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Lectures in Calculus, Lecture notes of Calculus

Calculus is the mathematical study of change, much as geometry is the study of shape and algebra is the study of operations and how they are applied to solve problems.

Typology: Lecture notes

2021/2022

Uploaded on 04/24/2023

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Work
The concept of work is important to scientists and engineers for
determining the energy needed to perform various jobs. For instance, it
is useful to know the amount of work done when a crane lifts a steel
girder, when a spring is compressed, when a rocket is propelled into the
air, or when a truck pulls a load along a highway.
In general, work is done by a force when it moves an object. If the force
applied to the object is constant, then the definition of work is as
follows:
DEFINITION OF WORK DONE BY A CONSTANT FORCE
If an object is moved a distance D in the direction of an applied constant
force F, then the work W done by the force is defined as W = FD.
There are many types of forces centrifugal, electromotive, and
gravitational, to name a few. A force can be thought of as a push or pull;
a force changes the state of rest or state of motion of a body. For
gravitational forces on Earth, it is common to use units of measure
corresponding to the weight of an object.
Example 1. Lifting an Object
1. Determine the work done in lifting a 50-pound object 4 feet.
W = FD
= 50 (4)
W = 200 foot pounds
In the US Measurement system, work is typically expressed in foot-
pounds (ft lb), inch pounds, or foot tons. In the centimeter gram
second (C-G-S) system, the basic unit of force is dyne- the force
required to produce an acceleration of 1 centimeter per second per
second on a mass of 1 gram. In this system, work is typically expressed
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Work

The concept of work is important to scientists and engineers for determining the energy needed to perform various jobs. For instance, it is useful to know the amount of work done when a crane lifts a steel girder, when a spring is compressed, when a rocket is propelled into the air, or when a truck pulls a load along a highway. In general, work is done by a force when it moves an object. If the force applied to the object is constant, then the definition of work is as follows: DEFINITION OF WORK DONE BY A CONSTANT FORCE If an object is moved a distance D in the direction of an applied constant force F, then the work W done by the force is defined as W = FD. There are many types of forces – centrifugal, electromotive, and gravitational, to name a few. A force can be thought of as a push or pull ; a force changes the state of rest or state of motion of a body. For gravitational forces on Earth, it is common to use units of measure corresponding to the weight of an object. Example 1. Lifting an Object

  1. Determine the work done in lifting a 50-pound object 4 feet. W = FD = 50 (4) W = 200 foot – pounds In the US Measurement system, work is typically expressed in foot- pounds (ft – lb), inch – pounds, or foot – tons. In the centimeter – gram
  • second (C-G-S) system, the basic unit of force is dyne- the force required to produce an acceleration of 1 centimeter per second per second on a mass of 1 gram. In this system, work is typically expressed

in dyne – centimeters (ergs) or newton – meter (joules), where 1 joule = 𝟏𝟎𝟕^ ergs.

  1. A force of 112 Newton is required to slide a cement block 8 m in a construction project. Ans. 896 N-m If a variable force is applied to an object, calculus is needed to determine the work done, because the amount of force changes as the object changes position. DEFINITION OF WORK DONE BY A VARIABLE FORCE If an object is moved along a straight line by a continuously varying force F(x), then the work W done by the force as the object is moved from x = a to x = b is W = 𝑭 𝒙 𝒅𝒙 𝒃 𝒂 Example 3. Total work done in moving an object along an axis Suppose an object is moved along the x- axis from x = 1 cm to x = 5 cm, and suppose that the force exerted obeys the law F(x) = 𝒙𝟐^ + 𝒙 𝒅𝒚𝒏𝒆𝒔. Find the total work done. Ans. W = 𝟏𝟔𝟎 𝟑

Spring Related Problem

One of the simplest notions of work deals with the problems of extension or compression of elastic materials such as a rubber band, a helical spring, or an elastic cord.

container ( a cylindrical tank, a conical vessel, or a spherical tank, etc.) to the top of the container or to any point above its top.

𝒃 𝒂 where w is the weight density (mass per unit volume of the liquid) h is the height dV is the differential of the Volume

  1. How much work is required to pump all the water from a cylindrical tank 9 ft. tall and 8 ft. in diameter, if it is emptied at a point 1 ft. above the top of the tank? (use the fact that water weighs 62. 𝒍𝒃 𝒇𝒕𝟑 𝑨𝒏𝒔. 𝑾 = 𝟒𝟗, 𝟓𝟎𝟎 𝒇𝒕 − 𝒍𝒃𝒔.
  2. A spherical tank of radius 3 m has a spout on the very top that is 1 m tall. The tank is full of water. Find the work required to pump the water out of the spout. Ans. 𝑾 = 𝟏, 𝟒𝟏𝟏, 𝟐𝟎𝟎 𝝅 J or = 4, 433, 415.553 J
  3. Find the work done in pumping all the water out of a conical tank up to a point 2 meters above the top of the tank. The tank has height 4 m and radius 2 m above the top. 𝑨𝒏𝒔. 𝑾 = 𝟒𝟗𝟐, 𝟔𝟎𝟏. 𝟕𝟐𝟖𝟏 𝑱

Seatwork.

1. A 100 – pound bag of sugar is lifted 20 ft.

Determine the work done by the constant force.

2. Find the work done in lifting a rock of mass 8 kg a

distance of 4 m.

𝒎 𝒔𝟐

3. A particle is moving along the x- axis under

the action of a force of 𝒇(𝒙) pounds when the

particle is 𝒙 feet from the origin.

If 𝒇 𝒙 = 𝒙

𝟐

+ 4 pounds, find the work done as

the particle moves from the point where

𝒙 = 𝟐 and 𝒙 = 𝟒.

𝟐 𝟑

ft. – lb.