation between resultant and equilibrant? (Equal magnitude, opposite direction)
• Why should the board be vertical? (To ensure forces act in a single plane)
• Why should the pulleys be frictionless? (So that tension in the thread equals the weight)
• What is the purpose of the mirror strip? (To avoid parallax error while marking directions)
• What is meant by equilibrium? What are the conditions for equilibrium of concurrent forces? (Net force is zero)
• State Lami's Theorem. Can it be applied here? (Yes, for verification)
• What are the units of force in SI and CGS systems? (Newton (N) and dyne)
• How do you convert gram-weight (gwt) to Newton (N)? (Multiply by g ≈ 9.8 m/s²)

Multiple Choice Questions (MCQs):

1. Which of the following is a vector quantity?
   a) Mass
   b) Speed
   c) Work
   d) Force

2. The Parallelogram Law of Vector Addition is used to find the:
   a) Product of two vectors
   b) Vector sum (resultant) of two vectors
   c) Angle between two vectors
   d) Components of a vector

3. In the Gravesand's apparatus experiment for vector addition, the unknown weight 'S' acts as the:
   a) Resultant of P and Q
   b) Equilibrant of P and Q
   c) Vector sum of P, Q, and R
   d) Component of P

4. If two forces P and Q act at an angle θ, the magnitude of their resultant R is given by:
   a) P + Q
   b) √(P² + Q²)
   c) √(P² + Q² + 2PQcosθ)
   d) √(P² + Q² - 2PQcosθ)

5. The equilibrant of two forces P and Q is equal in magnitude to their resultant R and acts:
   a) In the same direction as R
   b) Perpendicular to R
   c) In the opposite direction to R
   d) At an angle of 45° to R

6. In the vector addition experiment, friction in the pulleys is a source of:
   a) Random error
   b) Systematic error
   c) Zero error
   d) Parallax error

7. To avoid parallax error while marking the direction of forces, one should use a:
   a) Protractor
   b) Set square
   c) Mirror strip
   d) Plumb line

8. If three concurrent forces keep a body in equilibrium, Lami's theorem states that each force is proportional to the:
   a) Cosine of the angle between the other two
   b) Sine of the angle between the other two
   c) Tangent of the angle between the other two
   d) Angle opposite to it

9. Two equal forces F act at a point. If the angle between them is 120°, what is the magnitude of their resultant?
   a) F
   b) 2F
   c) F√2
   d) F/2

10. A student performs the parallelogram law experiment. Force P = 100 gwt, Force Q = 150 gwt, and the angle θ = 60°. The length of the diagonal representing the resultant is measured as 4.3 cm using a scale of 1 cm = 50 gwt. What is the experimental value of the resultant force?
    a) 150 gwt
    b) 215 gwt
    c) 250 gwt
    d) 4.3 gwt

Answers to MCQs:

1. d) Force
2. b) Vector sum (resultant) of two vectors
3. b) Equilibrant of P and Q
4. c) √(P² + Q² + 2PQcosθ)
5. c) In the opposite direction to R
6. b) Systematic error (It consistently opposes motion/tension)
7. c) Mirror strip
8. b) Sine of the angle between the other two
9. a) F (Use R = √(F² + F² + 2F²cos120°) = √(2F² + 2F²(-1/2)) = √(2F² - F²) = √F² = F)
10. b) 215 gwt (R = Length × Scale = 4.3 cm × 50 gwt/cm = 215 gwt)

Study these notes thoroughly. Pay attention to the theory, procedure, precautions, and sources of error as questions are often framed around these aspects in competitive exams. Good luck!