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17-Phys-A1 Classical Mechanics · December 2013

Question 5 of 6: Cable-Pulley System — Dependent Motion Analysis

Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)

Notes on this paper

Paper format. 98-Phys-A1 Classical Mechanics, National Exams December 2013 — a three-hour closed-book examination; one of two approved calculator models (Casio or Sharp) is permitted, plus a single 8.5″ × 11″ aid sheet. The cover page states that five (5) questions constitute a complete exam paper and that only the first five as they appear in the answer book are marked, and that each question is of equal value — so each of the six printed questions is worth 20 marks against a 100-mark paper. The candidate is invited to submit a clear statement of any assumptions made where a question is open to interpretation; this paper needs that licence in Questions 3 and 6, both flagged below in a Check box. The cover page also warns that most questions require an essay-format answer, where clarity and organisation carry marks. All six questions are worked here, because the set is a study resource rather than a timed attempt.

Reference texts. H. Goldstein, C. Poole and J. Safko, Classical Mechanics, 3rd ed. (Lagrange multipliers and constraint forces, holonomic versus nonholonomic constraints, Hamiltonian mechanics, cyclic coordinates and conservation laws); R. C. Hibbeler, Engineering Mechanics: Dynamics, 14th ed. (rolling without slipping, relative-motion analysis using rotating axes, dependent-motion pulley analysis, impulse and momentum for a system of particles, planar kinetics of a rigid body).

Question 5: Cable-Pulley System — Dependent Motion Analysis (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. One continuous cable runs from block A, up and over a fixed pulley, down and under a movable pulley that carries block B, up and over a second fixed pulley, then down to block C.

Given data
QuantitySymbolValue
Speed of block A$v_A$5 m/s, upward
Speed of block C$v_C$2.5 m/s, downward
ABCv_A = 5 m/s upv_B = ?v_C = 2.5 m/s down
Figure 5 -- cable-pulley system: blocks A, B (movable pulley), C.

Find. The speed (and direction) of block B.

Approach. Write the single scalar constraint that the total cable length is constant, using downward-positive position coordinates for the three blocks, then differentiate once.

  1. Set up the length constraint. Let $s_A, s_B, s_C$ be the (downward positive) positions of the three blocks measured from the ceiling. Block B hangs from a MOVABLE pulley, so it is supported by two cable runs, both of length $s_B - \text{const}$; block A and block C each hang from a single run. The total cable length is $$L = s_A + 2s_B + s_C - \text{const} \quad\Rightarrow\quad \dot s_A + 2\dot s_B + \dot s_C = 0$$
  2. Substitute the given speeds (downward positive, so "up" is negative): $\dot s_A = -5$ m/s, $\dot s_C = +2.5$ m/s. $$-5 + 2\dot s_B + 2.5 = 0$$ $$\boxed{\dot s_B = 1.25 \text{ m/s (positive} \Rightarrow \text{downward)}}$$
Question 5 — results
QuantityValue
Speed of block B, $v_B$1.25 m/s, downward