Physics 111

In Physics 111 (Classical Mechanics), tension ($\mathbf{T}$ or $F_T$) is the pulling contact force transmitted axially through a flexible connector, such as a string, rope, cable, or cord.

1. Key Characteristics & Ideal Approximations

• Direction: Tension always acts along the line of the rope and always pulls away from the object to which it is attached (strings cannot push).

• Ideal (Massless) String: If the cord has negligible mass ($m_{\text{string}} \approx 0$), the tension magnitude $T$ is uniform throughout the entire length of the string, even if the string is accelerating.

• Inextensible Cord: Strings are typically assumed not to stretch, meaning connected objects share the same magnitude of acceleration ($a_1 = a_2 = a$).

• Ideal Pulley: Frictionless and massless pulleys redirect the direction of the tension force without altering its magnitude.

2. Common Applications & Equations

A. Vertical Hanging Mass

For a mass $m$ suspended stationary from a vertical cable: $\sum F_y = T - mg = 0 \implies T = mg$

If the mass accelerates upward with acceleration $a$: $\sum F_y = T - mg = ma \implies T = m(g + a)$

If accelerating downward with acceleration $a$: $\sum F_y = mg - T = ma \implies T = m(g - a)$

B. Atwood Machine

Two masses ($m_1 > m_2$) connected over an ideal pulley:

• Equation for $m_1$ (moving down): $m_1 g - T = m_1 a$

• Equation for $m_2$ (moving up): $T - m_2 g = m_2 a$

• System acceleration: $a = \left(\frac{m_1 - m_2}{m_1 + m_2}\right)g$

• Tension in the string: $T = \left(\frac{2 m_1 m_2}{m_1 + m_2}\right)g$

C. Static Equilibrium at Angles

When a suspended weight is supported by cables at angles $\theta_1$ and $\theta_2$: $\sum F_x = T_2 \cos(\theta_2) - T_1 \cos(\theta_1) = 0$ $\sum F_y = T_1 \sin(\theta_1) + T_2 \sin(\theta_2) - mg = 0$

D. Circular Motion (Conical Pendulum / Vertical Circle)

• Vertical loop at bottom: $T - mg = \frac{m v^2}{r} \implies T = mg + \frac{m v^2}{r}$

• Vertical loop at top: $T + mg = \frac{m v^2}{r} \implies T = \frac{m v^2}{r} - mg$ (critical minimum speed occurs when T = 0$

In Physics 111 (Classical Mechanics), tension (\mathbf{T}$or$F_T) is the pulling contact force transmitted axially through a flexible connector, such as a string, rope, cable, or cord.

1. Key Characteristics & Ideal Approximations

• Direction: Tension always acts along the line of the rope and always pulls away from the object to which it is attached (strings cannot push).

• Ideal (Massless) String: If the cord has negligible mass (m_{\text{string}} \approx 0$), the tension magnitude$T$is uniform throughout the entire length of the string, even if the string is accelerating.</p></li><li><p><strong>Inextensible Cord</strong>: Strings are typically assumed not to stretch, meaning connected objects share the same magnitude of acceleration ($a_1 = a_2 = a).

• Ideal Pulley: Frictionless and massless pulleys redirect the direction of the tension force without altering its magnitude.

2. Common Applications & Equations (Rearranged for Mass)

A. Vertical Hanging Mass

For a mass m suspended stationary from a vertical cable: \sum F_y = T - mg = 0 Solving for mass: m = \frac{T}{g}$</p><p>If the mass accelerates upward with acceleration$a: \sum F_y = T - mg = ma \implies T = m(g + a) Solving for mass: m = \frac{T}{g + a}$</p><p>If accelerating downward with acceleration$a: \sum F_y = mg - T = ma \implies T = m(g - a) Solving for mass: m = \frac{T}{g - a}$</p><p>B. Atwood Machine</p><p>Two masses ($m_1 > m_2$) connected over an ideal pulley:</p><ul><li><p>Equation for$m_1$(moving downward with acceleration$a): m_1 g - T = m_1 a Solving for mass: m_1 = \frac{T}{g - a}$</p></li><li><p>Equation for$m_2$(moving upward with acceleration$a): T - m_2 g = m_2 a Solving for mass: m_2 = \frac{T}{g + a}$</p></li><li><p>In terms of system acceleration$a and the companion mass: a = \left(\frac{m_1 - m_2}{m_1 + m_2}\right)g Solving for m_1: m_1 = m_2\left(\frac{g + a}{g - a}\right) Solving for m_2: m_2 = m_1\left(\frac{g - a}{g + a}\right)$</p></li><li><p>In terms of tension$T and the companion mass: T = \left(\frac{2 m_1 m_2}{m_1 + m_2}\right)g Solving for m_1: m_1 = \frac{T m_2}{2 m_2 g - T} Solving for m_2: m_2 = \frac{T m_1}{2 m_1 g - T}$</p></li></ul><p>C. Static Equilibrium at Angles</p><p>When a suspended mass$m$is supported by cables at angles$\theta_1$and$\theta_2: \sum F_x = T_2 \cos(\theta_2) - T_1 \cos(\theta_1) = 0 \sum F_y = T_1 \sin(\theta_1) + T_2 \sin(\theta_2) - mg = 0 Solving for mass: m = \frac{T_1 \sin(\theta_1) + T_2 \sin(\theta_2)}{g}

