Question

A particle of mass M rests on a straight groove along which it is constrained to move. A perfectly elastic rubber band of natural length l and uniform area of cross-section is attached with the particle. The other end of the band is suspended from a rigid support. A force K(l^'2 - l²)^1/2is required to stretch the band to a length l'. The particle is moved to a distance S (where S << 1) and then released. Taking K = $\frac { \mathrm { Mg } } { \mathrm { S } }$ and μas the coefficient of friction between the particle and the groove, the velocity of particle when passing through the initial position is

• A. $\frac { g S } { 31 } ( 2 S - 3 \mu 1 ) ^ { 1 / 2 }$
• B. $\left[ \frac { \mathrm { gS } } { 31 } ( 2 \mathrm {~S} - 3 \mu \mathrm { l } ) \right] ^ { 1 / 2 }$
• C. $\frac { g S } { 1 } ( 3 S - 2 \mu l ) ^ { 1 / 2 }$
• D. $\left[ \frac { \mathrm { gS } } { 21 } ( 3 \mathrm {~S} - 2 \mu \mathrm { l } ) \right] ^ { 1 / 2 }$

Answer 1

Answer: (A)

$\frac { g S } { 31 } ( 2 S - 3 \mu 1 ) ^ { 1 / 2 }$

Answer 2

Let the particle be at a distance x at any instant and T be the tension in the string then

T = K(l'² – l²)^1/2 = Kx

Net force tending to make the particle move further through dx.

Work done = $\left[ \frac { \mathrm { Kx } ^ { 2 } } { 1 } - \mu ( \mathrm { Mg } - \mathrm { Kx } ) \right] \mathrm { dx }$

$\frac { 1 } { 2 } \mathrm { Mv } ^ { 2 } = \int _ { 0 } \left[ \frac { K x ^ { 2 } } { 1 } - \mu ( M g - K x ) \right] d x$

= $\frac { \operatorname { Mg } } { 31 } \left( 3 - \frac { 3 } { 2 } \mu 1 \right)$

or v = $\left[ \frac { \mathrm { gS } } { 31 } ( 2 \mathrm {~S} - 3 \mu \mathrm { l } ) \right] ^ { 1 / 2 }$