BackFilling 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 water trough A water trough has a semicircular cross section with a radius of 0.25 m and a length of 3 m (see figure).
a. How much work is required to pump the water out of the trough (to the level of the top of the trough) when it is full?
Emptying a water trough A water trough has a semicircular cross section with a radius of 0.25 m and a length of 3 m (see figure).
b. If the length is doubled, is the required work doubled? Explain.
Emptying a water trough A water trough has a semicircular cross section with a radius of 0.25 m and a length of 3 m (see figure).
c. If the radius is doubled, is the required work doubled? Explain.
46–50. Force on dams The following figures show the shapes and dimensions of small dams. Assuming the water level is at the top of the dam, find the total force on the face of the dam.
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.
b. How much work is done in stretching the spring from its equilibrium position (x=0) to x=1.5?
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?
Lifting problem A 4-kg mass is attached to the bottom of a 5-m, 15-kg chain. If the chain hangs from a platform, how much work is required to pull the chain and the mass onto the platform?
Work in a gravitational field For large distances from the surface of Earth, the gravitational force is given by F(x) = GMm / (x+R)², where G = 6.7×10^−11 N m²/kg² is the gravitational constant, M = 6×10^24 kg is the mass of Earth, m is the mass of the object in the gravitational field, R = 6.378×10⁶ m is the radius of Earth, and x≥0 is the distance above the surface of Earth (in meters).
a. How much work is required to launch a rocket with a mass of 500 kg in a vertical flight path to a height of 2500 km (from Earth’s surface)?
70–72. Variable density in one dimension Find the mass of the following thin bars.
A bar on the interval 0≤x≤6 with a density ρ(x) = {1 if 0 ≤ x < 2
2 if 2 ≤ x < 4
4 if 4 ≤ x ≤ 6
Pumping water A water tank has the shape of a box that is 2 m wide, 4 m long, and 6 m high.
b. If the water in the tank is 2 m deep, how much work is required to pump the water to a level of 1 m above the top of the tank?
46–50. Force on dams The following figures show the shapes and dimensions of small dams. Assuming the water level is at the top of the dam, find the total force on the face of the dam.
Force on a triangular plate A plate shaped like an isosceles triangle with a height of 1 m is placed on a vertical wall 1 m below the surface of a pool filled with water (see figure). Compute the force on the plate.
What is the pressure on a horizontal surface with an area of 2 m² that is 4 m underwater?
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?