8.01SC | Fall 2016 | Undergraduate

Classical Mechanics

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In the figure, pulley A is fixed to the ceiling. An ideal rope is attached at one end to block 1 of mass \(m_1\), it passes around a second pulley, labelled B, and its other end is fixed to the ceiling. A second ideal rope attaches pulley B to a second block of mass \(m_2\). As a result, pulley B and block 2 move together.

The goal of the problem is to calculate the accelerations of blocks 1 and 2.

The solution of this problem is divided into four parts:

  • Part I : Set up the system of equations.
  • Part II: Constraint condition - find the relationship between the accelerations.
  • Part III: Constraint condition using a virtual displacement argument.
  • Part IV: Solving the system of equations.

The video below will help you to set up the equations needed to answer the question.

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Blocks 1 and 2 lie on a frictionless surface and are connected by three massless ropes strung over massless, frictionless pulleys as shown above. A force of magnitude \(F\) is applied to Block 1.

(Part a) Draw free-body diagrams for block 1 and block 2 as they move accross the table.

(Part b) Draw free-body diagrams for the right and the left pulley.

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A rope of mass \(\displaystyle m \) hangs between two trees, making an angle \(\displaystyle \theta \) with the vertical at each end. (It is attached to to each tree at the same height.) Your goal in this problem will be to find the magnitude of the tension force in the rope at the ends and midpoint of the rope. Begin by drawing a free-body diagram just for the left half of the rope.

Calculate \(\displaystyle T_{mid}\), the tension in the rope at the midpoint. Express your answer in terms of \(\displaystyle m\), \(\displaystyle g\), and \(\displaystyle \theta\).

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Problem Set 4 contains the following problems:

  1. Block and Pulley System
  2. Velocity Dependent Force
  3. Tension in Massive Rotating Rope with Object
  4. Tension in Rope Wrapped Around a Rod
  5. Drag Force at Low Speeds
  6. Two Pulleys, Two Strings and Two Blocks

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