10. Physics Applications of Integrals

A spring requires $12J$ of work to stretch the spring from $1.1m$ to $1.4m$ past its equilibrium. What is the spring constant?

A spring requires a force of $8N$ to stretch the spring to $5\mathrm{cm}$ past its equilibrium point. How much work could it take to stretch the spring from $2\mathrm{cm}$ to $9\mathrm{cm}$ past equilibrium?

Suppose a force of $10N$ is required to stretch a spring $0.5m$ from its equilibrium position. How much work is required to compress the spring $0.2m$ from its equilibrium position?

A $120m$ chain hangs freely from the side of a building. The chain weighs $15\mathrm{kg}$/$m$. How much work is done to pull $80m$ of the chain to the top of the building?

Find the work done by fully winding up a cable of length $30\mathrm{ft}$ and weight-density $2\mathrm{lb}$/$\mathrm{ft}$.

A $40m$ rope hangs freely over a ledge. The density of the rope is $12\mathrm{kg}$/$m$. If a $5\mathrm{kg}$ bucket is attached to the end of the rope, how much work is done to pull the rope and the bucket to the ledge?

A $60\mathrm{ft}$ cable is attached to a cylinder that is attached to a winch. If the cable weighs 300 lbs, how much work is needed to wind $20\mathrm{ft}$ of the cable onto the cylinder using the winch? Hint: Divide cable weight by cable length to get density.

A water trough for horses has a triangular cross section with a height of $3m$ and horizontal side lengths of $2m$. The length of the trough is $12m$. How much work is required to pump the water to the top of the trough when it is half full.

A swimming pool has the shape of a rectangular prism with abase that measures 30 $\mathrm{ft}$ by 20 $\mathrm{ft}$ and is 5 $\mathrm{ft}$ deep. The top of the pool is 1 $\mathrm{ft}$ above the surface of the water. How much work is required to pump all the water out? Assume the density of water is 62.4 $\mathrm{lb}$/$f{t}^3$.

90. Work Let R be the region in the first quadrant bounded by the curve y = √(x⁴ - 4)

and the lines y = 0 and y = 2. Suppose a tank that is full of water has the shape of a solid of revolution obtained by revolving region R about the y-axis. How much work is required to pump all the water to the top of the tank? Assume x and y are in meters.

Explain how to find the mass of a one-dimensional object with a variable density ρ.

Why is integration used to find the work required to pump water out of a tank?

9–12. Consider the cylindrical tank in Example 4 that has a height of 10 m and a radius of 5 m. Recall that if the tank is full of water, then ∫₀¹⁰ 25 π ρg(15−y) dy equals the work required to pump all the water out of the tank, through an outflow pipe that is 15 m above the bottom of the tank. Revise this work integral for the following scenarios. (Do not evaluate the integrals.)

The work required to empty the top half of the tank

9–12. Consider the cylindrical tank in Example 4 that has a height of 10 m and a radius of 5 m. Recall that if the tank is full of water, then ∫₀¹⁰ 25 π ρg(15−y) dy equals the work required to pump all the water out of the tank, through an outflow pipe that is 15 m above the bottom of the tank. Revise this work integral for the following scenarios. (Do not evaluate the integrals.)

The work required to empty the tank through an outflow pipe at the top of the tank

Emptying a cylindrical tank A cylindrical water tank has height 8 m and radius 2m (see figure).

a. If the tank is full of water, how much work is required to pump the water to the level of the top of the tank and out of the tank?