Work: Videos & Practice Problems

Let $Q$ be the region in the first quadrant enclosed by $y=\sqrt{x^4-16}$, $y=0$, and $y=4$. This region is revolved around the $y$-axis to form a tank, which is then completely filled with water. How much work is required to pump all the water up to the top of the tank? Assume that the variables $x$ and $y$ are measured in meters. Use ${\rho }_{\text{water}}=1000\frac{\text{kg}}{{\text{m}}^3}$ and $g=9.8\frac{\text{m}}{{\text{s}}^2}$.

A wire stretches from $x=0$ to $x=6$ meters and has a density function $\rho \left(x\right)=3x^2$ $\text{kg/m}$. What is the mass of the wire?

When calculating the work needed to pump oil from the bottom of a cylindrical tank to a point above the tank, why is the process of integration used?

A cylindrical tank has a height of $12$ m and a radius of $4$ m. If the tank is full of water, the work required to pump all the water to a point $16$ m above the bottom is given by ${\int }_0^{12}16\pi \rho g(16-y)dy$. Write the work integral needed to empty the top half of the tank (do not evaluate the integral).

A cylindrical tank has a height of $12$ m and a radius of $4$ m. If the tank is full of water, write the integral (do not evaluate) that represents the work required to pump all the water out through an outflow pipe located at the very top of the tank ($12$ m above the bottom).

A cylindrical tank with a height of $6$ $\text{m}$ and a radius of $1.5$ $\text{m}$ is filled with water (density $\rho =1000$ ${\text{kg/m}}^3$). How much work is required to pump all the water to the top of the tank and out? (Take $g=9.8$ ${\text{m/s}}^2$).

A vertical cylindrical vessel with a height of $6$ m and radius of $1.5$ m is filled with water. Does pumping out all of the water when the vessel is half full require exactly half the work compared to when the vessel is full?

Find the work done in moving an object from $x=0$ to $x=5$ meters if the force acting on it is $F\left(x\right)=x$ newtons along the $x$-axis.

Calculate the work required to move an object from $x=0$ to $x=4$ meters if the force acting on it along the $x$-axis is $F\left(x\right)=4x^2$ newtons.

If it takes $25\text{ N}$ to hold a spring stretched $0.1\text{ m}$ beyond its equilibrium, how much work is required to stretch the spring $0.2\text{ m}$ from equilibrium?

It takes a force of $24\text{ N}$ to stretch a spring $0.12\text{ m}$ from its natural length. What is the work required to stretch the spring from $0.12\text{ m}$ to $0.24\text{ m}$ beyond equilibrium?

A wire has a density (mass per unit length) function . Calculate the total mass of the wire from $x=0$ to $x=4$.

A force of $24\text{ N}$ is needed to stretch a spring $0.3\text{ m}$ from its natural length. How much work is required to compress the spring $0.15\text{ m}$ from equilibrium?

A spring requires a force of $60\text{ N}$ to stretch it $0.4\text{ m}$ from its equilibrium position. How much work is required to compress the spring $0.4\text{ m}$ from equilibrium? Use calculus in your solution.

A force of $90\text{ N}$ is needed to stretch a spring $0.45\text{ m}$ from equilibrium. Find the work required to stretch the spring $0.9\text{ m}$ from its equilibrium position. Solve using calculus.

It takes $4\text{ J}$ of work to stretch a spring $0.15\text{ m}$ beyond its equilibrium. How much work is needed to stretch it $0.6\text{ m}$ from equilibrium? Use calculus.

A force of $60\text{ J}$ is needed to stretch a spring $0.4\text{ m}$ beyond its equilibrium. How much work is required to stretch the spring $0.8\text{ m}$ from equilibrium? Use calculus.

A $40$-$\text{m}$ long cable hangs vertically from a platform. The cable has a linear mass density of $4\text{ kg/m}$. How much work is required to wind the entire cable onto the platform, assuming the acceleration due to gravity is $9.8{\text{ m/s}}^2$ and there is no friction?

Find the mass of a thin rod with the density (mass per unit length) function $\rho \left(x\right)=2+\cos x$ for $0\le x\le \frac{\pi }{2}$.

The density (mass per unit length) of a bar is given by $\rho \left(x\right)=6e^{-3x}$ for $0\le x\le 2$. Calculate the mass of the bar.

A bar has density (mass per unit length) . Find the mass of the bar with length $x=3$ units.

A water tank is $5\text{ m}$ deep and has a rectangular observation window $0.7\text{ m}$ wide and $0.4\text{ m}$ high, with its bottom edge $1\text{ m}$ above the base of the tank. Find the total force exerted by the water on the window. Take $\rho =1000{\text{kg/m}}^3$ and $g=9.8{\text{m/s}}^2$.

Is the work required to lift a $12$$\text{-kg}$ object $8\text{ m}$ vertically the same as the work required to lift a $24$$\text{-kg}$ object $4\text{ m}$ vertically? Use calculus to justify your answer.

A spring requires a force of $80\text{ N}$ to be stretched $0.4\text{ m}$ from equilibrium. How much work is done in stretching the spring from $0.4\text{ m}$ to $1.0\text{ m}$?

Let two beams of length $\text{M}$ have linear densities (in units of mass per unit length) of ${\lambda }_1\left(x\right)=7e^{-x}$ and ${\lambda }_2\left(x\right)=10e^{-2x}$ for $0\le x\le M$. For which values of $\text{M}$ is beam 1 heavier than beam 2?

A certain nonlinear spring has a restoring force $F\left(x\right)=10x-0.05x^3$ for $\mid x\mid \le 5$. Calculate the work required to stretch the spring from $x=0$ to $x=3$. Round your answer to the nearest whole number.

A nonlinear spring has a restoring force described by $F\left(x\right)=10x-0.05x^3$ for $\mid x\mid \le 5$. How much work is done in compressing the spring from $x=0$ to $x=-3$? Round your answer to the nearest whole number.

A $6$$\text{-kg}$ object is attached to the end of an $8\text{-m}$, $12\text{-kg}$ rope hanging vertically from a ledge. How much work is needed to pull both the rope and the object up onto the ledge?

An aquarium shaped like an inverted cone has a height of $8$ $\text{m}$ and a base radius of $2$ $\text{m}$. If the aquarium is full of water, how much work is required to pump all the water to the top of the aquarium and out? (Use $1000{\text{kg/m}}^3$ for water density and $g=9.8{\text{m/s}}^2$).

An aquarium shaped like an inverted cone has a height of $8\text{ m}$ and a base radius of $2\text{ m}$. Does pumping all the water to the top of the aquarium and out when it is half full require exactly half the work as when it is full?