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A-sphere-is-rolling-without-slipping-on-a-fixed-horizontal-plane-surface-In-the-Figure-A-is-a-point-of-contact-B-is-the-centre-of-the-sphere-and-C-is-its-topmost-point-Then-a-v-C-v-A-




Question Number 20724 by Tinkutara last updated on 01/Sep/17
A sphere is rolling without slipping on  a fixed horizontal plane surface. In the  Figure, A is a point of contact, B is the  centre of the sphere and C is its topmost  point. Then  (a) v_C ^→  − v_A ^→  = 2(v_B ^→  − v_C ^→ )  (b) v_C ^→  − v_B ^→  = v_B ^→  − v_A ^→   (c) ∣v_C ^→  − v_A ^→ ∣ = 2∣v_B ^→  − v_C ^→ ∣  (d) ∣v_C ^→  − v_A ^→ ∣ = 4∣v_B ^→ ∣
$$\mathrm{A}\:\mathrm{sphere}\:\mathrm{is}\:\mathrm{rolling}\:\mathrm{without}\:\mathrm{slipping}\:\mathrm{on} \\ $$$$\mathrm{a}\:\mathrm{fixed}\:\mathrm{horizontal}\:\mathrm{plane}\:\mathrm{surface}.\:\mathrm{In}\:\mathrm{the} \\ $$$$\mathrm{Figure},\:\mathrm{A}\:\mathrm{is}\:\mathrm{a}\:\mathrm{point}\:\mathrm{of}\:\mathrm{contact},\:{B}\:\mathrm{is}\:\mathrm{the} \\ $$$$\mathrm{centre}\:\mathrm{of}\:\mathrm{the}\:\mathrm{sphere}\:\mathrm{and}\:{C}\:\mathrm{is}\:\mathrm{its}\:\mathrm{topmost} \\ $$$$\mathrm{point}.\:\mathrm{Then} \\ $$$$\left({a}\right)\:\overset{\rightarrow} {{v}}_{\mathrm{C}} \:−\:\overset{\rightarrow} {{v}}_{\mathrm{A}} \:=\:\mathrm{2}\left(\overset{\rightarrow} {{v}}_{\mathrm{B}} \:−\:\overset{\rightarrow} {{v}}_{\mathrm{C}} \right) \\ $$$$\left({b}\right)\:\overset{\rightarrow} {{v}}_{\mathrm{C}} \:−\:\overset{\rightarrow} {{v}}_{\mathrm{B}} \:=\:\overset{\rightarrow} {{v}}_{\mathrm{B}} \:−\:\overset{\rightarrow} {{v}}_{\mathrm{A}} \\ $$$$\left({c}\right)\:\mid\overset{\rightarrow} {{v}}_{\mathrm{C}} \:−\:\overset{\rightarrow} {{v}}_{\mathrm{A}} \mid\:=\:\mathrm{2}\mid\overset{\rightarrow} {{v}}_{\mathrm{B}} \:−\:\overset{\rightarrow} {{v}}_{\mathrm{C}} \mid \\ $$$$\left({d}\right)\:\mid\overset{\rightarrow} {{v}}_{\mathrm{C}} \:−\:\overset{\rightarrow} {{v}}_{\mathrm{A}} \mid\:=\:\mathrm{4}\mid\overset{\rightarrow} {{v}}_{\mathrm{B}} \mid \\ $$
Commented by Tinkutara last updated on 01/Sep/17
Commented by ajfour last updated on 01/Sep/17
(b) and (c)  because  v_C ^(→)  = 2v_B ^(→)            and v_A ^(→)  =0 .
$$\left({b}\right)\:{and}\:\left({c}\right) \\ $$$${because}\:\:\overset{\rightarrow} {{v}_{{C}} }\:=\:\mathrm{2}\overset{\rightarrow} {{v}_{{B}} } \\ $$$$\:\:\:\:\:\:\:\:\:{and}\:\overset{\rightarrow} {{v}_{{A}} }\:=\mathrm{0}\:. \\ $$
Commented by Tinkutara last updated on 02/Sep/17
Thank you very much Sir!
$$\mathrm{Thank}\:\mathrm{you}\:\mathrm{very}\:\mathrm{much}\:\mathrm{Sir}! \\ $$

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