LAB TORQUES

Experiment: Torques, Equilibrium, and Center of Gravity

Introduction and Objectives

• In introductory physics, forces act on objects, with the assumption that they can be treated as particles responding linearly to forces.

• Reality depicts objects as extended collections of particles, where the point of force application significantly influences their motion.

• Rotational motion is essential when analyzing solid extended objects or rigid bodies.

  • Definition of a Rigid Body: An object or system of particles with fixed distances between its particles that remain constant under standard conditions.

  • Example: Liquid water is not a rigid body, but ice is.

  • Acknowledgement of idealization: While the model implies rigidity, solid particles (atoms/molecules) do vibrate and can undergo deformation.

Static Equilibrium of Rigid Bodies

• Static Equilibrium: An important condition for rigid bodies where they are at rest or not rotating.

• Types of Equilibrium discussed:

  • Translational Equilibrium: Ensured when the sum of all forces acting on the body equals zero:
    $\Sigma F = 0$

  • Rotational Static Equilibrium: Ensured when the sum of torques acting on the rigid body equals zero:
    $\Sigma \tau = 0$

  • If a rigid body is even momentarily rotating, it may still be static if the net torque also sums to zero.

Equipment Needed

1. Meterstick

2. Support stand

3. Laboratory balance

4. String (and corresponding clamps)

5. Four hooked weights:

   • Two weights of 50 g

   • Two weights of 100 g

   • One weight of 200 g

   • One unknown mass with a hook

Theory

A. Equilibrium Conditions

• Critical equations for static equilibrium in rigid bodies:

  • Translational: $\Sigma F = 0$ (11.1a)

  • Rotational: $\Sigma \tau = 0$ (11.1b)

• The rigid body (e.g., meterstick) has no linear displacement constraints, hence the rotational conditions dominate.

B. Torque

• Torque Definition: Torque (or moment of force) arises from a force applied at a distance from an axis of rotation.

  • Mathematical Representation:
    $\tau = rF$
    where:

  • $r$: Perpendicular distance from the axis to the force's line of action (lever arm).

  • $F$: Magnitude of the force.

  • Units: The unit of torque is the meter-newton (m-N, same as joules but indicated differently for clarity).

C. Determining Directions of Torque

• Torque is considered positive if it rotates counterclockwise and negative if clockwise relative to the axis of rotation.

• Example torque analysis involves organizing torques from various forces acting at specified distances, balancing torques influences by masses:

  • $\Sigma \tau<em>{cc} = \Sigma \tau</em>{cw}$ (11.3)

• Convert specific force applications into balanced torque conditions:
  $m<em>1g r</em>1 + m<em>2g r</em>2 = m<em>3g r</em>3 + m<em>4g r</em>4$ (11.4)

Example Calculations

Example 11.1: Calculating Torque to Find Unknown Mass

• Given masses:

  • $m<em>1 = m</em>3 = 50 ext{ g}$

  • $m_2 = 100 ext{ g}$

• Tasks involve balancing torques:

  • Counterclockwise forces $F<em>1, F</em>2$ and clockwise forces $F<em>3, F</em>4$ lead to finding positions needed for static equilibrium.

Center of Gravity and Center of Mass

• Center of Gravity: Point about which the gravitational torques balance; defined as the balance point.

  • Expression of gravitational balance:
    $\Sigma(mg) r<em>{cc} = \Sigma(mg) r</em>{cw}$

Linear Mass Density Concept

• Linear mass density (C) derived as: $\mu = \frac{m}{L}$

  • Units in grams per centimeter or kilograms per meter.

• Example calculation for a meterstick of length 100 cm with mass 50 g yields a linear density of 0.50 g/cm:
  $\mu = \frac{50 ext{ g}}{100 ext{ cm}} = 0.50 ext{ g/cm}$

Experimental Procedure

A. Apparatus with Support Point at Center of Gravity

1. Setup including supports and clamps and procedure to record mass of the meterstick.

2. Adjust balance point recording displacement of meterstick to find balancing point.

   • Use cases of known weights (100 g at 15 cm, etc.) to establish static equilibrium and measurement.

3. Detailed calculations for counterclockwise and clockwise torque comparisons recorded.

4. Explore the effect of mass distribution on the balance point and center of gravity.

B. Apparatus Supported at Different Pivot Points

• Acknowledged need for recalculation when the support is not at the center of gravity; requires comprehensive torque calculations involving the average mass distribution and lever arms for theoretical support.

Cases 1-6: Systematic Investigation

1. Multiple scenarios are explored through gradually complex setups, focusing on position variations of masses (1, 2, etc.) on the meterstick leading to center of gravity study and numerical predictions.

Conclusion

• Analyze data by calculating percent differences between predicted and experimental results, focusing on accuracy in the setups made for understanding torques and statics in real-world applications.

• Assuring learning about moments, equilibrium, and forces as foundational principles for mechanical physics.