Back82–84. Fluid Forces Suppose the following plates are placed on a vertical wall so that the top of the plate is 2 m below the surface of a pool that is filled with water. Compute the force on each plate.
A circular plate with a radius of 2 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?
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
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
Work from force How much work is required to move an object from x=0 to x=3 (measured in meters) in the presence of a force (in N) given by F(x)=2x acting along the x-axis?
Drinking juice A glass has circular cross sections that taper (linearly) from a radius of 5 cm at the top of the glass to a radius of 4 cm at the bottom. The glass is 15 cm high and full of orange juice. How much work is required to drink all the juice through a straw if your mouth is 5 cm above the top of the glass? Assume the density of orange juice equals the density of water.
Work
Pumping a conical tank A right-circular conical tank, point down, with top radius 5 ft and height 10 ft, is filled with a liquid whose weight-density is 60lb/ft³. How much work does it take to pump the liquid to a point 2 ft above the tank? If the pump is driven by a motor rated at 275ft-lb/sec (1/2 hp), how long will it take to empty 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.
How much work is done by a person lifting a bucket off the ground?
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.
b. How much work is required to wind the chain onto the cylinder if a 50-kg block is attached to the end of the chain?
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
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?
A swimming pool has the shape of a rectangular prism with abase that measures 30 by 20 and is 5 deep. The top of the pool is 1 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 /.
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?
Compressing and stretching a spring Suppose a force of 30 N is required to stretch and hold a spring 0.2 m from its equilibrium position.
c. How much work is required to stretch the spring 0.3 m from its equilibrium position?