Chapter 5 Lecture: Introduction to Tension (Massless String & Massless Pulley)

Fundamentals of Tension and Mechanical Forces

• Context of Fundamental Contact and Field Forces:

  • Contact forces analyzed in classical mechanics include the normal force, which always acts perpendicular to the surface of contact and is electromagnetic at the microscopic level.

  • Gravitational force represents the non-contact field attraction that the Earth exerts on an object, defining the object's weight.

  • Rotated coordinate systems are standardly employed for resolving gravitational and normal force components on inclined planes.

  • Tension is introduced as a third mechanical force, which also originates from fundamental electromagnetic interactions.

• Microscopic and Physical Nature of Tension:

  • Tension arises from the electromagnetic bonds holding the constituent atoms of a material together.

  • Macroscopically, tension is directly derived from the elasticity of the material, whether structured as a rope, string, or cable.

  • The word "tensile" pertains to stretching and strictly implies pulling forces.

  • Flexible media such as strings or ropes cannot sustain compressive (pushing) forces; when pushed, a string simply collapses or folds without resisting.

  • By physical definition, tension is a pulling force transmitted along the entire continuous length of an object (typically ropes, strings, or cables).

The Massless String Approximation

• Physical Justification of Massless Strings:

  • Real physical ropes and cables possess non-zero mass.

  • In physics, setting a parameter to zero signifies that its contribution is negligible relative to the desired precision or significant figures of the measurement.

  • When the mass of the accelerated object ($m_b$) is substantially greater than the mass of the string ($m_s$), the string mass can be treated as zero ($m_s \to 0$) without altering the outcome within relevant significant figures.

  • If a heavy rope is used to pull a load whose mass is comparable to that of the rope, the mass of the rope cannot be neglected; doing so would introduce significant physical error into dynamic calculations.

• Ideal String Assumptions in Mechanics:

  • The string is treated as an idealized transmitter of tensile force.

  • The tension is transmitted uniformly from one end of a massless string to the other.

Dynamics of a Hand-String-Box System

• System Definitions and Coordinate System:

  • Consider a system consisting of a box ($b$), an intervening string ($s$), and a pulling hand ($h$).

  • A horizontal coordinate system is established where motion to the right corresponds to the positive $x$-direction, and motion to the left corresponds to the negative $x$-direction.

  • The surface supporting the box is assumed to be completely frictionless.

  • Vertical forces acting on the box (the downward gravitational force and the upward normal force from the floor) balance completely and are omitted from horizontal dynamic analysis.

• Free Body Diagram and Equations for the Box:

  • The box of mass $m_b$ interacts exclusively with the string horizontally.

  • The force exerted by the string on the box is denoted by $F_{sb}$, oriented along the positive $x$-direction.

  • Applying Newton's second law along the horizontal axis yields:     $\sum F_x^{(b)} = F_{sb} = m_b a_{bx}$

• Free Body Diagram and Equations for the Massless String:

  • The string is modeled as a point particle situated between the hand and the box.

  • The string is in direct physical contact with two entities: the box at its left end and the hand at its right end.

  • The force exerted by the hand on the string is denoted by $F_{hs}$, pointing in the positive $x$-direction.

  • The force exerted by the box on the string is denoted by $F_{bs}$, pointing in the negative $x$-direction.

  • By Newton's third law, the force exerted by the box on the string ($F_{bs}$) and the force exerted by the string on the box ($F_{sb}$) constitute an action-reaction pair:     $|F_{sb}| = |F_{bs}|$

  • Applying Newton's second law to the string along the horizontal axis yields:     $\sum F_x^{(s)} = F_{hs} - F_{bs} = m_s a_{sx}$

• Force Transmission Through the Massless Medium:

  • Substituting the condition for a massless string ($m_s = 0$) simplifies Newton's second law equation:     $F_{hs} - F_{bs} = 0$     $F_{hs} = F_{bs}$

  • Because $F_{bs} = F_{sb}$ by Newton's third law, equating the terms yields:     $F_{hs} = F_{sb}$

  • Consequently, the pulling force exerted by the hand is transmitted without modification directly to the box:     $F_{hs} = m_b a_{bx}$

  • A massless string behaves as a pure force transmitter that transfers the external applied force directly to the load.

Foundational Assumptions in Coupled Cable-Pulley Systems

• Point Mass Assumption:

  • Objects in translation are treated as point masses without spatial volume, removing rotational dynamics and angular accelerations from consideration.

• Uniform Massless Cable Assumption:

  • Massless cables transmit tension uniformly throughout their entire length, maintaining constant tension magnitude at all internal segments.

• Ideal Massless, Frictionless Pulley Assumption:

  • Massless, frictionless pulleys alter the orientation of the tension force vector without altering its scalar magnitude.

  • If a tension vector directed horizontally along the negative $x$-direction passes over an ideal pulley to become a vertical vector, the scalar magnitude of tension remains identical in both segments.

• Inextensible Constraint Condition:

  • When two objects are joined by a taut, inextensible cable, motion of one body is rigidly coupled to the motion of the second.

  • While their velocity and acceleration directional vectors may differ according to the geometry of the pulley, their scalar acceleration magnitudes are identical:     $|a_1| = |a_2| = a$

Full Derivation of a Modified Atwood Machine

• Physical Configuration:

  • A mass $m_1$ rests on a frictionless horizontal table.

