Laboratory Exercise 4: Vector Analysis and Static Equilibrium

Laboratory Objectives and Apparatus

• Core Laboratory Objectives:

  • Master the analytical addition of vectors through the resolution of their orthogonal components.
  • Experimentally verify theoretical vector calculations using a physical force table apparatus.
  • Evaluate the conditions necessary to establish static translational equilibrium among concurrent coplanar forces.

• Required Equipment and Supplies:

  • Force Table: A circular horizontal surface calibrated with angular degree markings from $0^{\circ}$ to $360^{\circ}$.
  • Low-Friction Pulleys: Three specialized clamp-on pulleys designed to minimize resistive frictional forces along the perimeter of the force table.
  • Mass Hangers: Three calibrated hangers, each possessing an inherent mass of exactly $5\,\text{g}$ ($0.005\,\text{kg}$).
  • Mass Set: A standardized assortment of slotted weights used to apply gravitational tension to the strings.
  • Central Ring and String Assembly: A small central ring (or knot) attached to strings radiating across the pulleys to represent concurrent force vectors.

Theoretical Principles of Vector Resolution

• Planar Vector Representation:

  • A vector $\mathbf{A}$ situated in a two-dimensional Cartesian plane can be uniquely resolved into the sum of two mutually perpendicular components directed along the coordinate axes.
  • By projecting perpendicular dashed lines from the terminus of vector $\mathbf{A}$ onto the horizontal $x$-axis and vertical $y$-axis, the orthogonal components $A_x$ and $A_y$ are isolated.

• Component Formulations:

  • For a vector $\mathbf{A}$ with magnitude $A$ oriented at an angle $\theta$ measured counterclockwise from the positive horizontal axis, the scalar components are given by:
  • Horizontal component:       $A_x = A \cos(\theta)$
  • Vertical component:       $A_y = A \sin(\theta)$

Analytical Vector Addition by Components

• Resultant Vector Definition:

  • The geometric and algebraic sum of two independent coplanar vectors $\mathbf{A}$ and $\mathbf{B}$ produces a single resultant vector $\mathbf{C}$:     $\mathbf{C} = \mathbf{A} + \mathbf{B}$(Equation 1)

• Orthogonal Component Relations:

  • The algebraic components of the resultant vector $\mathbf{C}$ equal the scalar sum of the individual vector components along each respective axis:     $C_x = A_x + B_x$(Equation 2)$C_y = A_y + B_y$(Equation 3)

• Expanded Component Equations:

  • Given vector $\mathbf{A}$ oriented at angle $\theta_a$ and vector $\mathbf{B}$ oriented at angle $\theta_b$:     $A_x = A \cos(\theta_a)$(Equation 4)$A_y = A \sin(\theta_a)$(Equation 5)$B_x = B \cos(\theta_b)$(Equation 6)$B_y = B \sin(\theta_b)$(Equation 7)

• Resultant Magnitude and Direction:

  • The total magnitude of resultant vector $\mathbf{C}$ is calculated using the Pythagorean theorem:     $C = \left(C_x^2 + C_y^2\right)^{1/2}$(Equation 8)
  • The directional angle $\theta_c$ of the resultant vector relative to the positive horizontal axis is determined by the inverse tangent function:     $\theta_c = \tan^{-1}\left(\frac{C_y}{C_x}\right)$(Equation 9)

Static Equilibrium on a Force Table

• Translational Equilibrium Condition:

  • When multiple coplanar forces act concurrently on a central point, translational static equilibrium requires the vector sum of all forces to equal zero.
  • When equilibrium is achieved, the central knot (or ring) remains stationary and perfectly centered around the center pin at the origin:     $\mathbf{A} + \mathbf{B} + \mathbf{C} = \mathbf{0}$(Equation 10)

• Equilibrant Determination:

  • When forces $\mathbf{A}$ and $\mathbf{B}$ are applied by known hung masses, force $\mathbf{C}$ acts as the balancing force (the equilibrant).
  • The equilibrant vector $\mathbf{C}$ is exactly equal in magnitude and opposite in direction to the resultant of $\mathbf{A}$ and $\mathbf{B}$:     $\mathbf{C} = -(\mathbf{A} + \mathbf{B}) = (-\mathbf{A}) + (-\mathbf{B})$(Equation 11)
  • The orthogonal components for the balancing force $\mathbf{C}$ satisfy:     $C_x = -(A_x + B_x)$$C_y = -(A_y + B_y)$

