Pulleys and connected objects Knowledge-Based Study

Study notes

1. Connected Objects and Constraints

   In classical mechanics, when two or more objects are linked together by ropes, cables, or strings over pulleys, they form a connected system. The motion of one body directly determines or constrains the motion of the others.

   To analyze these systems accurately, physicists often begin with idealizing assumptions:

   • Light (massless) string: The mass of the string is negligible compared to the connected objects. Consequently, the magnitude of the tension $T$ is uniform along any continuous segment of the string between two objects or across an ideal pulley.
   • Inextensible string: The string does not stretch. This kinematic constraint ensures that if two objects are connected directly by a taut string passing over fixed pulleys, their speeds and the magnitudes of their accelerations must be identical: $a_1 = a_2 = a$.
   • Ideal pulley: The pulley is assumed to be frictionless and massless. It alters only the direction of the tension force, not its magnitude.

   Two Hanging Masses over a Fixed Pulley

   An Atwood machine consists of two masses, $m_1$ and $m_2$, connected by a light, inextensible cord passing over a massless, frictionless pulley. Assume $m_2 > m_1$, so $m_2$ accelerates downward while $m_1$ accelerates upward at the same rate $a$.

   Isolating Each Body

   We apply Newton's second law, $\Sigma F = ma$, to each mass along its direction of motion:

   • For mass $m_1$ (upward motion as positive):
     $T - m_1 g = m_1 a$
   • For mass $m_2$ (downward motion as positive):
     $m_2 g - T = m_2 a$

   Solving for System Acceleration and Tension

   Adding the two equations eliminates the internal tension $T$:

   $(m_2 - m_1)g = (m_1 + m_2)a$

   Thus, the magnitude of acceleration is:

   $$a = \frac{m_2 - m_1}{m_1 + m_2} g$$

   Substituting this acceleration back into either equation yields the string tension:

   $$T = \frac{2 m_1 m_2}{m_1 + m_2} g$$

   Common Misconception: The tension $T$ is neither equal to $m_2 g$ nor $m_1 g$. For $m_2$ to accelerate downward, $T$ must be strictly less than $m_2 g$, and for $m_1$ to accelerate upward, $T$ must be strictly greater than $m_1 g$.

   Mass on a Surface Connected to a Hanging Mass

   Consider mass $m_1$ resting on a horizontal table and connected by a horizontal string over a frictionless pulley at the table edge to a vertically hanging mass $m_2$.

   Frictionless Surface

   Along the horizontal axis for $m_1$, the only force is tension: $T = m_1 a$. For the hanging mass $m_2$, moving downward: $m_2 g - T = m_2 a$.

   Combining these gives:

   $$a = \frac{m_2 g}{m_1 + m_2}$$

   $$T = \frac{m_1 m_2 g}{m_1 + m_2}$$

   Surface with Kinetic Friction

   If kinetic friction $f_k = \mu_k N = \mu_k m_1 g$ opposes the motion of $m_1$, Newton's second law for each body becomes:

   • For $m_1$: $T - \mu_k m_1 g = m_1 a$
   • For $m_2$: $m_2 g - T = m_2 a$

   Summing both equations yields:

   $$a = \frac{(m_2 - \mu_k m_1)g}{m_1 + m_2}$$

   Motion occurs only if the gravitational pulling force $m_2 g$ exceeds the maximum static friction force $f_{s,\max} = \mu_s m_1 g$.

   Treating Connected Objects as a Combined System

   While drawing separate free-body diagrams for each mass is always reliable, you can also treat the entire connected chain as a single composite object of total mass $M = \Sigma m_i$.

   • Internal Forces: Forces between connected parts, such as tension in the connecting string, cancel out when summing forces over the entire system.
   • Net Driving Force: The acceleration is found directly by dividing the net external force acting along the line of motion by the total mass: $$a = \frac{\Sigma F_{\text{ext, parallel}}}{m_{\text{total}}}$$
   • Finding Internal Tension: Once $a$ is determined, isolate a single mass and apply $\Sigma F = ma$ to compute the tension.

   Step-by-Step Procedure:

   1. Define a consistent coordinate direction along the path of motion (for example, clockwise or along the string).
   2. Identify all external forces acting on each object (gravity, normal force, friction).
   3. Set up equations of motion for each body or for the entire system.
   4. Verify that calculated accelerations satisfy $0 \le a < g$ for free-hanging systems and that tensions are non-negative.