  • A light, inextensible string connects mass $m_1$ horizontally to an ideal pulley ($p$) at the edge of the table.

  • The string routes over the pulley and hangs vertically, terminating at suspended mass $m_2$.

  • The entire system is frictionless (table surface and pulley axle).

• Forces Acting on Mass 1 ($m_1$):

  • Downward gravitational force exerted by the Earth:     $w_{e1} = m_1 g$

  • Upward normal contact force exerted by the table:     $N_{t1}$

  • Tension force pulling $m_1$ horizontally to the right toward the pulley, denoted as $T_{p1}$.

  • Vertical equilibrium on the table:     $\sum F_{1y} = N_{t1} - m_1 g = 0$

  • Horizontal dynamic equation along the $+x$-direction:     $\sum F_{1x} = T_{p1} = m_1 a_{1x}$

• Forces Acting on the Pulley ($p$):

  • The horizontal segment of string exerts a tension pulling leftward toward mass 1, denoted as $T_{1p}$.

  • The vertical segment of string exerts a tension pulling downward toward mass 2, denoted as $T_{2p}$.

  • By Newton's third law, these forces form action-reaction pairs with the forces acting on the respective masses:     $|T_{p1}| = |T_{1p}|$     $|T_{p2}| = |T_{2p}|$

  • Applying Newton's second law to the pulley in vector form:     $\mathbf{T}_{1p} + \mathbf{T}_{2p} = m_p \mathbf{a}_p$

  • Setting the pulley mass to zero ($m_p = 0$):     $\mathbf{T}_{1p} + \mathbf{T}_{2p} = 0$

  • Hence, the scalar magnitude of tension transmitted from mass 1 equals the scalar magnitude transmitted to mass 2:     $|T_{1p}| = |T_{2p}|$

  • Therefore, across the ideal massless pulley:     $T_{p1} = T_{1p} = T_{2p} = T_{p2} = T$

• Forces Acting on Mass 2 ($m_2$):

  • Downward gravitational force exerted by the Earth:     $w_{e2} = m_2 g$

  • Upward tension force exerted by the pulley/string:     $T_{p2}$

  • Defining the downward vertical direction as negative, the acceleration is $-a_{2y}$.

  • Dynamic equation from Newton's second law:     $\sum F_{2y} = T_{p2} - m_2 g = -m_2 a_{2y}$

  • Isolating upward tension $T_{p2}$:     $T_{p2} = m_2 g - m_2 a_{2y}$

  • For downward acceleration to occur, the gravitational vector $m_2 g$ must exceed the upward tension vector $T_{p2}$.

• Mathematical Coupling and Acceleration Solution:

  • Setting coordinate acceleration magnitudes equal based on the string constraint:     $a_{1x} = a_{2y} = a$

  • Setting the tension magnitudes equal across the massless pulley:     $T_{p1} = T_{p2} = T$

  • Equating the dynamic equations for mass 1 and mass 2:     $m_1 a = m_2 g - m_2 a$

  • Grouping acceleration terms to the left-hand side:     $m_1 a + m_2 a = m_2 g$     $(m_1 + m_2) a = m_2 g$

  • Solving explicitly for the system acceleration $a$:     $a = \frac{m_2}{m_1 + m_2} g$

System-Level Derivation Method

• Concept of Composite System Analysis:

  • Rather than evaluating free body diagrams for individual constituent masses, the coupled assembly is treated as a unified composite system of total inertia $m_{\text{total}}$.

  • Because strings and pulleys are massless, the combined mass of the system is:     $m_{\text{total}} = m_1 + m_2$

• Internal Versus External Forces:

  • Tensile forces connecting $m_1$ to $m_2$ act strictly as internal forces within the system boundary and cancel out under summation.

  • The only net external driving force acting parallel to the path of motion is the gravitational attraction on the hanging mass:     $F_{\text{net, ext}} = w_{e2} = m_2 g$

• Direct Computation:

  • Applying Newton's second law directly to the composite system:     $F_{\text{net, ext}} = m_{\text{total}} a$     $m_2 g = (m_1 + m_2) a$

  • Solving directly for $a$:     $a = \frac{m_2}{m_1 + m_2} g$

  • While this intuitive shortcut provides a rapid check, comprehensive academic derivations require explicit multi-body free body diagrams and Newton's third law pairs.

Limiting Case Analysis and Validation

• Limiting Case 1: Vanishing Mass on the Table ($m_1 \to 0$):

  • Evaluated algebraically:     $a = \lim_{m_1 \to 0} \left( \frac{m_2}{m_1 + m_2} g \right) = \frac{m_2}{0 + m_2} g = \frac{m_2}{m_2} g = g$

  • Physical interpretation: With zero mass resting on the table to provide horizontal inertia, the hanging mass falls completely unimpeded, accelerating at the free-fall rate of gravity $g$.

• Limiting Case 2: Vanishing Hanging Mass ($m_2 \to 0$):

  • Evaluated algebraically:     $a = \lim_{m_2 \to 0} \left( \frac{m_2}{m_1 + m_2} g \right) = \frac{0}{m_1 + 0} g = 0$

  • Physical interpretation: With no hanging mass to supply a gravitational accelerating force, mass $m_1$ experiences zero net horizontal force and remains at rest with zero acceleration.