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Calculating the work you do to push a lawn mower across a large lawn

How much work is done on the lawn mower by the person in [link] (a) if he exerts a constant force of 75 . 0 N size 12{"75" "." 0" N"} {} at an angle 35 º size 12{"35"°} {} below the horizontal and pushes the mower 25 . 0 m size 12{"25" "." 0" m"} {} on level ground? Convert the amount of work from joules to kilocalories and compare it with this person’s average daily intake of 10 , 000 kJ size 12{"10","000"" kJ"} {} (about 2400 kcal size 12{"2400"" kcal"} {} ) of food energy. One calorie (1 cal) of heat is the amount required to warm 1 g of water by 1 º C size 12{1°C} {} , and is equivalent to 4 . 184 J size 12{4 "." "184"" J"} {} , while one food calorie (1 kcal) is equivalent to 4184 J size 12{"4184"" J"} {} .

Strategy

We can solve this problem by substituting the given values into the definition of work done on a system, stated in the equation W = Fd cos θ size 12{W= ital "Fd"" cos"θ} {} . The force, angle, and displacement are given, so that only the work W size 12{W} {} is unknown.

Solution

The equation for the work is

W = Fd cos θ . size 12{W= ital "Fd"" cos"θ} {}

Substituting the known values gives

W = 75.0 N 25.0 m cos 35.0º = 1536 J = 1.54 × 10 3 J. alignl { stack { size 12{W= left ("75" "." "0 N" right ) left ("25" "." "0 m" right )"cos " left ("35" "." 0° right )} {} #size 12{" "="1536"" J"=1 "." "54" times "10" rSup { size 8{3} } " J" "." } {} } } {}

Converting the work in joules to kilocalories yields W = ( 1536 J ) ( 1 kcal / 4184 J ) = 0 . 367 kcal size 12{W= \( "1536"`J \) \( 1`"kcal"/"4184"`J \) =0 "." "367"`"kcal"} {} . The ratio of the work done to the daily consumption is

W 2400 kcal = 1 . 53 × 10 4 . size 12{ { {W} over {"2400"`"kcal"} } =1 "." "53" times "10" rSup { size 8{ - 4} } "." } {}

Discussion

This ratio is a tiny fraction of what the person consumes, but it is typical. Very little of the energy released in the consumption of food is used to do work. Even when we “work” all day long, less than 10% of our food energy intake is used to do work and more than 90% is converted to thermal energy or stored as chemical energy in fat.

Applying the science practices: boxes on floors

Plan and design an experiment to determine how much work you do on a box when you are pushing it over different floor surfaces. Make sure your experiment can help you answer the following questions: What happens on different surfaces? What happens if you take different routes across the same surface? Do you get different results with two people pushing on perpendicular surfaces of the box? What if you vary the mass in the box? Remember to think about both your effort in any given instant (a proxy for force exerted) and the total work you do. Also, when planning your experiments, remember that in any given set of trials you should only change one variable.

You should find that you have to exert more effort on surfaces that will create more friction with the box, though you might be surprised by which surfaces the box slides across easily. Longer routes result in your doing more work, even though the box ends up in the same place. Two people pushing on perpendicular sides do less work for their total effort, due to the forces and displacement not being parallel. A more massive box will take more effort to move.

Applying the science practices: force-displacement diagrams

Suppose you are given two carts and a track to run them on, a motion detector, a force sensor, and a computer that can record the data from the two sensors. Plan and design an experiment to measure the work done on one of the carts, and compare your results to the work-energy theorem. Note that the motion detector can measure both displacement and velocity versus time, while the force sensor measures force over time, and the carts have known masses. Recall that the work-energy theorem states that the work done on a system (force over displacement) should equal the change in kinetic energy. In your experimental design, describe and compare two possible ways to calculate the work done.

Sample Response: One possible technique is to set up the motion detector at one end of the track, and have the computer record both displacement and velocity over time. Then attach the force sensor to one of the carts, and use this cart, through the force sensor, to push the second cart toward the motion detector. Calculate the difference between the final and initial kinetic energies (the kinetic energies after and before the push), and compare this to the area of a graph of force versus displacement for the duration of the push. They should be the same.

Practice Key Terms 3

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Source:  OpenStax, Work and energy. OpenStax CNX. Nov 09, 2015 Download for free at http://legacy.cnx.org/content/col11902/1.1
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