• Gravitational Force Conversion:

  • The downward gravitational force (tension along the string in Newtons) is generated by suspended masses.
  • Every applied force must include both the added mass and the mass of the hanger:     $\text{Total Mass } (m) = m_{\text{added}} + m_{\text{hanger}}$$m_{\text{hanger}} = 5\,\text{g} = 0.005\,\text{kg}$
  • Force calculation equation:     $F = m \times g$     where $g = 9.8\,\text{m/s}^2$.
  • Definition of unit force:     $1\,\text{N} = 1\,\text{kg} \times 9.8\,\text{m/s}^2$

Experimental Procedures

• Procedure A: Initial Setup of Apparatus

  • Position the force table on an entirely level, horizontal plane to eliminate gravitational bias.
  • Clamp three low-friction pulleys securely to the circular outer rim of the force table.
  • Thread strings from the central ring over each pulley and attach mass hangers to the loose ends.

• Procedure B: Fixed Angle Balancing

  • Step 1 — Configure Force A:
  • Suspend an added mass of $20\,\text{g}$ onto the hanger (total mass $= 25\,\text{g} = 0.025\,\text{kg}$).
  • Secure the pulley clamp at an angular position of $0^{\circ}$.
  • Step 2 — Configure Force B:
  • Suspend an added mass of $10\,\text{g}$ onto the hanger (total mass $= 15\,\text{g} = 0.015\,\text{kg}$).
  • Secure the pulley clamp at an angular position of $270^{\circ}$.
  • Step 3 — Balance with Force C:
  • Clamp the third pulley at an angular position of $150^{\circ}$.
  • Place an initial added mass of $20\,\text{g}$ on the hanger (initial total mass $= 25\,\text{g}$).
  • Incrementally add or subtract small masses on this third hanger until the central knot is centered at the origin pin.
  • Step 4 — Friction Compensation:
  • Due to residual friction within the pulley bearings, gently tap the strings to overcome static friction and initiate equilibrium motion.
  • Step 5 — Recording Data:
  • Record the experimental mass required to achieve center equilibrium and the experimental angle of $150^{\circ}$.

• Procedure C: Trial and Error Balancing

  • Step 1 — Configure Two Initial Forces:
  • Clamp the first pulley at an angle of $0^{\circ}$ with an added mass of $200\,\text{g}$ on the hanger (total mass $= 205\,\text{g} = 0.205\,\text{kg}$).
  • Clamp the second pulley at an angle of $210^{\circ}$ with an added mass of $250\,\text{g}$ on the hanger (total mass $= 255\,\text{g} = 0.255\,\text{kg}$).
  • Step 2 — Determine Force C by Trial and Error:
  • Move the third pulley along the circumference and adjust the suspended mass simultaneously through iterative trial and error.
  • Adjust both the angular position and suspended mass until the central knot centers at the origin.
  • Tap the strings lightly between mass and angle adjustments to overcome pulley friction.
  • Step 3 — Recording Data:
  • Record the final balancing angle and the total suspended mass (including the $5\,\text{g}$ hanger).

Calculations, Analysis, and Data Organization

• Analytical Calculation Steps:

  • Convert all masses from grams to kilograms ($\text{kg} = \frac{\text{g}}{1000}$).
  • Compute the tension force in Newtons for each vector by multiplying total mass in kilograms by $9.8\,\text{m/s}^2$.
  • Resolve known vectors $\mathbf{A}$ and $\mathbf{B}$ into horizontal ($A_x, B_x$) and vertical ($A_y, B_y$) components.
  • Sum components algebraically to find resultant components $R_x = A_x + B_x$ and $R_y = A_y + B_y$.
  • Calculate theoretical equilibrant components $C_x = -R_x$ and $C_y = -R_y$.
  • Compute theoretical equilibrant magnitude $C = \sqrt{C_x^2 + C_y^2}$ and direction $\theta_c = \tan^{-1}\left(\frac{C_y}{C_x}\right)$, adjusting the angle for the correct quadrant.
  • Convert theoretical force $C$ back into equivalent mass in kilograms ($m = \frac{C}{9.8\,\text{m/s}^2}$).

