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AP Calculus AB: 10.8.1 An Introduction to Work

Mathematics12 CardsCreated 8 months ago

Work is the energy transferred when a force causes an object to move over a distance in the direction of that force. For a constant force, the work done is simply the product of the force magnitude and the distance moved.

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An Introduction to Work

  • Work is the energy used when applying a force over a distance.

  • For a constant force F, work is the product of the force and the change in distance.

  • For a changing or variable force F(x) on [a, b], work is given by the integral.

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Key Terms

Term
Definition

An Introduction to Work

  • Work is the energy used when applying a force over a distance.

  • For a constant force F, work is the product of the force and ...

note

  • Most people have a good idea of what work means to them, but in physics and mathematics work has a very specific meaning.

  • Wo...

Which of the following is the work done by a non-constant force F (x) = −x applied to an object moving along the x‑axis from 0 to 2?

−2

Which of the following statements about work is not correct?

For a changing force, work is the product of the force and the change in distance.

Which of the following is the work done by a constant force F moving an object along a straight line from a to b ?

W=F⋅(b−a)

Which of the following is the work done by a non-constant force F = F (x) = x moving an object along the x‑axis from 0 to 2?

2

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AP Calculus AB: 10.8.1 An Introduction to Work
TermDefinition

An Introduction to Work

  • Work is the energy used when applying a force over a distance.

  • For a constant force F, work is the product of the force and the change in distance.

  • For a changing or variable force F(x) on [a, b], work is given by the integral.

note

  • Most people have a good idea of what work means to them, but in physics and mathematics work has a very specific meaning.

  • Work is the energy used when moving an object. For work to be done, a force must be applied to an object and that object must move in the direction of the force. Thus, pushing against a wall might take a lot of effort, but doesn’t do any work.

  • When a force is constant, work can be found by multiplying the magnitude of the force by the change in distance.

  • But forces do not have to be constant. To find the work done using a variable, or changing, force you have to use calculus. F(x) means the force at a particular x-value. It’s a notational way of saying “changing force.”

  • Notice that the integral looks a lot like the formula for a constant force. The force is multiplied by a change in distance, this time dx. The integral sums up all the infinite little pieces.

Which of the following is the work done by a non-constant force F (x) = −x applied to an object moving along the x‑axis from 0 to 2?

−2

Which of the following statements about work is not correct?

For a changing force, work is the product of the force and the change in distance.

Which of the following is the work done by a constant force F moving an object along a straight line from a to b ?

W=F⋅(b−a)

Which of the following is the work done by a non-constant force F = F (x) = x moving an object along the x‑axis from 0 to 2?

2

Suppose you lift a weight of 20 lb a vertical distance of 35 ft. Which of the following shows the work that has been done?

700 foot-pounds

Which of the following is the work done by a non-constant force F(x)=sinx moving an object along the x-axis from 0 to π?

2

Which of the following is the work done by a changing force F = F (x) moving an object along a straight line from a to b ?

W=∫baF(x)dx

If a horizontal force of 60 newtons is applied to a heavy box in order to push it a distance of 20 meters along a rough floor, which of the following is the work done by the force?

1200 newton-meters

Which of the following is the work done by a non-constant force F (x) = x moving an object along the x‑axis from 2 to 0?

−2

Let W1 be the work done by a constant force F moving an object along a straight line for a distance D. Let W2 be the work done by a constant force 3 F moving an object along a straight line for a distance D/3. Compare W1 and W2.

W1 = W2