Question

Let μbe the coefficient of friction between blocks of mass m and M, The pulleys are frictionless and strings are massless. Acceleration of mass m is

• A. $\frac { \sqrt { 5 } \mathrm { mg } } { \mathrm { M } + \sqrt { 5 } \mathrm {~m} + 2 \mu \mathrm { m } }$
• B. $\frac { 2 m g } { M + 5 m + 2 \mu m }$
• C. $\frac { 2 \sqrt { 5 } \mathrm { mg } } { \mathrm { M } + 5 \mathrm {~m} + 2 \mu \mathrm { m } }$
• D. $\frac { 5 \sqrt { 2 } m g } { M + \sqrt { 5 } m + \sqrt { 2 } \mu \mathrm { m } }$

Answer 1

Answer: (C)

$\frac { 2 \sqrt { 5 } \mathrm { mg } } { \mathrm { M } + 5 \mathrm {~m} + 2 \mu \mathrm { m } }$

Answer 2

acceleration of mass m is a = $\sqrt { \mathrm { a } _ { \mathrm { x } } ^ { 2 } + \mathrm { a } _ { \mathrm { y } } ^ { 2 } }$

Various force equations are

2T – N = ma_x

N = ma_x

and mg - μN – T = ma_y

Solving a_x = $\frac { 2 m g } { M + 5 m + 2 \mu m }$

and a_y = $\frac { 4 m g } { M + 5 m + 2 \mu m }$

∴ a = $\sqrt { \mathrm { a } _ { \mathrm { x } } ^ { 2 } + \mathrm { a } _ { \mathrm { y } } ^ { 2 } }$

= $\frac { 2 \sqrt { 5 } \mathrm { mg } } { \mathrm { M } + 5 \mathrm {~m} + 2 \mu \mathrm { m } }$