Question 771: Centre of Mass and Rotational Motion

Question

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

Accepted Answer

We have
$$\overrightarrow {{V_C}}  = \overrightarrow {{V_{CM}}}  + \overrightarrow {r\omega }  = 2\overrightarrow {{V_{CM}}} $$
$$\overrightarrow {{V_B}}  = \overrightarrow {{V_{CM}}} $$
$$\overrightarrow {{V_A}}  = 0$$
Therefore,
$$\overrightarrow {{V_C}}  - \overrightarrow {{V_B}}  = \overrightarrow {r\omega }  = \overrightarrow {{V_{CM}}} $$
$$\overrightarrow {{V_B}}  - \overrightarrow {{V_A}}  = \overrightarrow {{V_{CM}}} $$
$$\overrightarrow {{V_C}}  - \overrightarrow {{V_A}}  = \overrightarrow {{V_{CM}}}  + \overrightarrow {r\omega }  = 2\overrightarrow {{V_{CM}}} $$
$$\overrightarrow {{V_B}}  - \overrightarrow {{V_C}}  =  - \overrightarrow {r\omega } $$
For pure rolling, we have
$$|\overrightarrow {{V_{CM}}} | = |\overrightarrow {r\omega } |$$
Therefore,
$$|\overrightarrow {{V_C}}  - \overrightarrow {{V_A}} | = 2{V_{CM}}$$
$$ \Rightarrow 2|\overrightarrow {{V_B}}  - \overrightarrow {{V_C}} | = 2r\omega  = 2{V_{CM}}$$

Answer

$${\overrightarrow V _C} - {\overrightarrow V _A} = 2({\overrightarrow V _B} - {\overrightarrow V _C})$$

Answer

$${\overrightarrow V _C} - {\overrightarrow V _B} = {\overrightarrow V _B} - {\overrightarrow V _A}$$

Answer

$$|{\overrightarrow V _C} - {\overrightarrow V _A}| = 2|{\overrightarrow V _B} - {\overrightarrow V _C}|$$

Answer

$$|{\overrightarrow V _C} - {\overrightarrow V _A}| = 4|{\overrightarrow V _B}|$$

Centre of Mass and Rotational Motion

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

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