# Section 3.5 Calculus 2

Work

## 3.5 Work

### 3.5.1 Work by a Constant Force

• In physics, the work $$W$$ done by a force of constant magnitude $$F$$ over a displacement $$d$$ by the formula $$W=Fd$$.
• Example Calculate the work done by a crane in lifting a $$3000$$ pound wrecking ball $$25$$ feet.
• Example Estimate the work done in lifting a leaky bucket of water $$1$$ meter off the ground if it weighs approximately $$4$$ newtons on the ground, $$3.8$$ newtons at $$25$$ cm, $$3.5$$ newtons at $$50$$ cm, and $$2.3$$ newtons at $$75$$ cm.

### 3.5.2 Work by a Variable Force

• If the force $$F(x)$$ acting on an object varies with respect to the position $$x$$ of the object, then work done in moving the object from $$a$$ to $$b$$ is defined by $$W=\int_a^b F(x)\,dx$$.
• Example Find the work done in lifting a leaky bucket of water $$1$$ meter off the ground if it weighs $$4-3x^2$$ newtons when it is $$x$$ meters above the ground.
• Example How much work is done in pulling up $$20$$ feet of hanging chain if it weighs $$1$$ pound per $$4$$ feet?
• Example Hooke’s Law states that the force required to hold a stretched or compressed spring is directly proportional to its natural length. That is, $$F(x)=kx$$ where $$x$$ is the difference between the spring’s natural length and its current length. If a spring has natural length $$10$$ inches, and it requires $$15$$ pounds of force to hold the spring at $$13$$ inches, how much work is required to stretch the spring an additional $$2$$ inches?

### 3.5.3 Work and Pumping Liquid

• To compute the work in pumping liquid, we proceed by computing a work differential $$dW$$ for each infintesimal cross-section of liquid at height $$y$$, and then evaluating $$W=\int_{y=a}^{y=b} dW$$ where $$y=a$$ is the lowest point of liquid and $$y=b$$ is the highest.
• $$dV = (\text{area})dy$$
• $$dF = (\text{density})dV$$
• $$dW = (\text{distance})dF$$
• Example Assume salt water weighs $$10,000$$ newtons per cubic meter. How much work is required to pump out a conical tank pointed downward of height $$6$$ meters and radius $$3$$ meters, if it is initially filled with $$4$$ feet of salt water?

### Review Exercises

1. Estimate the work done in pushing a plow $$6$$ meters through increasingly packed dirt; this movement requires $$1$$ newton of force at the beginning, $$5$$ newtons of force after $$2$$ meters, and $$9$$ netwons of force after $$4$$ meters.
2. Compute the exact amount of work done in pushing a plow $$6$$ meters through increasingly packed dirt; this movement requires $$F(x)=1+2x$$ newtons of force after $$x$$ meters.
3. Find the work done in lifting a leaky bucket from the ground to a height of four feet, assuming it weighs $$25-x$$ pounds at $$x$$ feet above the ground.
4. A cable weighing $$4$$ pounds per foot holds a $$500$$ pound bucket of coal at the bottom of a $$300$$ foot mine shaft. Show that the total work done in lifting the cable and bucket is $$330,000$$ foot-pounds.
5. Hooke’s Law states that the force required to hold a stretched or compressed spring is directly proportional to its natural length. That is, $$F(x)=kx$$ where $$x$$ is the difference between the spring’s natural length and its current length. Show that if a spring has natural length $$20$$ cm, and it requires $$25$$ newtons of force to hold the spring at $$15$$ cm, then the work required to stretch the spring from its natural length to $$26$$ cm is $$90$$ N-cm.
6. A uniformly weighted $$100$$ foot rope weighs $$50$$ pounds. Suppose it is fully extended into a well, tied to a leaky bucket of water. This bucket weighs $$10$$ pounds and initially holds $$30$$ pounds of water, but loses $$1$$ pound of water every $$2$$ feet. Show that the work done in lifting the rope and bucket is $$4400$$ ft-lbs. (Hint: When does the bucket run out of water?)
7. Assume salt water weighs $$10$$ kilonewtons (kN) per cubic meter. A cylindrical tank with a radius of $$3$$ meters and a height of $$10$$ meters holds $$8$$ meters of salt water. Show that the work required to pump out the salt water to the top of the tank is $$4320\pi$$ kN-m (kJ).
8. Assume salt water weighs $$10000$$ newtons per cubic meter. A pyramid-shaped tank of height $$4$$ meters is pointed upward, with a square base of side length $$4$$ meters, and is completely filled with salt water. Show that the work done in completely pumping all the water in the tank up to the point of the pyramid is $$10000\int_0^4(4-y)^3\,dy$$ J.
9. Assume that a cubic inch of Juicy Juice™ weighs $$D$$ oz. Suppose a perfectly spherical coconut-shaped cup with radius $$R$$ inches is completely filled with Juicy Juice™. Show that drinking the entire beverage using a straw which extends $$S$$ inches above the top of the container requires $$\frac{4}{3}D\pi R^3(R+S)$$ inch-ounces of work.
10. What is the work required to push a heavy box $$3$$ meters over an irregular surface, assuming it requires $$F(x)=3+2x-x^2$$ newtons of force to move at $$x$$ meters?
11. What integral gives the work in ft-lbs required to pull up a hanging $$30$$-pound $$15$$-foot chain?
12. What integral gives the work in kN-m required to pump out all salt-water to the top of a cubical tank with side length $$4$$ meters, if it is initially half-full? Assume the density of salt water is $$10$$ kilonewtons per cubic meter.

Solutions 1-6

Solutions 7-12

#### Textbook References

• University Calculus: Early Transcendentals (3rd Ed)
• 6.5