10. Physics Applications of Integrals

13–20. Mass of one-dimensional objects Find the mass of the following thin bars with the given density function.

ρ(x) = {1 if 0≤x≤2 {2 if 2<x≤3

Compressing and stretching a spring Suppose a force of 15 N is required to stretch and hold a spring 0.25 m from its equilibrium position.

b. How much work is required to compress the spring 0.2 m 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?

Calculating work for different springs Calculate the work required to stretch the following springs 0.4 m from their equilibrium positions. Assume Hooke’s law is obeyed.

b. A spring that requires 2 J of work to be stretched 0.1 m from its equilibrium position

Calculating work for different springs Calculate the work required to stretch the following springs 1.25 m from their equilibrium positions. Assume Hooke’s law is obeyed.

a. A spring that requires 100 J of work to be stretched 0.5 m from its equilibrium position

Winding a chain A 30-m-long chain hangs vertically from a cylinder attached to a winch. Assume there is no friction in the system and the chain has a density of 5kg/m.

a. How much work is required to wind the entire chain onto the cylinder using the winch?

13–20. Mass of one-dimensional objects Find the mass of the following thin bars with the given density function.

ρ(x)=1+sin x, for 0≤x≤π

13–20. Mass of one-dimensional objects Find the mass of the following thin bars with the given density function.

ρ(x) = 5e^-2x,for 0≤x≤4

13–20. Mass of one-dimensional objects Find the mass of the following thin bars with the given density function.

ρ(x) = {x² if 0≤x≤1 {x(2-x) if 1<x≤2

52–54. Force on a window A diving pool that is 4 m deep and full of water has a viewing window on one of its vertical walls. Find the force on the following windows. 

The window is a square, 0.5 m on a side, with the lower edge of the window on the bottom of the pool.

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

c. The work required to lift a 10-kg object vertically 10 m is the same as the work required to lift a 20-kg object vertically 5 m.

Spring work

b. It takes 50 N of force to stretch a spring 0.2 m from its equilibrium position. How much work is needed to stretch it an additional 0.5 m?

Mass of two bars Two bars of length L have densities ρ₁(x) = 4e^−x and ρ₂(x) = 6e^−2x, for 0≤x≤L.

a. For what values of L is bar 1 heavier than bar 2?

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?

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).

b. Is it true that it takes half as much work to pump the water out of the tank when it is filled to half its depth as when it is full? Explain.