Find the unknown weight using the vector addition method.

Experiment to find unknown weight using vector addition of forces with Gravesand’s apparatus, applying equilibrium conditions and rectangular components.

Apparatus

Gravesand’s apparatus, two slotted weights with hangers, a plane mirror strip, thread, drawing pins, plain paper, a protractor, and a body of unknown weight.

---

Concepts to Remember

Vectors

Vectors are physical quantities that have both magnitude and direction, such as force, torque, and momentum.
They can be represented symbolically (A, B, C, etc.) or graphically by arrows drawn to scale.

Resultant Vector

The resultant of several vectors is a single vector that produces the same effect as all the individual vectors combined.

Resolution of a Vector

Breaking a vector into its components is called resolution. It is the reverse of vector addition.
For example, two forces, P and Q, acting in different directions can be resolved into x and y components.

Rx=Px+QxandRy=Py+QyR_x = P_x + Q_x \quad \text{and} \quad R_y = P_y + Q_yRx​=Px​+Qx​andRy​=Py​+Qy​

---

Equilibrium

A body is in equilibrium when the vector sum of all forces acting on it is zero or when it moves with uniform velocity.

Types of Equilibrium

1. Static Equilibrium: When the body is at rest.

2. Dynamic Equilibrium: When the body moves with uniform velocity.

According to the first condition of equilibrium:

∑F=0orW=Psin⁡θ1+Qsin⁡θ2\sum F = 0 \quad \text{or} \quad W = P \sin \theta_1 + Q \sin \theta_2∑F=0orW=Psinθ1​+Qsinθ2​

---

Procedure

1. Check that the pulleys are frictionless and the Gravesand’s apparatus is vertical.

2. Attach two hangers to the ends of a thread and another thread to the unknown weight.

3. Place equal weights on both hangers and tie the third thread between the pulleys, forming a Y-shaped setup.

4. Fix a paper sheet so the knot lies at its center. Ensure weights do not touch the board.

5. Place a mirror strip under each thread and mark points where the thread and its image coincide.

6. Join the points with straight lines that meet at point O. Draw a horizontal line through O.

7. Select a suitable scale and mark forces P and Q on the paper.

8. Draw perpendiculars to find rectangular components:

   Px=Pcos⁡θ1,Py=Psin⁡θ1,Qx=Qcos⁡θ2,Qy=Qsin⁡θ2P_x = P \cos \theta_1, \quad P_y = P \sin \theta_1, \quad Q_x = Q \cos \theta_2, \quad Q_y = Q \sin \theta_2Px​=Pcosθ1​,Py​=Psinθ1​,Qx​=Qcosθ2​,Qy​=Qsinθ2​

   Since the knot is in equilibrium:

   W=R=Psin⁡θ1+Qsin⁡θ2W = R = P \sin \theta_1 + Q \sin \theta_2W=R=Psinθ1​+Qsinθ2​

---

Observation and Calculation

$$\text{Mean Weight, }W=\dots \text{ N}$$

---

Result

$$\text{Unknown Weight, }W=\dots \text{ N}$$

---

Exercise

Find the weight of a body using the graphical method or the head-to-tail rule of vector addition.

---

Precautions

1. Pulleys should be frictionless.

2. The board must be perfectly vertical.

3. Hangers should not touch the board.

4. The knot should be at the center of the paper.

5. Always draw arrowheads on vector components.

---

Viva Voce

Q1. What are rectangular components?
A. Components of a vector that are perpendicular to each other.

Q2. What is a resultant vector?
A. A single vector that produces the same effect as several vectors acting together.

Q3. How can the unknown weight be found?
A.

W=T1sin⁡60∘+T2sin⁡20∘=50+27=77 NW = T_1 \sin 60^\circ + T_2 \sin 20^\circ = 50 + 27 = 77 \, \text{N}W=T1​sin60∘+T2​sin20∘=50+27=77N

Thus, the unknown weight W = 77 N.