Informative line

Work And Its Sign

Learn work definition & formula in physics, practice positive and negative work equation. Practice for calculating work done formula with example, dot product of two vectors and angle between vectors.

Work

  • Work is defined as the amount of energy transferred  to a body by application of force.
  • Work is done on an object only when there is force applied on it.
  • If more than one force is acting on the body, then work done by different forces are to be calculated separately.  
  • Work done by a constant force is calculated as

              \(W_F = \vec F.\vec s\) 

              \(= F\,s \,cos \,\theta\)  

where    F=  Force applied 

s = displacement from the point at which force is applied 

\(\theta =\)  Angle between \(\vec F\) and \(\vec s\)

Illustration Questions

A constant force \(\vec F\) is applied at the free end of a spring so that it is compressed by a distance s. What is the work done by force \(\vec F\)? 

A \(Fs\)

B \(-Fs\)

C \(\dfrac{Fs}{2}\)

D \(2\,Fs\)

×

Work done,  \(W=\vec F. \vec s\)

\(W = Fs\,cos \,\theta \)

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Here, \(\theta = 0°\)

image

\(W = F\,s \,cos \,0°\)

\(W = Fs\)

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A constant force \(\vec F\) is applied at the free end of a spring so that it is compressed by a distance s. What is the work done by force \(\vec F\)? 

image
A

\(Fs\)

.

B

\(-Fs\)

C

\(\dfrac{Fs}{2}\)

D

\(2\,Fs\)

Option A is Correct

Work as DOT Product 

  • Work done by a force is the dot product of force and displacement from point of its application.

DOT Product 

  • Consider two vectors 

     \(\vec A = x_1 \hat i + y _1 \, \hat j +z_1 \,\hat k\) 

     \(\vec B = x_2 \hat i + y _2 \, \hat j +z_2 \,\hat k\)

\(\vec A . \vec B = ( x_1 \hat i + y _1 \, \hat j +z_1 \,\hat k).(x_2 \hat i + y _2 \, \hat j +z_2 \,\hat k)\)

\(= x_1 \,x_2 +y_1\,y_2 + z_1\,z_2\)

  • \(\vec A . \vec B \)  is a scalar quantity

 Displacement  in Component from

Consider, a particle is displaced from its position vector

\(\vec r_1 = (x_1 \,\hat i +y_1 \,\hat j +\,z_1\, \hat k)\) to 

\(\vec r_2 = (x_2 \,\hat i +y_2 \,\hat j +\,z_2\, \hat k)\)

Displacement 

\(\Rightarrow \Delta \vec r = \vec r_2 -\vec r_1\)

\(\Rightarrow (x_2 - x_1 )\hat i + (y_2 - y_1 )\hat j + (z_2 - z_1 )\hat k \)

\(\Rightarrow \Delta \vec r = \Delta x\,\hat i +\Delta y\,\hat j +\Delta z\,\hat k \)

 

Illustration Questions

If a particle is displaced  \(\vec d = \hat i+2\,\hat j - 3\,\hat k \) by a force \(\vec F = (2\,\hat i +3\,\hat j +\hat k )\) then, calculate the work done.

A \(10 \,J\)

B \(5 \,J\)

C \(12 \,J\)

D \(8 \,J\)

×

Work done,  \(W= \vec F . \vec d\)

\(\Rightarrow W= (2\,\hat i +3\, \hat j+\hat k) . (\hat i + 2\,\hat j -3\,\hat k)\)

\(\Rightarrow 2+ 6-3\)

\(= 5 \, J\)

\(\therefore\) Option (B) is correct. 

If a particle is displaced  \(\vec d = \hat i+2\,\hat j - 3\,\hat k \) by a force \(\vec F = (2\,\hat i +3\,\hat j +\hat k )\) then, calculate the work done.

A

\(10 \,J\)

.

B

\(5 \,J\)

C

\(12 \,J\)

D

\(8 \,J\)

Option B is Correct

Sign of Work

  • Work done by a force may be positive, negative or zero depending upon the angle between force and displacement vector.

Positive Work 

Consider a force \(\vec F\) is applied on a body making an angle \(\theta\) with the displacement vector, as shown in figure.

\( W = \vec F_1 . \vec s_1\)

\(W= |\vec F_1||\vec s_1| \, cos\,\theta \)

  • For work to be positive, \(cos\,\theta\)  must be positive. 

   \(\therefore\)    \( 0\leq \theta<90°\)

Important point

If angle between applied force and displacement vector is less then 90°, then work done by the applied force is always positive.

Illustration Questions

Which of the following force is responsible for positive work done?

A \(\vec F_1\)

B \(\vec F_2\)

C \(\vec F_3\)

D \(\vec F_4\)

×

The angle between \(\vec F_1\) and \(\vec s\) is less than 90°.

image

Work done by \(\vec F_1\) is positive.

image

\(\therefore\) Option (A) is correct.

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Which of the following force is responsible for positive work done?

image
A

\(\vec F_1\)

.