D. Circular Motion (Conical Pendulum / Vertical Circle)

• Vertical loop at bottom: T - mg = \frac{m v^2}{r} \implies T = m\left(g + \frac{v^2}{r}\right) Solving for mass: m = \frac{T}{g + \frac{v^2}{r}}

• Vertical loop at top: T + mg = \frac{m v^2}{r} \implies T = m\left(\frac{v^2}{r} - g\right) Solving for mass: m = \frac{T}{\frac{v^2}{r} - g}$</p></li></ul><p></p><p>In Physics 111 (Classical Mechanics), tension ($\mathbf{T}$or$F_T) is the pulling contact force transmitted axially through a flexible connector, such as a string, rope, cable, or cord.

  1. Key Characteristics & Ideal Approximations

  • Direction: Tension always acts along the line of the rope and always pulls away from the object to which it is attached (strings cannot push).

  • Ideal (Massless) String: If the cord has negligible mass (m_{\text{string}} \approx 0$), the tension magnitude$T$is uniform throughout the entire length of the string, even if the string is accelerating.</p></li><li><p><strong>Inextensible Cord</strong>: Strings are typically assumed not to stretch, meaning connected objects share the same magnitude of acceleration ($a_1 = a_2 = a).

  • Ideal Pulley: Frictionless and massless pulleys redirect the direction of the tension force without altering its magnitude.

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  2. Common Applications & Equations (Rearranged for Weight)

  A. Vertical Hanging Mass

  For an object with weight W = mg suspended stationary from a vertical cable: \sum F_y = T - W = 0 Solving for weight: W = T$</p><p>If the object accelerates upward with acceleration$a: \sum F_y = T - W = ma = \left(\frac{W}{g}\right)a Solving for weight: W = \frac{T g}{g + a}$</p><p>If accelerating downward with acceleration$a: \sum F_y = W - T = ma = \left(\frac{W}{g}\right)a Solving for weight: W = \frac{T g}{g - a}$</p><p>B. Atwood Machine</p><p>Two suspended objects with weights$W_1$and$W_2$($W_1 > W_2$) connected over an ideal pulley:</p><ul><li><p>Equation for$W_1$(moving downward with acceleration$a): W_1 - T = \left(\frac{W_1}{g}\right)a Solving for weight: W_1 = \frac{T g}{g - a}$</p></li><li><p>Equation for$W_2$(moving upward with acceleration$a): T - W_2 = \left(\frac{W_2}{g}\right)a Solving for weight: W_2 = \frac{T g}{g + a}$</p></li><li><p>In terms of system acceleration$a and the companion weight: a = \left(\frac{W_1 - W_2}{W_1 + W_2}\right)g Solving for W_1: W_1 = W_2\left(\frac{g + a}{g - a}\right) Solving for W_2: W_2 = W_1\left(\frac{g - a}{g + a}\right)$</p></li><li><p>In terms of tension$T and the companion weight: T = \frac{2 W_1 W_2}{W_1 + W_2} Solving for W_1: W_1 = \frac{T W_2}{2 W_2 - T} Solving for W_2: W_2 = \frac{T W_1}{2 W_1 - T}$</p></li></ul><p>C. Static Equilibrium at Angles</p><p>When a suspended weight$W$is supported by cables at angles$\theta_1$and$\theta_2: \sum F_x = T_2 \cos(\theta_2) - T_1 \cos(\theta_1) = 0 \sum F_y = T_1 \sin(\theta_1) + T_2 \sin(\theta_2) - W = 0 Solving for weight: W = T_1 \sin(\theta_1) + T_2 \sin(\theta_2)

