## Work Done By Forces That Vary

In general, forces may vary in magnitude and direction at points in space, and paths between two points may be curved. The infinitesimal work done by a variable force can be expressed in terms of the components of the force and the displacement along the path,

Here, the components of the force are functions of position along the path, and the displacements depend on the equations of the path. Equation \ref defines the total work as a line integral, or the limit of a sum of infinitesimal amounts of work. The physical concept of work is straightforward: you calculate the work for tiny displacements and add them up. Sometimes the mathematics can seem complicated, but the following example demonstrates how cleanly they can operate.

##### Example \: Work Done by a Variable Force over a Curved Path

An object moves along a parabolic path y = x2 from the origin A = to the point B = under the action of a force \ = y \ + x \ ). Calculate the work done.

**Strategy**

The components of the force are given functions of x and y. We can use the equation of the path to express y and dy in terms of x and dx namely,

Then, the integral for the work is just a definite integral of a function of x.

###### Solution

The infinitesimal element of work is

The integral of x2 is \, so

**Significance**

##### Exercise \

Find the work done by the same force in Example \ over a cubic path, y = x3, between the same points A = and B = .

The components of the force, in terms of y, are

**Strategy**

###### Solution

**Significance**

## Work Definition In Physics And Work Formula

In our everyday life, work is any kind of effort, job, or action that we perform. Work in physics, however, has a very strict and clear definition. As it happens normally, **the work definition in physics is almost the same as the work equation**.

In physics, to have work we need an object to move as a result of a force applied to it. The work equation is:

W = F * d

- m is the mass of the object and
- a the acceleration it experiences.

Remember that when the object is being lifted at a constant speed , the acceleration on the object is *g* .

Work and power are two related concepts, you can **calculate the power** from the work done if you know the time that it took to do such work. Simply use the equation:

P = W / t

## What Is Maent By Work

Work is a type of effort that we perform everyday. In physics, work is defined as the force that requires to occur a displacement or movement in the object. We can also say it as the product of displacement and force acting on the body. The SI unit of work is Joule .

The formulas to calculate the work are along the lines

w = f * d

v0 is the initial velocity

v1 is the final velocity

m is the mass of the object

w is the work

**Example**

**Question: A cycle having a mass of 50 kgs moves with an initial velocity of 10 m/s and reaches the destination with a final velocity of 15 m/s. Find the work done by the cycle?**

**Solution:**

Mass of the cycle m = 50 kgs

Initial velocity v0 = 10 m/s

Final velocity v1 = 15 m/s

Work = ?

Work formula is w = *

Substitute the given values

w = *

= 25 *

= 2875

Therefore, the work is 2875 J

Perform and understand physics concepts within seconds by taking the help of online calculator tools provided at Physicscalc.Com

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## The Angle Between The Force Vector And The Displacement Vector

The work done by a force on an object can be positive, negative, or zero, depending upon the direction of displacement of the object with respect to the force. For an object moving in the opposite direction to the direction of force, such as friction acting on an object moving in the forward direction, the work is done due to the force of friction is negative.

Similarly, an object experiences a zero force when the displacement angle is perpendicular to the direction of the force. Consider an example of a coolie lifting a mass on his head moving at an angle of 90 with respect to the force of gravity. Here, the work done by gravity on the object is zero.

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## Work Done By Constant Forces And Contact Forces

The simplest work to evaluate is that done by a force that is constant in magnitude and direction. In this case, we can factor out the force the remaining integral is just the total displacement, which only depends on the end points A and B, but not on the path between them:

We can also see this by writing out Equation \ref in Cartesian coordinates and using the fact that the components of the force are constant:

Figure \ shows a person exerting a constant force \ along the handle of a lawn mower, which makes an angle \ with the horizontal. The horizontal displacement of the lawn mower, over which the force acts, is \. The work done on the lawn mower is

which the figure also illustrates as the horizontal component of the force times the magnitude of the displacement.

Figure \ shows a person holding a briefcase. The person must exert an upward force, equal in magnitude to the weight of the briefcase, but this force does no work, because the displacement over which it acts is zero. So why do you eventually feel tired just holding the briefcase, if youâre not doing any work on it? The answer is that muscle fibers in your arm are contracting and doing work inside your arm, even though the force your muscles exert externally on the briefcase doesnât do any work on it.

