Mechanical Advantage of Pulleys

Rohan Singh

Ryerson University

The purpose of this lab is to measure the advantage and efficiency of different pulley arrangements. Two different pulley arrangement were set up and an increasing amount of load was placed on the system. Masses were used as effort to balance the load placed on the system. A graph of load versus effort was plotted to determine the Actual Mechanical Advantage (AMA) of the system. For the system with a Theoretical Mechanical Advantage (TMA) of 3, the AMA was found to be 2.96. Giving the system and efficiency of 0.987. For the system with a TMA of 4, the AMA was found to be 3.62. Giving the system and efficiency of 0.905. A possible source of error came from friction between then axels and the pulleys. In conclusion the advantage and efficiency of the different pulley systems was measured.

Introduction

The purpose of this experiment is to study various pulley arrangements determine the advantage and efficiency of each of the arrangements.

Industry wants machines to be as efficient as possible, meaning they have to output as much as possible with as little input as possible. Therefore, it is important for engineers to understand the concept of efficiency in machines.

A simple machine exerts output force/torque great that the input force/torque. The factor by which the input force is multiplied by is the Mechanical Advantage. The Theoretical Mechanical Advantage (TMA) is the ratio of output force (ideal) by ideal input force (deal force):

TMA = F_o/F_i^ideal (1) where:

F_o = output force in Newtons [N]

F_i^ideal = idea input force in Netwons [N].

In real situations friction al losses occur. Actual Mechanical Advantage (AMA) is defined as:

AMA = F_o/F_i^real(2) where:

F_i^real = real output force in Newtons [N]

F_o = output force in Newtons [N].

In the ideal case for work:

F_od_o = F_i^ideald_i(3) where:

F_o = output force in Newtons [N]

d_o = distance traveled by load [m]

F_i^ideal = idea input force in Netwons [N]

d_i = distance traveled by the effort [m].

Assuming work force and displacement have the same direction:

TMA = F_o/F_i^ideal = d_i/d_o(4)

In the real case output work plus lost energy equals input work, therefore:

W_friction + F_od_o = F_i^ideald_i (5) where:

W_friction = work done due to friction [J].

Efficiency is a measure of input energy compared to output energy, thus:

ε = W_out/W_in = AMA/TMA (6) where:

ε = efficiency.

Procedure and Observations

Two systems of pulleys were set up in a support frame as shown in Figure 3 and Figure 4.

The following procedure was used to gather the data for this experiment^(1):

Determine the mass of each pulley type.
Set-up the pulleys as shown in Figure 3.
Use a load mass starting at 0.250kg and determine the mass you have to place at the other end of the string to balance the pulley arrangement.
Repeat step 2 for loads up to 0.500kg in 0.050kg intervals.
Plot the graph of the load plus the pulley mass versus the effort and determine the slope of the best fit line. The value obtained is the AMA of the pulley arrangement.
Determine the efficiency of the system.
Repeat steps 3 to 6 for system shown in Figure 4.

Mass of pulleys in system 1 = 106.2g

= 0.1062kg

Mass of pulleys in system 2 = 137.9g

= 0.1379kg

Load (kg)	Effort (kg)	Load + pulley mass (kg)
0.250	0.100	0.3562
0.300	0.120	0.4062
0.350	0.139	0.4562
0.400	0.149	0.5062
0.450	0.167	0.5562
0.500	0.185	0.6062

Figure 1: Load/Effort Data for System 1.

Load (kg)	Effort (kg)	Load + pulley mass (kg)
0.250	0.082	0.3879
0.300	0.096	0.5258
0.350	0.112	0.6637
0.400	0.123	0.8016
0.450	0.137	0.9395
0.500	0.151	1.0774

Figure 2: Load/Effort Data for System 2.

Figure 3: Diagram of Pulley System 1.

Figure 4: Diagram of Pulley System 2.

Figure 5: Graph of Load versus Effort for System 1.

Figure 6: Graph of Load versus Effort for System 2.
Data Analysis and Results

For the first system of pulleys the AMA was found to be 2.96, the TMA equaled 3. This gave an efficiency of 0.987. For the second system, the AMA was found to be 3.62 and the TMA equaled 4. This gave an efficiency of 0.905. See Appendix A for calculations.

The AMA was found by calculating the slope of the line of best fit for the graph of load versus effort as seen in Figure 3 and Figure 4. The efficiency was calculate by dividing the AMA by the TMA.

Conclusion

The objectives of this lab were achieved. The TMA and AMA as well as the efficiency was measured for each of the pulley systems.

Friction was the major source of error in the lab. The work done by friction affects the real input force, which results in a change in the AMA. The chance in the AMA causes a lower efficiency. To overcome some of the friction, pulleys could be mounted on bearings rather than a straight axel.

The AMA values was higher for the system with more pulleys, however they were still close to their respective TMA values. The efficiency value for System 2 was lower than that of System 1. The drop in efficiency can be attributed to increased friction in System 2 due to more pulleys.

References

1. Ryerson University Department of Mathematics, Physics and Computer Science, Physics Laboratory Manual for Engineering Students in the First Year PCS 211, S2-3,2002.

Appendix A

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