— ## What are Systems of Objects? A system of objects refers to two or more objects that are physically connected or interacting in such a way that the motion of one affects the motion of the others. Analyzing these systems often involves: 1. Treating the entire system as one object: Sometimes, if all objects move together with the same acceleration, you can consider the total mass and the external forces acting on the whole system. 2. Analyzing each object individually: More commonly, you draw a separate free-body diagram (FBD) for each object in the system and apply Newton’s Second Law ($\Sigma F = ma$) to each one. The forces connecting the objects (like tension in a rope or contact forces between blocks) become internal forces for the whole system but external forces for individual objects. — ## Key Concepts for Systems * Internal vs. External Forces: * External forces: Forces acting on the system from outside (e.g., gravity, applied pushes/pulls, friction with the ground). These cause the system to accelerate. * Internal forces: Forces between objects within the system (e.g., tension in a rope connecting two blocks, the normal force between two stacked blocks). These do not affect the acceleration of the entire system, but they do affect the acceleration of individual objects. * Common Acceleration: If objects are rigidly connected (e.g., by a taut, massless, inextensible string), they will often have the same magnitude of acceleration. * Tension: The force transmitted through a string, rope, cable, or wire when it is pulled tight by forces acting from opposite ends. Tension is always a pulling force. * Pulleys: Ideal (massless, frictionless) pulleys change the direction of a force but do not change its magnitude. — ## Interactive: Connected Blocks over a Pulley Observe two connected blocks over a pulley. Adjust their masses and see how they accelerate and what the tension in the string is! <div class="animator-container"> <div class="input-controls"> </div> <div style="margin-bottom: 0.8rem;"> </div> Connected Blocks System Simulator A simulation of two connected blocks over a pulley, demonstrating forces, tension, and system acceleration. Surface M1 M2 $F_{g1}$ $F_{N1}$ $F_{T}$ $F_{f1}$ $F_{g2}$ $F_{T}$ Acceleration: 0.00 m/s² Tension: 0.0 N State: At Rest <div id="animationExplanation" class="animation-explanation" aria-live="polite"> <p>Adjust masses and friction, then click Play to see the system accelerate!

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Problem-Solving Strategy for Systems

  1. Draw FBDs for Each Object: This is the most critical step. Isolate each object and draw all external forces acting on that object.
  2. Choose a Coordinate System: For each FBD, choose a coordinate system. Align one axis with the direction of acceleration (or potential motion). For a pulley system, it’s often helpful to “unroll” the string and consider the direction of motion as positive.
  3. Apply Newton’s Second Law to Each Object: Write $\Sigma F = ma$ for each object along each chosen axis.
  4. Identify Connecting Forces: Recognize that internal forces (like tension) are equal in magnitude but opposite in direction between connected objects (Newton’s Third Law).
  5. Solve the System of Equations: You will typically have a system of equations with unknowns like acceleration and tension. Solve these simultaneously.

Example: Horizontal and Hanging Masses

Consider a mass $m_1$ on a horizontal surface connected by a string over a pulley to a hanging mass $m_2$.

  • For $m_1$ (horizontal):
    • Vertical: $F_N - F_{g1} = 0 \implies F_N = m_1g$
    • Horizontal: $F_T - F_{f1} = m_1a$ (if $m_1$ moves right)
      • Where $F_{f1} = \mu_k F_N = \mu_k m_1 g$
  • For $m_2$ (hanging):
    • Vertical: $F_{g2} - F_T = m_2a$ (if $m_2$ moves down)
      • Where $F_{g2} = m_2g$

You then solve these equations for $a$ and $F_T$.

Audio Explanation

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💡 Quick Concept Check:

In a system with two blocks connected by a string over a pulley, if the string is massless and inextensible, what can you say about the acceleration of the two blocks?

Click to Reveal Answer
If the string is massless and inextensible, the two blocks will have the **same magnitude of acceleration**. Their directions might be different (e.g., one moves horizontally, one vertically), but their speeds will change at the same rate.

Ready to put your understanding of systems of objects into practice? Check out these related skills:

Practice Problems

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