10. Physics Applications of Integrals

How much work is required to push a chair across the floor with a force of $F=8\mathrm{lb}$ from $x=6\mathrm{ft}$ to $x=9\mathrm{ft}$ along the $x$-axis?

How much work is done by a person lifting a $10\mathrm{lb}$ bucket $4\mathrm{ft}$ off the ground?

How much work is required to move an object with a force of $F\left(x\right)=4x^{3}N$ acting along the $x$-axis from $x=0m$ to $x=2m$?

Compute the work done by a force of $F=\frac{3}{x^2}N$ from $x=2m$ to $x=6m$.

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 $\mathrm{lb}$/$f{t}^3$.