Two Blocks on Inclined plane

“Physics Principles with Applications”, Douglas C. Giancoli, 7th Edition, Global Edition

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44. Two blocks are connected by a light string passing over a pulley of radius 0.15 m and moment of inertia I. The blocks move (toward the right) with an acceleration of 1.00 m/s2 along their frictionless inclines (See figure below). (a) Draw free-body diagrams for each of the two blocks and the pulley. (b) Determine FTA and FTB, the tensions in the two parts of the string. (c) Find the net torque acting on the pulley, and determine its moment of inertia, I.

Giancoli 8-44

Analyze the problem:

The problem has two blocks on inclined surfaces and they are attached by an inelastic string (so they should have the same acceleration) the pulley’s mass is not ignored (we know that because the problem mentioned that there is a moment of inertia).

What are the given information?

  1. Block A mass, mA=8 kg
  2. Block B mass, mB=10 kg
  3. Acceleration, a = 1 m/s2
  4. Inclination Angle 1, θ1=32º
  5. Inclination Angle 2, θ2=61º
  6. Radius of pulley, R= 0.15 m

What are the needed information?

  1. Body-diagram draw for the two blocks and the pulley.
  2. FTA and FTB tension forces
  3. The net torque on the pulley
  4. The moment of inertia of the pulley

SOLUTION:

Giancoli 8-44-1

Giancoli 8-44-2

Atwood Machine

Source:
“Physics Principles with Applications”, Douglas C. Giancoli, 7th Edition, Global Edition

This is a common physics problem and it mentioned in several textbooks.

45. An Atwood machine (shown in figure below) consists of two masses, mA=65 kg and mB=75 kg, connected by a massless inelastic cord that passes over a pulley free to rotate, The pulley is a solid cylinder of radius R=0.45 m and mass 6 kg. (a) Determine the acceleration of each mass. (b) What % error would be made if the moment of inertia of the pulley is ignored? [Hint: the tensions FTA and FTB aare not equal]

Analyze the problem:

It is an Atwood typical problem but here the pulley’s mass  is not ignored and the tensions forces are different for the same cord. Having the pulley’s mass ignored and the tension being the same for the single cord simplify the problem, so here we have to approach the problem in different way but using the same methods (Newton’s laws!).

What are the given information?

  1. Block A mass, mA = 65 Kg
  2. Block B mass, mB=75 Kg
  3. Pulley mass, M=6 kg
  4. Pulley Radius, R=0.45 m

What are needed information?

  1. Acceleration of both blocks.
  2. Percentage error of the two cases (the pulley’s moment of inertia is ignored and the pulley’s moment of inertia is not ignored).

 

SOLUTION:

  1. Finding the acceleration of the two blocks:

Giancoli 8-45

2. Finding the percentage error:

Giancoli 8-45-2