Rolling motion

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Two wheels A and C are connected by a belt B as shown in figure. The radius of C is three times the radius of A. What would be the ratio of the rotational inertias $(I_{C} / I_{A})$ if both the wheels have the same rotational kinetic energy?

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Solution

As the belt does not slip, $v_{P} = v_{Q}$
i.e., $r_{A} ω_{A} = r_{C} ω_{C} [as v = rω]$ …(i)
According to the given problem if $r_{A} = r, r_{C} = 3 r$ , so Eq. (i) becomes
$ω_{A} = 3 ω_{C}$ …(ii)
If both the wheels have the same rotational kinetic energy, then
$\frac{1}{2} I_{A} ω_{A}^{2} = \frac{1}{2} I_{C} ω_{C}^{2}$
$\frac{I_{A}}{I_{C}} = {[\frac{ω_{C}}{ω_{A}}]}^{2} = {[\frac{1}{3}]}^{2} = \frac{1}{9}$

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Rolling motion

Remember concepts with our Masterclasses.

Two wheels A and C are connected by a belt B as shown in figure. The radius of C is three times the radius of A. What would be the ratio of the rotational inertias IC/IA if both the wheels have the same rotational kinetic energy?

As the belt does not slip, vP=vQi.e., rAωA=rCωC [ as v=rω] …(i)According to the given problem if rA=r, rC=3r, so Eq. (i) becomesωA=3ωC …(ii)If both the wheels have the same rotational kinetic energy, then12IAωA2=12ICωC2IAIC=ωCωA2=132=19

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Two wheels A and C are connected by a belt B as shown in figure. The radius of C is three times the radius of A. What would be the ratio of the rotational inertias $(I_{C} / I_{A})$ if both the wheels have the same rotational kinetic energy?