MCAT Physics and Math Review

Chapter 2: Work and Energy

2.3 Mechanical Advantage

Would it make a difference whether Sisyphus lifted the rock vertically to its final position or rolled it there along an incline? The difference between these two scenarios is mechanical advantage, a measure of the increase in force accomplished by using a tool. Sloping inclines, such as hillsides and ramps, make it easier to lift objects because they distribute the required work over a larger distance, decreasing the required force. For a given quantity of work, any device that allows for work to be accomplished through a smaller applied force is thus said to provide mechanical advantage. In addition to the inclined plane, five other devices are considered the classic simple machines designed to provide mechanical advantage: wedge (two merged inclined planes), wheel and axle, lever, pulley, and screw (rotating inclined plane). Of these, the inclined plane, lever, and pulley are most frequently tested on the MCAT.

Mechanical advantage is the ratio of magnitudes of the force exerted on an object by a simple machine (Fout) to the force actually applied on the simple machine (Fin):

Equation 2.11

The mechanical advantage, because it is a ratio, is dimensionless.

Reducing the force needed to accomplish a given amount of work does have a cost associated with it, however: the distance through which the smaller force must be applied in order to do the work must be increased. Inclined planes, levers, and pulleys do not magically change the amount of work necessary to move an object from one place to another. Because displacement is pathway independent, the actual distance traveled from the initial to final position does not matter, assuming all forces are conservative. Therefore, applying a lesser force over a greater distance to achieve the same change in position (displacement) accomplishes the same amount of work. We’ve already considered the dynamics of incline planes and levers in Chapter 1 of MCAT Physics and Math Review. Here, we look at the work associated with inclined planes.

Example:

A block weighing 100 N is pushed up a frictionless incline over a distance of 20 m to a height of 10 m as shown below.

Find:

1.    The minimum force required to push the block

2.    The work done by the force

3.    The force required and the work done by the force if the block were simply lifted vertically 10 m

Solution:

1.    To find the minimum force required to push the block, we must draw a free body diagram of the situation:

The minimum force needed is a force that will push the block with no acceleration parallel to the surface of the incline. This means the magnitude of the applied force is equal to that of the parallel component of gravity:

F = mg sin θ

mg represents the weight of the object, which is 100 N. Using trigonometry, sin θ is ratio of the length of the opposite side to the hypotenuse, which is  Therefore,

2.    The work done by F is

W = Fd cos θ

In this case, θ represents the angle between the force and displacement vectors, not the angle of the inclined plane. Because the force and displacement vectors are parallel, θ = 0 and cos θ = 1. Therefore,

W = (50 N)(20 m)(1) = 1000 J

3.    To raise the block vertically, an upward force equal to the object’s weight (100 N) would have to be generated. The work done by the lifting force is

W = Fd cos θ = (100 N)(10 m)(1) = 1000 J

The same amount of work is required in both cases, but twice the force is needed to raise the block vertically compared with pushing it up the incline.

PULLEYS

Pulleys utilize the same paradigm to provide mechanical advantage as the inclined plane: a reduction of necessary force at the cost of increased distance to achieve a given value of work or energy transference. In practical terms, pulleys allow heavy objects to be lifted using a much-reduced force. Simply lifting a heavy object of mass m to a height of h will require an amount of work equal to mgh—its change in gravitational potential energy. If the displacement occurs over a distance equal to the displacement, then the force required to lift the object will equal mg. If, however, the distance through which the displacement is achieved is greater than the displacement (an indirect path), then the applied force will be less than mg. In other words, we’ve been able to lift this heavy object to the desired height by using a smaller force, but we’ve had to apply that smaller force through a greater distance in order to lift this heavy object to its final height.

Before examining how pulleys create this mechanical advantage, let’s consider first the heavy block in Figure 2.3, suspended from two ropes. Because the block is not accelerating, it is in translational equilibrium, and the force that the block exerts downward (its weight) is cancelled by the sum of the tensions in the two ropes. For a symmetrical system, the tensions in the two ropes are the same and are each equal to half the weight of the block.

