A sphere is rolling without slipping on a fixed horizontal plane …

Question

A sphere is rolling without slipping on a fixed horizontal plane surface. In the figure $A$ is the point of contact, $B$ is the centre of sphere and $C$ is its topmost point, then

$(i){\vec V_C} - {\vec V_A} = 2\left( {{{\vec V}_B} - {{\vec V}_C}} \right)$

$(ii){\vec V_C} - {\vec V_B} = {\vec V_B} - {\vec V_A}$

$(iii)\left| {{{\vec V}_C} - {{\vec V}_A}} \right| = 2\left| {{{\vec V}_B} - {{\vec V}_C}} \right|$

$(iv)\left| {{{\vec V}_C} - {{\vec V}_A}} \right| = 4\left| {{{\vec V}_B}} \right|$

✓

Answer

$\overrightarrow{\mathrm{V}}_{\mathrm{C}}=(2 \overrightarrow{\mathrm{V}}) \hat{\mathrm{i}} \quad \overrightarrow{\mathrm{V}}_{\mathrm{A}}=0$

$\overrightarrow{\mathrm{V}}_{\mathrm{B}}=(\overrightarrow{\mathrm{V}}) \hat{\mathrm{i}}$

$(i)$ $\overrightarrow{\mathrm{V}}_{\mathrm{c}}-\overrightarrow{\mathrm{V}}_{\mathrm{A}}=2\left(\overrightarrow{\mathrm{V}}_{\mathrm{B}}-\overrightarrow{\mathrm{V}}_{\mathrm{C}}\right)$

$=2(\mathrm{V} \hat{\mathrm{i}}-2 \mathrm{V} \hat{\mathrm{i}})=-2 \mathrm{V} \hat{\mathrm{i}}$ incorrect

$(ii)$ $\overrightarrow{\mathrm{V}}_{\mathrm{C}}-\overrightarrow{\mathrm{V}}_{\mathrm{B}}=2 \overrightarrow{\mathrm{V}} \hat{\mathrm{i}}-\overrightarrow{\mathrm{V}} \hat{\mathrm{i}}$

$=\overrightarrow{\mathrm{V}}_{\mathrm{B}}-\overrightarrow{\mathrm{V}}_{\mathrm{A}}=\mathrm{V} \hat{\mathrm{i}}-0=\mathrm{V} \hat{\mathrm{i}}$ correct

$(iii)$ $\left|\overrightarrow{\mathrm{V}}_{\mathrm{C}}-\overrightarrow{\mathrm{V}}_{\mathrm{A}}\right|=2 \mathrm{V}, 2\left|\overrightarrow{\mathrm{V}}_{\mathrm{B}}-\overrightarrow{\mathrm{V}}_{\mathrm{C}}\right|=2 \mathrm{V}$ correct

$(iv)$ $\left|\overrightarrow{\mathrm{V}}_{\mathrm{C}}-\overrightarrow{\mathrm{V}}_{\mathrm{A}}\right|=2 \mathrm{V}, 4\left|\overrightarrow{\mathrm{V}}_{\mathrm{B}}\right|=4 \mathrm{V}$ incorrect