10.8 Work and power for rotational motion (Page 2/7)

University physics volume 1 Page 3 / 7

P = \frac{d W}{d t} .

If we have a constant net torque, [link] becomes $W = τ θ$ and the power is

P = \frac{d W}{d t} = \frac{d}{d t} (τ θ) = τ \frac{d θ}{d t}

P = τ ω .

Torque on a boat propeller

A boat engine operating at

9.0 \times 10^{4} W

is running at 300 rev/min. What is the torque on the propeller shaft?

Strategy

We are given the rotation rate in rev/min and the power consumption, so we can easily calculate the torque.

Solution

300.0 rev/min = 31.4 rad/s;

τ = \frac{P}{ω} = \frac{9.0 \times 10^{4} N \cdot m / s}{31.4 rad / s} = 2864.8 N \cdot m .

Significance

It is important to note the radian is a dimensionless unit because its definition is the ratio of two lengths. It therefore does not appear in the solution.

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Check Your Understanding A constant torque of $500 kN \cdot m$ is applied to a wind turbine to keep it rotating at 6 rad/s. What is the power required to keep the turbine rotating?

3 MW

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Rotational and translational relationships summarized

The rotational quantities and their linear analog are summarized in three tables. [link] summarizes the rotational variables for circular motion about a fixed axis with their linear analogs and the connecting equation, except for the centripetal acceleration, which stands by itself. [link] summarizes the rotational and translational kinematic equations. [link] summarizes the rotational dynamics equations with their linear analogs.

Rotational and translational variables: summary
Rotational	Translational	Relationship
$θ$	$x$	$θ = \frac{s}{r}$
$ω$	$v_{t}$	$ω = \frac{v_{t}}{r}$
$α$	$a_{t}$	$α = \frac{a_{t}}{r}$
	$a_{c}$	$a_{c} = \frac{v_{t}^{2}}{r}$

Rotational and translational kinematic equations: summary
Rotational	Translational
$θ_{f} = θ_{0} + \bar{ω} t$	$x = x_{0} + \bar{v} t$
$ω_{f} = ω_{0} + α t$	$v_{f} = v_{0} + a t$
$θ_{f} = θ_{0} + ω_{0} t + \frac{1}{2} α t^{2}$	$x_{f} = x_{0} + v_{0} t + \frac{1}{2} a t^{2}$
$ω_{f}^{2} = ω^{2}_{0} + 2 α (Δ θ)$	$v_{f}^{2} = v^{2}_{0} + 2 a (Δ x)$

Rotational and translational equations: dynamics
Rotational	Translational
$I = \sum_{i} m_{i} r_{i}^{2}$	m
$K = \frac{1}{2} I ω^{2}$	$K = \frac{1}{2} m v^{2}$
$\sum_{i} τ_{i} = I α$	$\sum_{i} {\vec{F}}_{i} = m \vec{a}$
$W_{A B} = \int_{θ_{A}}^{θ_{B}} (\sum_{i} τ_{i}) d θ$	$W = \int \vec{F} \cdot d \vec{s}$
$P = τ ω$	$P = \vec{F} \cdot \vec{v}$

Summary

The incremental work dW in rotating a rigid body about a fixed axis is the sum of the torques about the axis times the incremental angle $d θ$ .
The total work done to rotate a rigid body through an angle $θ$ about a fixed axis is the sum of the torques integrated over the angular displacement. If the torque is a constant as a function of $θ$ , then $W_{A B} = τ (θ_{B} - θ_{A})$ .
The work-energy theorem relates the rotational work done to the change in rotational kinetic energy: $W_{A B} = K_{B} - K_{A}$ where $K = \frac{1}{2} I ω^{2} .$
The power delivered to a system that is rotating about a fixed axis is the torque times the angular velocity, $P = τ ω$ .

