Reading Questions
- What is the formula for the work done by a constant force \(F\) over a distance \(d\)?
- How is work calculated when the force, \(F(x)\), is not constant over the interval from \(x=a\) to \(x=b\)?
- What is Hooke's Law, and what does the spring constant \(k\) represent?
- What is the key difference between a "point mass" and a "distributed mass" in the context of work problems?
- When setting up an integral to find the work required to pump a liquid out of a tank, what three quantities must be determined for a representative "slice" of the liquid?
- Why is the force required to wind up a hanging cable considered a variable force?
- What is the difference between mass density and weight density, and which one is used to calculate the force on a slice of water in a pumping problem?
Basic Work Problems
For exercises 1 - 6, find the work done.
1) Find the work done when a constant force \( F=12\) lb moves a chair from \( x=0.9\) to \( x=1.1\) ft.
2) How much work is done when a person lifts a \( 50\) lb box of comics onto a truck that is \( 3\) ft off the ground?
- Answer
- \(W = 150\) ft-lb
3) What is the work done lifting a \( 20\) kg child from the floor to a height of \( 2\) m? (Note that \( 1\) kg equates to \( 9.8\) N)
4) Find the work done when you push a box along the floor \( 2\) m, when you apply a constant force of \( F=100\) N.
- Answer
- \(W = 200\) J
5) Compute the work done for a force \( F=\dfrac{12}{x^2}\) N from \( x=1\) to \( x=2\) m.
6) What is the work done moving a particle from \( x=0\) to \( x=1\) m if the force acting on it is \( F=3x^2\) N?
- Answer
- \(W = 1\) J
Spring Work Problems
7) A \( 12\)-in. spring is stretched to \( 15\) in. by a force of \( 75\) lb. What is the spring constant?
8) A spring has a natural length of \( 10\) cm. It takes \( 2\) J to stretch the spring to \( 15\) cm. How much work would it take to stretch the spring from \( 15\) cm to \( 20\) cm?
- Answer
- \(W = 6\) J
9) A \( 1\)-m spring requires \( 10\) J to stretch the spring to \( 1.1\) m. How much work would it take to stretch the spring from \( 1\) m to \( 1.2\) m?
10) A spring requires \( 5\) J to stretch the spring from \( 8\) cm to \( 12\) cm, and an additional \( 4\) J to stretch the spring from \( 12\) cm to \( 14\) cm. What is the natural length of the spring?
- Answer
- The natural length is\( 5\) cm.
11) A shock absorber is compressed 1 in. by a weight of 1 ton. What is the spring constant?
12) A force of \( F=\left(20x−x^3\right)\) N stretches a nonlinear spring by \( x\) meters. What work is required to stretch the spring from \( x=0\) to \( x=2\) m?
- Answer
- \(W = 36\) J
Cable and Chain Work Problems
13) Find the work done by winding up a hanging cable of length \( 100\) ft and weight density \( 5\) lb/ft.
14) For the cable in the preceding exercise, how much work is done to lift the cable \( 50\) ft?
- Answer
- \(W = 18,750\) ft-lb
15) For the cable in the preceding exercise, how much additional work is done by hanging a \( 200\) lb weight at the end of the cable?
Pyramid & Satellite/Rocket Work Problems
16) [Technology Required] A pyramid of height \( 500\) ft has a square base \( 800\) ft by \( 800\) ft. Find the area \( A\) at height \( h\). If the rock used to build the pyramid weighs approximately \( w=100\,\text{lb/ft}^3\), how much work did it take to lift all the rock?
- Answer
- \(W= \frac{32}{3} \times 10^9\) ft-lb
17) [Technology Required] For the pyramid in the preceding exercise, assume there were \( 1000\) workers each working \( 10\) hours a day, \( 5\) days a week, \( 50\) weeks a year. If each of the workers, on average, lifted ten 100-lb rocks \( 2\) ft/hr, how long did it take to build the pyramid?
18) [Technology Required] The force of gravity on a mass \( m\) is \( F=−((GMm)/x^2)\) newtons. For a rocket of mass \( m=1000\) kg, compute the work to lift the rocket from \( x=6400\) to \( x=6500\) km. (Note: \( G=6 \times 10^{−17}\,\text{N m}^2/\text{kg}^2\) and \( M=6 \times 10^{24}\) kg.)
- Answer
- \(W = 8.65 \times 10^5\) J
19) [Technology Required] For the rocket in the preceding exercise, find the work to lift the rocket from \( x=6400\) to \( x= \infty \).
Pumping Work Problems
20) [Technology Required] Find the work required to pump all the water out of a cylinder that has a circular base of radius \( 5\)ft and height \( 200\) ft. Use the fact that the weight density of water is \( 62\) lb/ft3.
21) [Technology Required] Find the work required to pump all the water out of the cylinder in the preceding exercise if the cylinder is only half full.
- Answer
- \(W = 23.25 \pi \) million ft-lb
22) [Technology Required] How much work is required to pump out a swimming pool if the area of the base is \(800 \, \text{ft}^2\), the water is \(4\) ft deep, and the top is \(1\) ft above the water level? Assume that the weight density of water is \( 62\) lb/ft3.
23) A cylinder of depth \(H\) and cross-sectional area \(A\) stands full of water at density \( \rho \). Compute the work to pump all the water to the top.
- Answer
- \(W = \dfrac{A \rho H^2}{2}\)
24) For the cylinder in the preceding exercise, compute the work to pump all the water to the top if the cylinder is only half full.
25) A cone-shaped tank has a cross-sectional area that increases with its depth: \( A=\dfrac{ \pi r^2h^2}{H^3}\). Show that the work to empty it is half the work for a cylinder with the same height and base.
- Answer
- Answers may vary.
