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Topic 2.5: Practice Sheet — Newton's Second Law, Part 2

Name: _______________________________ Date: _______________ Period: ______

Directions: Label each equation "Whole system:" or "Isolate [object]:" before writing it, and solve symbolically before substituting any numbers.

Warm-Up Level (straightforward, one-step)

  1. Two blocks, masses 3 kg and 5 kg, are connected by a string on a frictionless floor. An external force of 16 N pulls the 3 kg block forward, with the 5 kg block trailing behind. Find the system's shared acceleration.
  2. Using your answer to problem 1, isolate the trailing (5 kg) block to find the tension in the connecting string.

Standard Level (multi-step, matches typical AP Classroom depth)

  1. Two objects, far from any other objects in space, orbit their shared center of mass in circles at constant speed. Is the center of mass of the two-object system accelerating? Explain using the concept of external force.
  2. Two blocks (masses $m_1$ and $m_2$) are connected by a string over a frictionless pulley, with $m_1$ resting on a horizontal frictionless table and $m_2$ hanging off the edge. Write the whole-system equation for the shared acceleration $a$, in terms of $m_1$, $m_2$, and $g$ (hint: the external force driving the system is $m_2 g$, the hanging block's weight).
  3. Using your answer to problem 4, isolate $m_1$ (on the table) to write an expression for the tension in the string.

AP-Level (multi-part: calculate + justify/represent/predict-a-change)

  1. Crate 1 (mass $2m$) and Crate 2 (mass $m$) are connected by a string on a rough floor. Crate 1 is pulled to the right by a constant force $F$. Crate 1 experiences friction $f_1$ and Crate 2 experiences friction $f_2$, both less than $F$. The string then breaks, and $F$ continues to act on Crate 1. a) Find an expression for Crate 2's acceleration right after the string breaks. b) Find an expression for the acceleration of the two-crate system's center of mass right after the string breaks, and explain why it may differ from Crate 2's individual acceleration.
  2. A system consists of two identical blocks connected by a rigid rod. At a certain instant, the system's center of mass has zero velocity but a nonzero acceleration to the right. a) Is the net force on the system zero at that instant? Explain your reasoning. b) Explain why "the center of mass is momentarily at rest" does not imply "the system is in equilibrium."

Progress Check Style (evaluate the method — AP Classroom format)

  1. Crate 1 (mass $m_1$) and Crate 2 (mass $m_2$) are connected by a string and pulled together by external force $F$ across a frictionless floor, sharing acceleration $a$. Which of the following correctly gives the tension in the connecting string if Crate 1 is the one being directly pulled by $F$, and Crate 2 trails behind? (A) $F_T = F$ (B) $F_T = m_1 a$ (C) $F_T = m_2 a$ (D) $F_T = (m_1+m_2)a$

Progress-Check-Aligned Practice (extra depth — same style as your unit test)

  1. Two identical blocks touch each other on a frictionless floor, connected only by contact (not a string). Equal-magnitude forces $F$ are applied in three separate trials: (A) a force $F$ pushes the left block to the right, and a separate force $F$ pushes the right block to the right as well (both forces pointing the same direction); (B) a force $F$ pulls the left block to the left (away from the right block) while a separate force $F$ pulls the right block to the right (away from the left block); (C) the left block has no applied force, while a force $F$ pushes the right block to the left, into the left block. For each trial, find the acceleration of the two-block system in terms of $F$ and the combined mass $2m$ (or explain why "the system's acceleration" isn't a meaningful single value in that trial), and rank the three results from largest to smallest.
  2. Two carts are connected by a spring with spring constant 1000 N/m. The cart on the right has mass 2.0 kg and is pulled by a rope exerting 100 N on it; the cart on the left has unknown mass $M$. Both carts share the same constant acceleration, and the spring is stretched 0.06 m beyond its natural length. Assuming negligible mass and friction in the wheels, find the value of $M$. (Hint: isolate the left cart — the only horizontal force on it is the spring.)
  3. Block A (mass $2m_0$) and Block B (mass $m_0$) sit on a frictionless horizontal surface and are connected to each other by a string. A second string, attached to Block B, passes over a frictionless, massless pulley and connects to a hanging block of mass $2m_0$. The system starts at rest and is then released. a) Find the acceleration of Block A after release, in terms of $g$. b) Let $T_1$ be the tension in the string connecting Block A and Block B in this setup. Now suppose Blocks A and B swap positions (so Block A is now the one directly connected to the string going over the pulley), and the system is again released from rest. Let $T_2$ be the tension in the A-B connecting string in this new arrangement. Is $T_1 = T_2$? Explain your reasoning without recalculating the whole problem from scratch — think about which block is now closer to the pulley.
  4. A block of mass $m$ is connected to a block of mass $2m$ below it by a vertical string, and the two-block system is in free fall near Earth's surface (falling together, nothing else touching either block). a) What is the tension in the connecting string during this free fall? Justify using Newton's second law applied to the whole two-block system, or to either block alone. b) Explain why the acceleration of the $2m$ block must be exactly $g$, even though it has "extra" weight compared to the $m$ block above it.

Keep this sheet — it's part of your semester study materials, same as your Concept Sheets.