B

\(\vec F_2\)

C

\(\vec F_3\)

D

\(\vec F_4\)

Option A is Correct

Concept of Negative Work

  • As  \(W=\vec F. \vec s\)

      \(W= |\vec F| |\vec s| \,cos \,\theta\)

  • For work to be negative \(\theta\) must be greater than 90 °

            \(i.e. \, \theta > 90 ° \) 

Illustration Questions

Which of the following force is responsible for negative work done ?

A \(\vec F_1\)

B \(\vec F_4\)

C \(\vec F_2 \,\,{\text {and }}\,\,\vec F_3\)

D \(\vec F_1 \,\,{\text {and }}\,\,\vec F_4\)

×

The angle of displacement vector with forces \(\vec F_2 \,\,{\text{and}}\,\,\vec F_3\) is more than 90° 

So the work done by \(\vec F_2 \,\,{\text {and }}\,\,\vec F_3\) is negative.

\(\therefore\) Option (C) is correct.

Which of the following force is responsible for negative work done ?

image
A

\(\vec F_1\)

.

B

\(\vec F_4\)

C

\(\vec F_2 \,\,{\text {and }}\,\,\vec F_3\)

D

\(\vec F_1 \,\,{\text {and }}\,\,\vec F_4\)

Option C is Correct

Zero Work Done by a Force

  • Work done by a force may be positive, negative and also zero.

Work is zero

Case 1 

When there is no displacement from point of application.

  • As work done, \(W = |\vec F| |\vec s| \,cos\,\theta\)  

 \(= |\vec F| \, cos\,\theta × 0\) 

     \(W=0\)

Example :  A child pushing a wall.  

Case 2 

Force is perpendicular to displacement

As work done,  \(W= |\vec F| |\vec s| \,cos\,\theta\)

\(W= |\vec F| |\vec s| \,cos\,90°\)

\(W= 0\)

Example :  The work done by the normal force as well as the work done by the gravitational force on the car  both are zero, because both forces are perpendicular to the displacement and have zero components along an axis in the direction of \(\vec s\).

Here the work done by gravitational force is zero.

Case 3 

When force is zero.

As \(W= |\vec F| |\vec s| \,cos\,\theta\)

\(W= 0× |\vec s| \,cos\,\theta\)

\(W= 0\)

Example :

  • A man jumps to a height 'h'. During the flight, the work done by normal contact force is zero.
  • Because when contact is lost, normal force is zero. 

Illustration Questions

Which of the following force is responsible for zero work done?

A \(\vec F_3\)

B \(\vec F_1\)

C \(\vec F_2\)

D \(\vec F_4\)

×

Only for \( F_3\), the angle between \(\vec s\) and \(\vec F_3\) is equal to 90°. 

Work done by \( F_3 = |\vec F_3||\vec s| \, cos \,90° \)

\(W_{F_3} =0\)

\(\therefore\) Option (A) is correct.

Which of the following force is responsible for zero work done?

image
A

\(\vec F_3\)

.

B

\(\vec F_1\)

C

\(\vec F_2\)

D

\(\vec F_4\)

Option A is Correct

Illustration Questions

Choose the correct combinations.

A \(\vec F _1, \vec F_2 \,\,{\text{Positive Work}}\) \(F_4 , \,F_5 \,\,{\text {Negative Work}}\)

B \(\vec F _1, \vec F_3 \,\,{\text{Positive Work}}\) \(\vec F_4 , \,\vec F_5 \,\,{\text {Zero Work}}\)

C \(\vec F _1, \vec F_4 \,\,{\text{Zero Work}}\) \(\vec F_2 , \,\vec F_3 \,\,{\text {Negative Work}}\)

D \(\vec F _2, \vec F_5 \,\,{\text{Positive Work}}\) \(\vec F_1, \vec F_4 , \,\vec F_3 \,\,{\text {Negative Work}}\)

×

\(\vec F_1 \,\,{\text{and}}\,\,\vec F_2\) are making angle < 90° with the displacement, \(\vec s\) 

Hence, work done by \(\vec F_1\,\,{\text{and}}\,\,\vec F_2\) will be positive.

\(\vec F_3\) is making angle 90° with the displacement , \(\vec s\) 

Hence, work done by \(\vec F_3\) is zero.

\(\vec F_4\) and \(\vec F_5\) are making angle > 90° with the displacement , \(\vec s\) 

Hence, work done by \(\vec F_4 \,\,{\text{and}}\,\,\vec F_5\) will be negative.

\(\therefore \) Option (A) is correct.

Choose the correct combinations.

image
A

\(\vec F _1, \vec F_2 \,\,{\text{Positive Work}}\)

\(F_4 , \,F_5 \,\,{\text {Negative Work}}\)

.

B

\(\vec F _1, \vec F_3 \,\,{\text{Positive Work}}\)

\(\vec F_4 , \,\vec F_5 \,\,{\text {Zero Work}}\)

C

\(\vec F _1, \vec F_4 \,\,{\text{Zero Work}}\)

\(\vec F_2 , \,\vec F_3 \,\,{\text {Negative Work}}\)

D

\(\vec F _2, \vec F_5 \,\,{\text{Positive Work}}\)

\(\vec F_1, \vec F_4 , \,\vec F_3 \,\,{\text {Negative Work}}\)

Option A is Correct

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