Question

A 0.840- kg glider on a level air track is joined by strings to two hanging masses. As seen in the figure, the mass on the left is 4.85 kg and the one on the right is 3.62 kg The strings have negligible mass and pass over light, frictionless pulleys. Find the acceleration of the masses when the air flow is turned off and the coefficient of friction between the glider and the track is 0.45. Take positive to be an acceleration to the right. Find the tension in the string on the left between the glider and the 4.85- kg mass when the air flow is turned off and the coefficient of friction between the glider and the track is 0.45. Find the tension in the string on the right between the glider and the 3.62- kg mass when the air flow is turned off and the coefficient of friction between the glider and the track is 0.45. 

Answer 1

Mass on the left M = 4.85 kg

Mass on the right m = 3.62 kg

Mass of the glider be M' = 0.84 kg

Acceleration of both the masses would be same but opposite in direction. Let this acceleration be a.

Since the acceleration to the right is taken positive , acceleration and forces to the left are taken negative.

( 1 ) For the mass on the right , weight acts downwards and tension T1 upwards. This mass m moves upwards as its mass is less.

mg - T1 = ma  

T1 = m ( g - a ) ............ ( 1 )

( 2 ) For the mass on the left , weight acts downwards and tension T2 upwards.

T2 - M g = M a

T2 = M ( g + a ) ............... ( 2 )

( 3 ) For the glider ,

T1 - T2 + f = M' a

where f is the friction between the glider and track

m ( g - a ) - M ( g + a ) + $\mu$ M' g = M'a

( m + $\mu$ M' - M ) g = ( M' + M + m ) a

a = ( m + $\mu$ M' - M ) / ( M' + M + m )

a = { 3.62 + ( 0.45 × 0.84 ) + 4.85 ) / ( 0.84 + 4.85 + 3.62 )

= 9.81 × ( 3.62 - 4.85 + 0.378 ) / 9.31

= 9.81 × ( - 0.852 ) / 9.31

= - 8.358 /9.31

= - 0.8977 m/s²

Here negative sign indicates that the acceleration is towards the heavier mass.

B) From the equation ( 2 ) , tension in the string on the left is

T2 = M ( g + a )

= 4.85 × ( 9.81 + ( - 0.8977 )

= 4.85 × ( 8.9123 )

= 43.22 N

( C ) From the equation ( 1 ) , tension in the string on the right is

T1 = m ( g - a )

= 3.62 × ( 9.81 - ( - 0.8977 )

= 3.62 × ( 9.81 + 0.8977 )

= 3.62 × ( 10.70 )

= 38.76 N