• Data Sheet Format:

• Tabulation Requirements:
  • Present experimental values alongside theoretically calculated values for both Procedure B and Procedure C.
  • Explicitly show full component breakdown calculations and trigonometric formulations in the designated reporting space.

According to Source 1: Laboratory Exercise 4: Vector Analysis and Static Equilibrium, here is the step-by-step procedure to perform the experiment:

Step 1: Apparatus Setup (Procedure A)

1. Place the circular force table on a completely flat, level horizontal surface to avoid gravitational tilt.
2. Clamp three low-friction pulleys along the outer rim of the force table.
3. Attach three strings to the central ring (or knot), drape each string over a pulley, and tie a calibrated mass hanger ($5\,\text{g}$ or $0.005\,\text{kg}$) to the loose end of each string.

Step 2: Fixed Angle Balancing (Procedure B)

1. Set up Force A:
   • Place $20\,\text{g}$ of added slotted mass onto the hanger (giving a total mass of $25\,\text{g} = 0.025\,\text{kg}$ including the $5\,\text{g}$ hanger).
   • Clamp this pulley at an angle of $0^{\circ}$.
2. Set up Force B:
   • Place $10\,\text{g}$ of added mass onto the hanger (total mass $= 15\,\text{g} = 0.015\,\text{kg}$).
   • Clamp this pulley at an angle of $270^{\circ}$.
3. Balance with Force C:
   • Clamp the third pulley at an angle of $150^{\circ}$.
   • Start with an initial added mass of $20\,\text{g}$ on the hanger (initial total mass $= 25\,\text{g}$).
   • Incrementally add or remove small masses until the central knot is centered directly over the origin pin.
4. Compensate for Pulley Friction:
   • Gently tap the strings to overcome static friction in the pulley bearings and verify that the knot remains centered.
5. Record Experimental Values:
   • Record the final balancing mass (including the $5\,\text{g}$ hanger) and the angle ($150^{\circ}$) in the Procedure B table.

Step 3: Trial and Error Balancing (Procedure C)

1. Set up the Two Known Forces:
   • Clamp the first pulley at $0^{\circ}$ with $200\,\text{g}$ added to the hanger (total mass $= 205\,\text{g} = 0.205\,\text{kg}$).
   • Clamp the second pulley at $210^{\circ}$ with $250\,\text{g}$ added to the hanger (total mass $= 255\,\text{g} = 0.255\,\text{kg}$).
2. Find Force C by Trial and Error:
   • Adjust both the angular position of the third pulley and the mass suspended from its hanger simultaneously by trial and error.
   • Tap the strings lightly after adjustments until the central ring remains centered at the origin.
3. Record Experimental Values:
   • Record the final balancing angle and the total suspended mass (including the hanger).

Step 4: Analytical Calculations and Comparison

1. Convert Masses to Tension Forces:
   • Ensure all masses are converted to kilograms ($\text{kg} = \frac{\text{g}}{1000}$), including the $0.005\,\text{kg}$ hanger mass.
   • Calculate force in Newtons: $F = m \times 9.8\,\text{m/s}^2$.
2. Resolve Vectors into Components:
   • $A<em>x = A\cos(\theta</em>a)$, $A<em>y = A\sin(\theta</em>a)$
   • $B<em>x = B\cos(\theta</em>b)$, $B<em>y = B\sin(\theta</em>b)$
3. Calculate Theoretical Equilibrant (Force C):
   • Sum the components and reverse the sign for static equilibrium ($\mathbf{A} + \mathbf{B} + \mathbf{C} = 0$):
     • $C<em>x = -(A</em>x + B_x)$
     • $C<em>y = -(A</em>y + B_y)$
   • Find theoretical magnitude: $C = \sqrt{C<em>x^2 + C</em>y^2}$
   • Find theoretical angle: $\theta<em>c = \tan^{-1}\left(\frac{C</em>y}{C<em>x}\right)$ (adjust for the correct quadrant based on signs of $C</em>x$ and $C_y$).
   • Calculate theoretical mass: $m = \frac{C}{9.8\,\text{m/s}^2}$.
4. Complete the Report:
   • Enter both the experimentally measured and theoretically calculated values in the data tables for Procedure B and Procedure C and show your component calculations.