Question

A, B, C and D are four masses each of mass m lying on the vertices of a square of side ‘a’. They always move along a common circle with velocity v. Find v so that they always remain on the vertices of the square.

|F₁| = |F₂| $\left| \overrightarrow { \mathrm { F } } _ { 1 } + \overrightarrow { \mathrm { F } } _ { 2 } \right|$ = $\sqrt { 2 }$ F₁

• A. $\sqrt { \frac { \operatorname { GM } ( 2 \sqrt { 2 } + 1 ) } { 2 \sqrt { 2 } a } }$
• B. $\sqrt { \frac { \mathrm { GM } ( \sqrt { 2 } + 1 ) } { 2 \mathrm { a } } }$
• C.
• D. None

Answer 1

Answer: (A)

$\sqrt { \frac { \operatorname { GM } ( 2 \sqrt { 2 } + 1 ) } { 2 \sqrt { 2 } a } }$

Answer 2

F_net =√2 F₁ + F₂ =

$\sqrt { 2 }$ a = 2R or R = $\frac { \mathrm { a } } { \sqrt { 2 } }$

$\sqrt { 2 }$ $\frac { \mathrm { Gm } ^ { 2 } } { \mathrm { a } ^ { 2 } } + \frac { \mathrm { Gm } ^ { 2 } } { 2 \mathrm { a } ^ { 2 } } = \frac { \mathrm { mv } ^ { 2 } } { \mathrm { a } } \sqrt { 2 }$

or v = $\sqrt { \frac { \operatorname { GM } ( 2 \sqrt { 2 } + 1 ) } { 2 \sqrt { 2 } a } }$