# Lecture 5: Applications of Newton's Laws ## Physics I — Mechanics & Thermodynamics --- ## Learning Objectives By the end of this lecture, students will be able to: 1. Solve multi-body problems involving strings, pulleys, and contact forces 2. Analyze motion on inclined planes with and without friction 3. Apply Newton's laws to systems with multiple interconnected objects 4. Solve problems involving tension in ropes and cables at various angles 5. Handle constraint equations for connected objects --- ## Lecture Content ### I. Strings, Ropes, and Tension An ideal string is massless and inextensible, meaning it transmits force without any loss. The tension is the **same** throughout such a string, and the string can only pull, never push. If a string has mass, tension varies along its length, but that complication lies beyond the introductory scope. When a string passes over an **ideal (massless, frictionless) pulley**, the pulley merely redirects the tension without changing its magnitude, so the tension on both sides remains the same. For a real pulley with mass or friction, the tensions on each side can differ, a situation addressed later with rotational dynamics. ### II. The Atwood Machine The Atwood machine consists of two masses m_1 and m_2 connected by a string over a frictionless, massless pulley. Assuming m_2 > m_1, the heavier mass accelerates downward while the lighter one accelerates upward. The free-body diagram for m_1 gives T - m_1 g = m_1 a, and for m_2 it gives m_2 g - T = m_2 a. Solving these simultaneously yields a = (m_2 - m_1) g / (m_1 + m_2) and T = 2 m_1 m_2 g / (m_1 + m_2). Examining special cases provides valuable physical insight. When m_1 = m_2, the acceleration is zero and the tension equals m_1 g, confirming that the system is in equilibrium. When m_2 >> m_1, the acceleration approaches g and the tension approaches 2 m_1 g, meaning the lighter mass accelerates upward at nearly the free-fall rate. ### III. Inclined Planes For a block on a frictionless incline at angle theta, the most effective approach is to choose axes with x along the incline (positive down the slope) and y perpendicular to the surface. The weight decomposes into mg sin(theta) along the incline and mg cos(theta) into the surface. Newton's second law in the y-direction, where a_y = 0, gives the normal force N = mg cos(theta). Along the incline, the acceleration is a = g sin(theta). The steeper the incline, the greater the component of gravity driving the motion. At the two extremes, when theta = 0 (flat surface), the acceleration is zero and N = mg, while at theta = 90 (vertical), the acceleration equals g and N = 0. A block of mass m on a frictionless inclined plane at angle theta. The coordinate system is tilted with x-axis along the incline and y-axis perpendicular. The weight vector mg points straight down and is decomposed into mg sin(theta) along the negative x-direction (down the slope) and mg cos(theta) along the negative y-direction (into the surface). The normal force N points in the positive y-direction. All components are clearly labeled with their expressions. ### IV. Connected Objects on Surfaces and Inclines When two blocks are connected by a string on a horizontal surface, both share the same acceleration due to the string constraint. Drawing separate free-body diagrams for each block reveals that the tension in the connecting string is an internal force of the system. Treating the system as a whole gives F_net = (m_1 + m_2) a, which determines the acceleration. To find the tension, apply Newton's second law to one block individually. A common variation involves a block on a table connected via a string over a pulley to a hanging block. The constraint is that both blocks have the same magnitude of acceleration, with the table block moving horizontally and the hanging block moving vertically. The system equation is m_hanging g - f_friction = (m_table + m_hanging) a. Solving for a first and then substituting back yields the tension. ### V. Multiple Pulleys and Mechanical Advantage A pulley system can provide **mechanical advantage**, allowing a heavy load to be lifted with a smaller applied force. For an ideal pulley system, if n rope segments support the load, the tension in the rope is W/n. The trade-off is that the rope must be pulled n times as far as the load moves, so the work done is the same either way, foreshadowing conservation of energy. For example, a single movable pulley has two rope segments supporting the load, so the required pulling force is W/2, but the rope must be pulled twice the distance the load rises. ### VI. Systems of Equations Approach For complex systems, the general strategy is to identify all objects and draw free-body diagrams for each, then write Newton's second law in component form for every object. The next step is to identify constraints: an ideal string means the same tension throughout and related acceleration magnitudes (for inextensible strings, |a_1| = |a_2|), while objects in contact share the same acceleration along the direction of contact. Count equations and unknowns to verify that you have enough independent equations, then solve the system algebraically and substitute numbers to check the result. Panel A: Two blocks stacked on a frictionless surface. A horizontal force F is applied to the bottom block. Free-body diagrams for each block are shown separately, with all forces labeled (applied force, weights, normal forces between blocks and from surface, friction between blocks). Panel B: A block on a frictionless table connected by a string over a pulley to a hanging block. FBDs for each block shown separately. Constraint equation (same magnitude of acceleration) is noted. The system of equations is written out step by step. ### VII. Apparent Weight and Accelerating Systems A person standing on a scale in an elevator reads the **normal force** N, which is the apparent weight. Applying Newton's second law gives N - mg = ma, so N = m(g + a). When the elevator accelerates upward (a > 0), N exceeds mg and the person feels heavier. When it accelerates downward (a < 0), N is less than mg and the person feels lighter. In free fall (a = -g), the normal force is zero, producing the sensation of weightlessness. At constant velocity (a = 0), the scale reads the person's true weight mg. This concept extends to any accelerating reference frame. ### VIII. Problem-Solving Tips Effective problem solving always begins with a clear diagram and free-body diagrams. Choose coordinate systems that simplify the math, typically aligning one axis with the direction of acceleration. A useful strategy is to apply the system approach first to find the overall acceleration, then isolate individual objects to find internal forces like tension. Checking limiting cases is an invaluable habit: does the answer make sense when masses are equal, when one mass is zero, or when the incline angle is 0 or 90 degrees? For Atwood machines, verify that the tension is less than the weight of the heavier object, and that the acceleration is less than g for any pulley-connected system. A summary diagram showing four classic Newton's law scenarios side by side: (1) Atwood machine with two hanging masses, (2) block on a frictionless incline, (3) two blocks connected on a horizontal surface with an applied force, (4) block on table with string over pulley to hanging mass. Each scenario shows the FBD, the key equations, and the solution for acceleration and tension in a compact reference-card format.