10. Physics Applications of Integrals

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

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?

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 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?

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?

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 $40m$ 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?

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

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 $x=2m$?

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.

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

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

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