Key equations

Angular position	$θ = \frac{s}{r}$
Angular velocity	$ω = lim_{Δ t \to 0} \frac{Δ θ}{Δ t} = \frac{d θ}{d t}$
Tangential speed	$v_{t} = r ω$
Angular acceleration	$α = lim_{Δ t \to 0} \frac{Δ ω}{Δ t} = \frac{d ω}{d t} = \frac{d^{2} θ}{d t^{2}}$
Tangential acceleration	$a_{t} = r α$
Average angular velocity	$\bar{ω} = \frac{ω_{0} + ω_{f}}{2}$
Angular displacement	$θ_{f} = θ_{0} + \bar{ω} t$
Angular velocity from constant angular acceleration	$ω_{f} = ω_{0} + α t$
Angular velocity from displacement and constant angular acceleration	$θ_{f} = θ_{0} + ω_{0} t + \frac{1}{2} α t^{2}$
Change in angular velocity	$ω_{f}^{2} = ω_{0}^{2} + 2 α (Δ θ)$
Total acceleration	$\vec{a} = {\vec{a}}_{c} + {\vec{a}}_{t}$
Rotational kinetic energy	$K = \frac{1}{2} (\sum_{j} m_{j} r_{j}^{2}) ω^{2}$
Moment of inertia	$I = \sum_{j} m_{j} r_{j}^{2}$
Rotational kinetic energy in terms of the moment of inertia of a rigid body	$K = \frac{1}{2} I ω^{2}$
Moment of inertia of a continuous object	$I = \int r^{2} d m$
Parallel-axis theorem	$I_{parallel-axis} = I_{initial} + m d^{2}$
Moment of inertia of a compound object	$I_{total} = \sum_{i} I_{i}$
Torque vector	$\vec{τ} = \vec{r} \times \vec{F}$
Magnitude of torque	$\| \vec{τ} \| = r_{⊥} F$
Total torque	$τ_{net} = \sum_{i} \| τ_{i} \|$
Newton’s second law for rotation	$\sum_{i} τ_{i} = I α$
Incremental work done by a torque	$d W = (\sum_{i} τ_{i}) d θ$
Work-energy theorem	$W_{A B} = K_{B} - K_{A}$
Rotational work done by net force	$W_{A B} = \int_{θ_{A}}^{θ_{B}} (\sum_{i} τ_{i}) d θ$
Rotational power	$P = τ ω$

Questions & Answers

Ayele, K., 2003. Introductory Economics, 3rd ed., Addis Ababa.

Widad Reply

can you send the book attached ?

Ariel

What is economics

Widad Reply

the study of how humans make choices under conditions of scarcity

AI-Robot

U(x,y) = (x×y)1/2 find mu of x for y

Desalegn Reply

U(x,y) = (xÃy)1/2 find mu of x for y

Desalegn

what is ecnomics

Jan Reply

this is the study of how the society manages it's scarce resources

Belonwu

what is macroeconomic

John Reply

macroeconomic is the branch of economics which studies actions, scale, activities and behaviour of the aggregate economy as a whole.

husaini

etc

husaini

difference between firm and industry

husaini Reply

what's the difference between a firm and an industry

Abdul

firm is the unit which transform inputs to output where as industry contain combination of firms with similar production 😅😅

Abdulraufu

Suppose the demand function that a firm faces shifted from Qd  120 3P to Qd  90  3P and the supply function has shifted from QS  20  2P to QS 10  2P . a) Find the effect of this change on price and quantity. b) Which of the changes in demand and supply is higher?

Toofiq Reply

explain standard reason why economic is a science

innocent Reply

factors influencing supply

Petrus Reply

what is economic.

Milan Reply

scares means__________________ends resources. unlimited

Jan

economics is a science that studies human behaviour as a relationship b/w ends and scares means which have alternative uses

Jan

calculate the profit maximizing for demand and supply

Zarshad Reply

Why qualify 28 supplies

Milan

what are explicit costs

Nomsa Reply

out-of-pocket costs for a firm, for example, payments for wages and salaries, rent, or materials

AI-Robot

concepts of supply in microeconomics

David Reply

economic overview notes

Amahle Reply

identify a demand and a supply curve

Salome Reply

i don't know

Parul

there's a difference

Aryan

Demand curve shows that how supply and others conditions affect on demand of a particular thing and what percent demand increase whith increase of supply of goods

Israr

Hi Sir please how do u calculate Cross elastic demand and income elastic demand?

Abari

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Source: OpenStax, University physics volume 1. OpenStax CNX. Sep 19, 2016 Download for free at http://cnx.org/content/col12031/1.5

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