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By the end of this section, you will be able to:
  • Use the work-energy theorem to analyze rotation to find the work done on a system when it is rotated about a fixed axis for a finite angular displacement
  • Solve for the angular velocity of a rotating rigid body using the work-energy theorem
  • Find the power delivered to a rotating rigid body given the applied torque and angular velocity
  • Summarize the rotational variables and equations and relate them to their translational counterparts

Thus far in the chapter, we have extensively addressed kinematics and dynamics for rotating rigid bodies around a fixed axis. In this final section, we define work and power within the context of rotation about a fixed axis, which has applications to both physics and engineering. The discussion of work and power makes our treatment of rotational motion almost complete, with the exception of rolling motion and angular momentum, which are discussed in Angular Momentum . We begin this section with a treatment of the work-energy theorem for rotation.

Work for rotational motion

Now that we have determined how to calculate kinetic energy for rotating rigid bodies, we can proceed with a discussion of the work done on a rigid body rotating about a fixed axis. [link] shows a rigid body that has rotated through an angle d θ from A to B while under the influence of a force F . The external force F is applied to point P , whose position is r , and the rigid body is constrained to rotate about a fixed axis that is perpendicular to the page and passes through O . The rotational axis is fixed, so the vector r moves in a circle of radius r , and the vector d s is perpendicular to r .

Figure shows the rigid body is constrained to rotate about a fixed axis that is perpendicular to the page and passes through a point labeled as O. The rotational axis is fixed, so the vector r moves in a circle of radius r, and the vector ds is perpendicular to vector r. An external force F is applied to point P and makes rigid body rotates through an angle dtheta.
A rigid body rotates through an angle d θ from A to B by the action of an external force F applied to point P .

From [link] , we have

s = θ × r .


d s = d ( θ × r ) = d θ × r + d r × θ = d θ × r .

Note that d r is zero because r is fixed on the rigid body from the origin O to point P . Using the definition of work, we obtain

W = F · d s = F · ( d θ × r ) = d θ · ( r × F )

where we used the identity a · ( b × c ) = b · ( c × a ) . Noting that ( r × F ) = τ , we arrive at the expression for the rotational work    done on a rigid body:

W = τ · d θ .

The total work done on a rigid body is the sum of the torques integrated over the angle through which the body rotates . The incremental work is

d W = ( i τ i ) d θ

where we have taken the dot product in [link] , leaving only torques along the axis of rotation. In a rigid body, all particles rotate through the same angle; thus the work of every external force is equal to the torque times the common incremental angle d θ . The quantity ( i τ i ) is the net torque on the body due to external forces.

Similarly, we found the kinetic energy of a rigid body rotating around a fixed axis by summing the kinetic energy of each particle that makes up the rigid body. Since the work-energy theorem W i = Δ K i is valid for each particle, it is valid for the sum of the particles and the entire body.

Work-energy theorem for rotation

The work-energy theorem for a rigid body rotating around a fixed axis is

W A B = K B K A


K = 1 2 I ω 2

and the rotational work done by a net force rotating a body from point A to point B is

W A B = θ A θ B ( i τ i ) d θ .

Questions & Answers

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Giorgi Reply
How does resonance occur
Rahim Reply
what is quantam
pamit Reply
quantum is a division of mechanics
what is friction
Muhammad Reply
a force act by surface between two bodies whose are always oppose the relative motion .....
when two rough bodies are placed in contact and try to slip each other ... than a force act them and it's ippse the relative motion between them
thats friction force and roughnes of both bodies is define friction of surface
what is a progressive wave
sheriff-deen Reply
What is the wake for therapist
Ife Reply
can u like explain your question with clear detail
who would teach me vectors?
Tintin Reply
what's chemistry
Esther Reply
branch of science dt deals with the study of physical properties of matter and it's particulate nature
Y acctually do u hav ur way of defining it? just bring ur iwn idear
well, it deals with the weight of substances and reaction behind them as well as the behavior
buh hope Esther, we've answered ur question
what's ohms law
ohms law states that, the current flowing through an electric circuit is directly proportional to the potential difference, provided temperature and pressure are kept constant
what is sound
ohms law states that the resistance of a material is directly proportional to the potential difference between two points on that material, if temperature and other physical conditions become constant
How do I access the MCQ
Abraham Reply
As I think the best is, first select the easiest questions for you .and then you can answer the remaining questions.
I mean I'm unable to view it
when I click on it, it doesn't respond
ohhh,try again and again ,It will be showed
what is centripetal force
Don Reply
هي قوة ناتجة من الحركة الدائرية ويكون اتجاهها إلى المركز دائماً
meaning of vector quantity
Felix Reply
vector quantity is any quantity that has both magnitude in terms of number (units) and direction in terms of viewing the quantity from an origin using angles (degree) or (NEWS) method
vector quantity is physical quantity has magnitude and direction
vector is a quantity that is use in measuring size of physical properties and their direction
what difference and similarities between work,force,energy and power?
Anes Reply
I need the best answer
enery is the ability to do work. work is job done, force is a pull or push. power has to do with potential. they belong to different categories which include heat energy, electricity.
force refers to a push or pull... energy refers to work done while power is work done per unit time
mathematically express angular velocity and angular acceleration
Mario Reply
it depends on the direction. an angular velocity will be linear and angular acceleration will be an angle of elevation.
The sonic range finder discussed in the preceding question often needs to be calibrated. During the calibration, the software asks for the room temperature. Why do you suppose the room temperature is required?
Shaina Reply
Suppose a bat uses sound echoes to locate its insect prey, 3.00 m away. (See [link] .) (a) Calculate the echo times for temperatures of 5.00°C5.00°C and 35.0°C.35.0°C. (b) What percent uncertainty does this cause for the bat in locating the insect? (c) Discuss the significance of this uncertainty an
give a reason why musicians commonly bring their wind instruments to room temperature before playing them.
The ear canal resonates like a tube closed at one end. (See [link]Figure 17_03_HumEar[/link].) If ear canals range in length from 1.80 to 2.60 cm in an average population, what is the range of fundamental resonant frequencies? Take air temperature to be 37.0°C,37.0°C, which is the same as body tempe
By what fraction will the frequencies produced by a wind instrument change when air temperature goes from 10.0°C10.0°C to 30.0°C30.0°C ? That is, find the ratio of the frequencies at those temperatures.
what are vector quantity
Aondover Reply
Quantities that has both magnitude and direction
what is lenses
vector quantities are those physical quantites which have magnitude as well as direction and obey the laws of vector algebra.
electric current has both magnitude and direction but it doesn't obey the laws of vector algebra, hence it is not a vector quantity.
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