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The figure shows angular position versus time graphs for six different objects. a) Rank these graphs on the basis of...

The figure shows angular position versus time graphs for six different objects.

a) Rank these graphs on the basis of the angular velocity of each object. Rank positive angular velocities as larger than negative angular velocities. Rank from largest to smallest. To rank items as equivalent, overlap them.

b) Rank these graphs on the basis of the angular acceleration of the object. Rank positive angular accelerations as larger than negative angular accelerations.Rank from largest to smallest. To rank items as equivalent, overlap them.

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Answer #1
Concepts and reason

The required concepts to solve these questions are angular velocity and angular acceleration.

Initially, find the slope of each of the line and then calculate the angular velocity for every point and rank from largest to smallest. Similarly, calculate the angular acceleration and then rank from largest to smallest.

Fundamentals

Angular velocity is defined as the rate of change of the angular position with respect to time.

The expression for angular velocity is,

ω=ΔθΔt\omega = \frac{{\Delta \theta }}{{\Delta t}}

Here, θ\theta is angular position and tt is time.

Angular acceleration is defined as the rate of change of the angular velocity with respect to time.

The expression for angular acceleration is,

α=ΔωΔt\alpha = \frac{{\Delta \omega }}{{\Delta t}}

Here, ω\omega is angular velocity and is tt the time.

The expression for slope is,

Slope=changeinquantityalongyaxischangeinquantityalongxaxis{\rm{Slope}} = \;\frac{{{\rm{change in quantity along }}y{\rm{ - axis}}}}{{{\rm{change in quantity along }}x{\rm{ - axis}}}}

(rad)
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12 1(S)

(a)

The expression for slope of the line (angular velocity) is,

Slope=ΔθΔt{\rm{Slope}} = \;\frac{{\Delta \theta }}{{\Delta t}}

The slope of the line A and F is zero because velocity is constant.

The slope of the line B is,

Slope=θ1θt1t{\rm{Slope}} = \;\frac{{{\theta _1} - \theta }}{{{t_1} - t}}

Substitute 2rad2{\rm{ rad}} for θ1{\theta _1} , 0rad0{\rm{ rad}} for θ\theta , 12s12{\rm{ s}} for t1{t_1} and 0s0{\rm{ s}} for tt in above equation.

Slope=2rad0rad12s0s=16rads\begin{array}{c}\\{\rm{Slope}} = \;\frac{{2{\rm{ rad}} - 0{\rm{ rad}}}}{{12{\rm{ s}} - 0{\rm{ s}}}}\\\\ = \frac{1}{6}{\rm{rad}} \cdot {\rm{s}}\\\end{array}

The slope of the line C is,

Slope=θ2θt2t{\rm{Slope}} = \;\frac{{{\theta _2} - \theta }}{{{t_2} - t}}

Substitute 1rad1{\rm{ rad}} for θ2{\theta _2} , 0rad0{\rm{ rad}} for θ\theta , 12s12{\rm{ s}} for t2{t_2} and 0s0{\rm{ s}} for tt in above equation.

Slope=1rad0rad12s0s=112rads\begin{array}{c}\\{\rm{Slope}} = \;\frac{{1{\rm{ rad}} - 0{\rm{ rad}}}}{{12{\rm{ s}} - 0{\rm{ s}}}}\\\\ = \frac{1}{{12}}{\rm{rad}} \cdot {\rm{s}}\\\end{array}

Slope of the line D is,

Slope=θ3θt3t{\rm{Slope}} = \;\frac{{{\theta _3} - \theta }}{{{t_3} - t}}

Substitute 2rad - 2{\rm{ rad}} for θ3{\theta _3} , 0rad0{\rm{ rad}} for θ\theta , 12s12{\rm{ s}} for t3{t_3} and 0s0{\rm{ s}} for tt in above equation.

Slope=2rad0rad12s0s=16rads\begin{array}{c}\\{\rm{Slope}} = \;\frac{{ - 2{\rm{ rad}} - 0{\rm{ rad}}}}{{12{\rm{ s}} - 0{\rm{ s}}}}\\\\ = - \frac{1}{6}{\rm{rad}} \cdot {\rm{s}}\\\end{array}

Slope of the line E is,

Slope=θ4θt4t{\rm{Slope}} = \;\frac{{{\theta _4} - \theta }}{{{t_4} - t}}

Substitute 3rad - 3{\rm{ rad}} for θ4{\theta _4} , 1rad - 1\;{\rm{rad}} for θ\theta , 12s12{\rm{ s}} for t3{t_3} and 0s0{\rm{ s}} for tt in above equation.

Slope=3rad(1)rad12s0s=16rads\begin{array}{c}\\{\rm{Slope}} = \;\frac{{ - 3{\rm{ rad}} - \left( { - 1} \right){\rm{ rad}}}}{{12{\rm{ s}} - 0{\rm{ s}}}}\\\\ = - \frac{1}{6}{\rm{rad}} \cdot {\rm{s}}\\\end{array}

Rank of angular velocities from largest to smallest is,

B>C>A=F>D=E{\rm{B}} > {\rm{C}} > {\rm{A}} = {\rm{F}} > {\rm{D = E}}

(b)

The expression for angular acceleration is,

α=dωdt\alpha = \;\frac{{d\omega }}{{dt}}

In the graph, they all have constant slopes. It means that all lines represent constant angular velocity. Thus,

α=d(constant)dt=0\begin{array}{c}\\\alpha = \;\frac{{d\left( {{\rm{constant}}} \right)}}{{dt}}\\\\ = 0\\\end{array}

Rank of angular acceleration from largest to smallest is D=E=B=C=A=F{\rm{D = E}} = {\rm{B}} = {\rm{C}} = {\rm{A}} = {\rm{F}} .

Ans: Part a

Rank of angular velocities from largest to smallest is B>C>A=F>D=E{\rm{B}} > {\rm{C}} > {\rm{A}} = {\rm{F}} > {\rm{D = E}} .

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Answer #3

b,c (a,f) (D,E)

source: mastering physics
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Answer #2

Here ,

angular velocity is given as

angular velocity , w = d(theta)/dt

d(theta)/dt is the slope of theta vs Time graph

from the graph ,

the order of slope of lines from largest to lowest is

1. B

2. C

3. A = F

5. D = E

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Answer #4
Rank these graphs on the basis of the angular acceleration of the object. Rank positive accelerations as larger than negative angular acceleration
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