  D. Circular Motion (Conical Pendulum / Vertical Circle)

  • Vertical loop at bottom: T - W = \frac{m v^2}{r} = \left(\frac{W}{g}\right)\frac{v^2}{r} \implies T = W\left(1 + \frac{v^2}{g r}\right) Solving for weight: W = \frac{T}{1 + \frac{v^2}{g r}} = \frac{T g}{g + \frac{v^2}{r}}

  • Vertical loop at top: T + W = \frac{m v^2}{r} = \left(\frac{W}{g}\right)\frac{v^2}{r} \implies T = W\left(\

  
  

  In Physics 111 (Classical Mechanics), friction ($\mathbf{f}$ or $F_f$) is the contact force that acts parallel to the contacting surfaces, opposing relative sliding motion or the tendency toward such motion.

  1. Fundamental Types & Governing Equations

  • Static Friction ($f_s$): Acts between surfaces at rest relative to each other, counteracting applied tangential forces to prevent slipping.

    • Inequality: $f_s \le f_{s,\text{max}}$

    • Maximum static friction (threshold of motion): $f_{s,\text{max}} = \mu_s F_N$

    • Note: $\mu_s$ is the dimensionless coefficient of static friction, and $F_N$ is the normal force.

  • Kinetic Friction ($f_k$): Acts between surfaces in active relative sliding motion.

    • Governing equation: $f_k = \mu_k F_N$

    • For nearly all material pairings: $\mu_k < \mu_s$

    • To an ideal approximation, $f_k$ is independent of the apparent contact area and the sliding speed.

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  2. Common Configurations & Applications

  A. Horizontal Surface with Applied Force

  • Horizontal pull $F$ on mass $m$:   $\sum F_y = F_N - mg = 0 \implies F_N = mg$$f_k = \mu_k mg$$\sum F_x = F - f_k = ma \implies a = \frac{F - \mu_k mg}{m}$

  • Pulling at an upward angle $\theta$ to the horizontal:   $\sum F_y = F_N + F\sin(\theta) - mg = 0 \implies F_N = mg - F\sin(\theta)$$f_k = \mu_k [mg - F\sin(\theta)]$$\sum F_x = F\cos(\theta) - f_k = ma \implies a = \frac{F\cos(\theta) - \mu_k [mg - F\sin(\theta)]}{m}$

  • Pushing at a downward angle $\theta$ to the horizontal:   $\sum F_y = F_N - F\sin(\theta) - mg = 0 \implies F_N = mg + F\sin(\theta)$$f_k = \mu_k [mg + F\sin(\theta)]$$\sum F_x = F\cos(\theta) - f_k = ma \implies a = \frac{F\cos(\theta) - \mu_k [mg + F\sin(\theta)]}{m}$

  B. Inclined Plane (Angle $\theta$)

  • Normal force perpendicular to incline:   $F_N = mg\cos(\theta)$

  • Maximum static friction limit:   $f_{s,\text{max}} = \mu_s mg\cos(\theta)$

  • Angle of Repose (critical angle $\theta_c$ where mass begins to slip unassisted):   $mg\sin(\theta_c) = f_{s,\text{max}} = \mu_s mg\cos(\theta_c)$$\mu_s = \tan(\theta_c)$

  • Sliding down the incline:   $\sum F_x = mg\sin(\theta) - f_k = ma$$a = g[\sin(\theta) - \mu_k \cos(\theta)]$

  • Sliding up the incline (decelerating):   $\sum F_x = -mg\sin(\theta) - f_k = ma$$a = -g[\sin(\theta) + \mu_k \cos(\theta)]$

  C. Circular Motion on an Unbanked Curve

  • Static friction provides the necessary centripetal force for a turning vehicle of mass $m$ and radius $r$:   $f_s = \frac{m v^2}{r} \le \mu_s F_N = \mu_s mg$

  • Maximum non-skid speed:   $v_{\text{max}} = \sqrt{\mu_s g r}$

  D. Work and Energy Dissipation

  • Work done by kinetic friction along displacement $d$:   $W_{\text{nc}} = -f_k d$

  • Thermal energy produced:   $$\Delta E_{\text{th}} = f_k d =