##### Example \: Calculating the Work You Do to Push a Lawn Mower

**Strategy**

###### Solution

The equation for the work is

Substituting the known values gives

**Significance**

##### Example \: Moving a Couch

**Strategy**

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## What Is The Formula For Work

**Definition:** In our daily life work implies an activity resulting in muscular or mental exertion. However, in physics the term work is used in a specific sense involves the displacement of a particle or body under the action of a force. work is said to be done when the point of application of a force moves. Work done in moving a body is equal to the product of force exerted on the body and the distance moved by the body in the direction of force.**Work = Force × Distance moved in the direction of force**.

**The work done by a force on a body depends on two factors** Magnitude of the force, and Distance through which the body moves

**Unit of Work** When a force of 1 newton moves a body through a distance of 1 metre in its own direction, then the work done is known as 1 joule. Work = Force × Displacement 1 joule = 1 N × 1 m or 1 J = 1 Nm

## Work Energy And Power

Work, Energy and Power are fundamental concepts of Physics. Work is said to be done when a force applied to an object causes a displacement of the object. We define the capacity to do the work as energy. Power is the work done per unit of time. This article discusses work, energy and power in detail.

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## Examples Using Formula For Work

**Example 1:** 10 Newton of force is applied on a body which displaces it by 2 meters. Calculate the work done by using the formula for work.**Solution: **

= 20Nm

**Answer: 20 Nm of work is done when a 10 Newton of force displaces an object by 2 meters.**

**Example 2: **A coolie at a railway station carries a bag weighing 100 N through some distance. Calculate the work done by the coolie on the bag by using the formula for work.**Solution: **

To find work done by the coolie.

Given: Weight of the bag = 100N

Also, the weight of the bag will be acting in the vertical direction and its motion is in the horizontal direction. So the displacement of the bag in the direction of the force is zero.

d = 0

= 0 J

**Answer: Work done by the coolie on the bag is zero.**

**Example 3: **Calculate the amount of work done by the force in moving the object through a distance of 7 m if an object is horizontally dragged across the surface by a 150 N force acting parallel to the surface.**Solution: **

To find: Work done by the force in moving the object through a distance of 7 m

Given: F = 150 N, d = 7 m

Since F and d are in the same direction,

= 0,

W = F × Cos × d

= 150× 7 × Cos 0

= 1050 J

**Answer: The amount of work done by the force in moving the object is 800 J.**

## Work Done By A Varying Force In One Dimension

If you are not familiar with integrals please follow this link before gong on.

The work done by a varying force F with only an x-component is defined as W = xixfFdx = limx–> 0xixfFx. We can plot the component of the force F acting on the object at position x versus the position x.The work done by the force is equal to the area under the curve.

A plot for a constant force acting from xito xf is shown on the right.The work done by the force is W = F.

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## What Is Work Done For The Motion Of A Block

Consider a block located on a frictionless horizontal surface. A constant force F is acted upon this block. The purpose of this force is to move the body through a certain distance in a straight path in the direction of the force.

Now, the total work done by this force is equal to the product of the magnitude of applied force and the distance traveled by the body. Scientifically Work done formula will be given as,

W = F * d

In this case, the force exerting on the block is constant, but the direction of force and direction of displacement influenced by this force is different. Here, force F reacts at an angle to the displacement d.

W = |d|

We know that work done is defined as the multiplication of magnitude of displacement d and the component of the force that is in the direction of displacement.

## How To Use The Work Calculator

The work calculator is very simple to use. In its basic form, you simply need to input the force and the distance from your scenario, and it will **automatically calculate the result** for you.

If you are an adventurous person, you can use the **advanced mode** to get to more… advanced calculations and alternative modes.

When you enter this mode, you will see 3 different sections that will allow you to calculate:

- Calculate work from mass, initial velocity, and final velocity
- Calculate work from Force and distance
- Calculate Force from mass and acceleration
- Calculate Acceleration from initial velocity, final velocity and time
- Calculate Power from work and time and…

This work calculator is smart. All you need to do is **input the values you know** and it will ~~do~~ calculate the work and all other possible values for you!

