## PYL 105

### Atwood"s machine

The figure listed below shows an Atwood"s machine, 2 unequal masses (m1 and also m2) linked by a string the passes end a pulley.

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Consider the pressures acting on every mass. Assume the the cable is massless and does not stretch and that pulley-block is massless and frictionless. Derive an expression for the acceleration; that should have the form

Of course, this is one idealized calculation, and also we cannot mean to uncover this acceleration experimentally. Let us store the other approximations however drop the frictionless approximation, identify friction together the cause of any deviation from the right acceleration provided above. We have the right to then relate the frictional pressure to the difference between the ideal and experimental accelerations. The ideal force leads to the ideal acceleration

Fideal = (m1+m2)aideal

Including the frictional pressure leads to the experimental acceleration

Fideal - Ffric = (m1 + m2) aexp

Solve these because that Ffric.

We can also take on an energy technique to this problem. Among the an initial steps in the energy strategy is to recognize the "system" for which we room calculating the energy. The "system" could be

m1 alone m1 and the planet m1, m2 and the earth

Determining the mechanism determines whether a pressure is identified as outside or internal and also in some instances whether us talk about work or potential energy. Because that the systems above

gravity and also tension are externalgravity is internal, tension is outside gravity and also tension room internal

If we consider m1 and also the planet as our system, then the stress is exterior to the system, and does work-related on m1. The occupational results in a change of energy, both kinetic and also gravitational potential energy.

Returning to the frictionless approximation, find an equation relating the work-related done by T1 come the change in energy of the m1-earth system. I think the massive starts native rest and is displaced a street x indigenous its early stage position. Likewise find one equation relating the job-related done by T2 to the readjust in energy of the m2-earth system. (They must be included in her report.) due to the fact that the tensions room equal in magnitude and also the displacements are similarly equal in magnitude, we could eliminate the occupational done by the tensions to uncover

This is the energy of the m1-m2-earth system and also ideally DE=0, that is, the energy of the m1-m2-earth mechanism is conserved. That is, ideally the power does not change from that is initial worth which deserve to be taken to be

E0 = m2 g h

again assuming that the mechanism starts from remainder in the position presented on the left above.

Of course, friction should be brought into the energy strategy as well. Just as friction accounted for the deviation indigenous the best acceleration in the previous approach, that should additionally account for any changes in the energy in the last approach. The is, the work connected with the frictional pressure (the job-related done versus the friction force) must equal the observed adjust in energy.

Measurements

Find the masses of 2 sinkers. Use two that are not the exact same size but are nearby in size. (The calculations room sensitive to this mass measurements, do them carefully.)
Set increase an Atwood"s machine, two uneven masses connected by a string that passes end a smart pulley. Set it up at the edge of the bench together shown listed below so that us can expand our speculative range.

collection up the interface to use the clever Pulley. Host the lighter sinker down at the floor level, note the height of the more heavier sinker, begin recording data and release. Copy and also paste both the position versus time and velocity matches time data end to Excel. Plot velocity versus time
. Recall that we carry out not want every one of this data. In ~ some allude the reduced sinker hits the floor or the higher sinker gets caught in the pulley. Toss out the undesired large-time data. right a brand-new graph (with negative data thrown away) to a line and extract the experimental acceleration. To compare this acceleration come the best one and also calculate the frictional force
Note the the times linked with the place data and also the times associated with the velocity data room different. The times are staggered.

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Calculate the average of 2 consecutive velocities. Because that example, you could enter a formula like

= (D2 + D3)/2

in cabinet E4. This will identify the velocities at roughly the times because that which we have actually position data. Next use Excel to calculate the state in the mechanical energy. Usage the velocity just obtained. Kinetic power of first mass (m1v2/2) kinetic energy of second mass (m2v2/2) gravitational potential energy of very first mass (m1gx) gravitational potential power of second mass(m2g(h-x))

(In the formulas above m2 was the heavier mass and started in ~ the greater position.) The equivalent Excel formula can be placed in columns F,G,H and also I as presented above.

In the following column (J), sum these four energy terms. This is the mechanical energy. Plot mechanical energy versus (position) time
. If there were no friction this would certainly be a horizontal line (why?) how much power is lost? If the change in power is due specifically to the presence of a friction force, climate we have the right to calculate the job-related done versus a consistent friction as follows

Wfric = Ffric x

Calculate the job-related done against friction (in shaft K).

add the job-related done against friction come the mechanical energy. Make a graph which plots at the same time mechanical power versus time and also mechanical energy plus work done versus friction versus time. (Highlight 3 columns: the moment column, the mechanical power column and the mechanical energy plus work done versus friction column, walk to chart Wizard and also proceed together usual.) The second curve need to be noticeably flatter 보다 the first. Why? Is all power now accounted for?