Figure 2.3. Block Suspended by Two Ropes If the block is in translational equilibrium, the tension in each rope is equal to half the weight of the block.

Now let’s imagine the heavy block in Figure 2.4 represents a heavy crate that must be lifted. Assuming that the crate is momentarily being held stationary in midair, we again have a system in translational equilibrium: the weight (the load) is balanced by the total tension in the ropes. The tensions in the two vertical ropes are equal to each other; if they were unequal, the pulleys would turn until the tensions were equal on both sides. Therefore, each rope supports one-half of the crate’s total weight. By extension, only half the force (effort) is required to lift the crate. This decrease in effort is the mechanical advantage provided by the pulley, but as we’ve already discussed, mechanical advantage comes at the expense of distance. To lift an object to a certain height in the air (the load distance), one must pull through a length of rope (the effort distance) equal to twice that displacement. If, for example, the crate must be lifted to a shelf 3 meters above the ground, then both sides of the supporting rope must shorten by 3 meters, and the only way to accomplish this is by pulling through 6 meters of rope.

Figure 2.4. Two-Pulley System The block is suspended from two ropes, each of which bears half of the block’s weight.

All simple machines can be approximated as conservative systems if we ignore the (usually) small amount of energy that is lost due to external forces, such as friction. The idealized pulley is massless and frictionless, and under these theoretical conditions, the work put into the system (the exertion of force through a distance of rope) will exactly equal the work that comes out of the system (the displacement of the mass to some height). Real pulleys—and all real machines, for that matter—fail to conform to these idealized conditions and, therefore, do not achieve 100 percentefficiency in conserving energy output to input. We can define work input as the product of effort and effort distance; likewise, we can define work output as the product of load and load distance. Comparing the two as a ratio defines the efficiency of the simple machine:

Equation 2.12

KEY CONCEPT

When considering simple machines, load and effort are both forces. The load determines the necessary output force. From the output force and mechanical advantage, we can determine the necessary input force.

Efficiency is often expressed as percentage by multiplying the efficiency ratio by 100 percent. The efficiency of a machine gives a measure of the amount of useful work generated by the machine for a given amount of work put into the system. A corollary of this definition is that the percentage of the work put into the system that becomes unusable is due to nonconservative or external forces.

The pulley system in Figure 2.5 illustrates the fact that adding more pulleys further increases mechanical advantage: for each additional pair of pulleys, we can reduce the effort further still. In this case, the load has been divided among six lengths of rope, so the effort required is now only one-sixth the total load. Remember that we would need to pull through a length of rope that is six times the desired displacement, and that efficiency will decrease due to the added weight of each pulley and the additional friction forces.

Figure 2.5. System of Six Pulleys Increasing the number of pulleys decreases the tension in each segment of rope; this leads to an increase in the mechanical advantage of the setup.

Example:

The pulley system in Figure 2.5 has an efficiency of 80 percent. A person is lifting a mass of 200 kg with the pulley.

Find:

1.    The distance through which the effort must move to raise the load a distance of 4 m

2.    The effort required to lift the load

3.    The work done by the person lifting the load through a height of 4 m

Solution:

1.    For the load to move through a vertical distance of 4 m, all six of the supporting ropes must shorten 4 m also. This may only be accomplished by pulling 6 × 4 = 24 m of rope through the setup. Therefore, the effort must move through a distance of 24 m.

2.    To calculate the effort required, the equation for efficiency should be used. The load is the weight of the object being lifted and is equal to the mass of the object times the acceleration due to gravity g. The effort distance, calculated in part A, is 24 m.

3.    The work done by the person is

MCAT Concept Check 2.3:

Before you move on, assess your understanding of the material with these questions.

1.    As the length of an inclined plane increases, what happens to the force required to move an object the same displacement?

2.    As the effort decreases in a pulley system, what happens to the effort distance to maintain the same work output?

3.    What accounts for the difference between work input and work output in a system that operates at less than 100% efficiency?

4.    What does it mean for a device to provide mechanical advantage?

5.    Name the six simple machines:

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