- Calculate the mass of linear and circular objects using their density distributions
- Compute the work required in various situations, such as pumping fluids or moving objects along a path
- Determine the force exerted by water on a vertical surface underwater
Mass and Density
The Main Idea
- Mass-Density Formula for One-Dimensional Objects:
- For a thin rod along the -axis from to :
- Mass-Density Formula for Two-Dimensional Disks:
- For a disk of radius with radial density:
- Concept of Density Functions:
- Linear density: mass per unit length
- Area density: mass per unit area
- Radial density: density varying along the radius
Problem-Solving Strategies
- One-Dimensional Case:
- Partition the rod into segments
- Approximate mass of each segment:
- Sum and take the limit as approaches infinity
- Two-Dimensional Case:
- Partition the disk into concentric washers
- Approximate area of each washer:
- Approximate mass of each washer:
- Sum and take the limit as approaches infinity
Find the mass of a rod with density over the interval .
Consider a thin rod oriented on the -axis over the interval If the density of the rod is given by what is the mass of the rod?
Calculate the mass of a disk with radius and radial density .
Let represent the radial density of a disk. Calculate the mass of a disk of radius .
Work Done by a Force
The Main Idea
- Definition of Work:
- Work is the energy transfer when a force moves an object
- Units: Joules (J); Force in Newtons (N), Distance in meters (m)
- Work Formula for Constant Force: (Force × Distance)
- Work Formula for Variable Force:
- Application to Springs:
- Hooke’s Law:
- is the spring constant (N/m)
Problem-Solving Strategy
- Partition the interval into segments
- Approximate work in each segment:
- Sum the work over all segments:
- Take the limit as to get the exact work:
Suppose it takes a force of lb to stretch a spring in. from the equilibrium position. How much work is done to stretch the spring ft from the equilibrium position?
A spring requires a N force to compress it m. How much work is done to stretch it m?
Work Done in Pumping
The Main Idea
- Work in Pumping:
- Calculating work required to pump water out of tanks
- Varies based on tank shape and water level
- Key Principle:
- Work = Force (Weight of water) × Distance (Height lifted)
- General Work Formula: , where depends on tank shape
- Weight-Density of Water:
- N/m³ (metric) or lb/ft³ (imperial)
Problem-Solving Strategy
- Sketch the tank and choose a coordinate system
- Calculate volume of a representative water layer
- Determine force using weight-density
- Calculate lifting distance for the layer
- Estimate work for the layer: Force × Distance
- Sum work for all layers (Riemann sum)
- Take limit and evaluate integral for exact work
Calculate work to pump water from a full inverted conical tank (height ft, base radius ft) until ft of water remains.
A tank is in the shape of an inverted cone, with height ft and base radius ft. The tank is filled to a depth of ft to start with, and water is pumped over the upper edge of the tank until ft of water remain in the tank. How much work is required to pump out that amount of water?
Hydrostatic Force and Pressure
The Main Idea
- Hydrostatic Pressure:
- Pressure exerted by a fluid at rest due to its weight
- , where is weight density and is depth
- Force on Horizontal Surface:
- , where is area
- Force on Vertical or Angled Surface:
- : width function
- : depth function
- Pascal’s Principle:
- Pressure at a given depth is the same in all directions
Problem-Solving Strategy
- Sketch the situation and choose a coordinate system
- Determine depth and width functions
- Identify the weight density of the liquid
- Set up and evaluate the force integral:
Calculate the force on one end of a water trough ft long, with ends shaped like inverted isosceles triangles (base ft, height ft).
When the reservoir is at its average level, the surface of the water is about 50 ft below where it would be if the reservoir were full. What is the force on the face of the dam under these circumstances?