10. Physics Applications of Integrals

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

Emptying a conical tank A water tank is shaped like an inverted cone with height 6 m and base radius 1.5 m (see figure).

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

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?

Work

Assume that a spring does not follow Hooke’s Law. Instead, the force required to stretch the spring x ft from its natural length is ƒ(𝓍) = 10𝓍³/² lb . How much work does it take to

a. stretch the spring 4 ft from its natural length?

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.

Orientation and force A plate shaped like an equilateral triangle 1 m on a side is placed on a vertical wall 1 m below the surface of a pool filled with water. On which plate in the figure is the force greater? Try to anticipate the answer and then compute the force on each plate.

Calculating work for different springs Calculate the work required to stretch the following springs 0.5m from their equilibrium positions. Assume Hooke’s law is obeyed.

a. A spring that requires a force of 50 N to be stretched 0.2 m from its equilibrium position

A nonlinear spring Hooke’s law is applicable to idealized (linear) springs that are not stretched or compressed too far from their equilibrium positions. Consider a nonlinear spring whose restoring force is given by F(x) = 16x−0.1x³, for |x|≤7. 

c. How much work is done in compressing the spring from its equilibrium position (x=0) to x=−2?

Filling a spherical tank A spherical water tank with an inner radius of 8 m has its lowest point 2 m above the ground. It is filled by a pipe that feeds the tank at its lowest point (see figure). Neglecting the volume of the inflow pipe, how much work is required to fill the tank if it is initially empty?

Emptying a partially filled swimming pool If the water in the swimming pool in Exercise 35 is 2 m deep, then how much work is required to pump all the water to a level 3 m above the bottom of the pool?

Force on the end of a tank Determine the force on a circular end of the tank in Figure 6.78 if the tank is full of gasoline. The density of gasoline is ρ = 737 kg/m³.

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?

Work done by a spring A spring on a horizontal surface can be stretched and held 0.5 m from its equilibrium position with a force of 50 N.

b. How much work is done in compressing the spring 0.5 m from its equilibrium position?