What is the maximum speed of a point on the outside of the wheel 15 cm from the axle?
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What is the maximum speed of a point on the outside of the wheel 15 cm from the axle?
Question: What is the maximum speed of a point on the outside of the wheel 15 cm from the axle?
Speed:
The linear speed of a point on the wheel increases as the axle's radial distance increases because it is proportional to rotation radius. The linear speed of that point is maximum at the outer surface of the wheel.
Answer and Explanation: 1
Given Data
The distance between a wheel and the axle is; {eq}r = 15\;{\rm{cm}} = 15 \times {10^{ - 2}}\;{\rm{m}} {/eq}
From the graph, the maximum rotation of the wheel at time {eq}{t_1} = 10\;{\rm{s}} {/eq} is {eq}{\theta _{\max }} \approx 75\;{\rm{rad}} {/eq}.
The minimum rotation of the wheel at time {eq}{t_2} = 0\;{\rm{s}} {/eq} is {eq}{\theta _{\min }} = 25\;{\rm{rad}} {/eq}.
Find the angular velocity of the wheel.
{eq}\begin{align*} \omega & = \dfrac{{d\theta }}{{dt}}\\ &= \dfrac{{{\theta _{\max }} - {\theta _{\min }}}}{{{t_1} - {t_2}}}\\ &= \dfrac{{75\;{\rm{rad}} - 25\;{\rm{rad}}}}{{10\;{\rm{s}} - 0\;{\rm{s}}}}\\ &= 5\;{\rm{rad/s}} \end{align*} {/eq}
Find the maximum speed of the wheel at a distance {eq}r = 15 \times {10^{ - 2}}\;{\rm{m}} {/eq} from the axle.
{eq}\begin{align*} v &= r\omega \\ &= 15 \times {10^{ - 2}}\;{\rm{m}} \times 5\;{\rm{rad/s}}\\ &= 0.75\;{\rm{m/s}} \end{align*} {/eq}
So, the maximum speed is {eq}0.75\;{\rm{m/s}} {/eq}.
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