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## What Is Formula For Work

The formula for work is used to calculate the work done to displace any object. Work is the product of the force applied and displacement in the direction of the force applied. Work is the dot product of the two vectors: force and displacement. Thus work is a scalar quantity. SI unit of work is Joule .

## How To Calculate Work

In physics, ** work** is the amount of energy required to perform a given task . We start by defining the scalar product of two vectors, which is an integral part of the definition of work, and then turn to defining and using the concept of work to solve problems.

**Key Terms**

o Recognize and use the scalar product of two vectors

o Understand the concept of work in the context of physics

o Calculate the work involved in moving objects from one location to another

**Let’s Begin!**

*More About Vectors*

In some physics problems or situations, the ability to calculate the component of one vector in the direction of another is helpful. We saw some of this in our earlier study of vectors in relation to unit vectors: we are able to break down a vector such as 3**x** + 2**y** into its component parts: 3**x** and 2**y** . But what if we want to calculate the component of some vector in the direction of another arbitrary vector? To this end, we define the **scalar product** of two vectors. Given two vectors **A** = *a*1**x** + *a*2**y** and **B** = *b*1**x** + *b*2**y,** the scalar product **A** ? **B** is the following:

Note that **A** ? **B = B** ? **A.** We can factor out the magnitudes of **A** and **B** and write the scalar product in terms of these magnitudes and the scalar product of two corresponding unit vectors, **a **and **b, **which are in the directions of **A** and **B,** respectively.

Now, let’s consider these two unit vectors by way of the diagram below. Note that we have defined the angle between the vectors as *?.*

*Work*

Then,

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## Finding Work With An Angled Force

**Find the force and displacement as normal.**Above, we dealt with work problems in which the object is moving in the same direction as the force being applied to it. In reality, this isn’t always the case. In cases where the force and the object’s motion are in two different directions, the difference between these two directions must also be factored into the equation for an accurate result. To begin, find the magnitude of the force and the object’s displacement as you normally would.

**10 newtons**and that it’s moved the same

**2 meters**forward as before.

**Find the angle between the force vector and the displacement.**Unlike in the examples above, with a force that’s in a different direction than the object’s motion, it’s necessary to find the difference between these two directions in the form of the angle between them. If this information isn’t provided to you, you may need to measure it yourself or deduce it from other information in the problem.XResearch source

## Derivation Of Work Formula

Consider a block placed on a frictionless horizontal floor acted upon by a constant force F due to which this block moves through a distance d in a straight line in the direction of the force.

In general, the work done by force F is equal to the change in kinetic energy

W = mv2 – mu2 = 1/2m

Applying v2-u2 = 2as

W = mas

Since F = ma , thus W = Fs.

Now, if the effective component of force along the direction of displacement is Fcos responsible for the displacement of any object in the given direction, then work done by the force F in displacing the body through displacement d is, W=|d|

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## Finding Work In One Dimension

**Find the direction of the force vector and the direction of motion.**To start, it’s important to first be able to identify both the direction the object is moving in and the direction from which force is being applied. Keep in mind that objects don’t always move in line with the force being applied to them for instance, if you pull a small wagon by its handle, you’re applying a diagonal force to move it forward. In this section, however, we’ll deal with situations in which the force and the object’s displacement

*do*have the same direction. For information on how to find the work when these things

*don’t*have the same direction, see below.XResearch source

**forward**. In the next few steps, we’ll use this information to help find the work done on the object.

**2 meters**for our “D” value in the formula.

## What Is Work In Physics

Generally, work is often defined as the product of the force over a distance of displacement. In other words, work is an energy transfer to or from an object by the application of force along with a displacement .

Joule is the SI unit for work. The joule is described as the work is done by the force of a newton acting over a distance of one meter . SI units of work is kgm2s-2. An online work calculator provides all calculations with SI work units.

According to the work definition the equation for work is:

$$W = F * d$$

d = distance covers by an object

F = force applied to an object

W = work done

Work equation in physics is:

W = Fd

Work = force X distance

The force can be written as:

F = work / displacement

Then the Displacement is:

d = Work / Force

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