A small roller of diameter 20 cm has an axle of diameter 10 cm (see figure below on the left). It is
on a horizontal floor and a meter scale is positioned horizontally on its axle with one edge of the
scale on top of the axle (see figure on the right). The scale is now pushed slowly on the axle so that
it moves without slipping on the axle, and the roller starts rolling without slipping. After the roller
has moved 50 cm, the position of the scale will look like (figures are schematic and not drawn to
scale)-
Options
A
B
C
D
✓ Correct! Well done.
✗ Incorrect. Try again or view the solution.
Solution
Sol. For no slipping at the ground,
$$
\begin{aligned}
& \mathrm{V}_{\text {centre }}=\omega \mathrm{R} \\
& \text { ( } \mathrm{R} \text { is radius of roller) } \\
& \text { ∴ Velocity of scale }=\left(\mathrm{V}_{\text {center }}+\omega \mathrm{r}\right) \quad[\mathrm{r} \text { is radius of axle }] \\
& \text { Given, } \mathrm{V}_{\text {center }} \cdot \mathrm{t}=50 \mathrm{~cm} \\
& \text { ∴ } \text { Distance moved by scale }=\left(\mathrm{V}_{\text {center }}+\omega \mathrm{r}\right) \mathrm{t} \\
& =\left(\mathrm{V}_{\text {center }}+\frac{\mathrm{V}_{\text {center }} \mathrm{r}}{\mathrm{R}}\right) \mathrm{t}=\frac{3 \mathrm{~V}_{\text {center }}}{2} \cdot \mathrm{t}=75 \mathrm{~cm}
\end{aligned}
$$
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