Three objects are connected on the table as show the mass are 6kg, 2kg, and 3kg.

Question

Three objects are connected on the table as show the mass are 6kg, 2kg, and 3kg. The talbe has a kinetic friction coefficient of 0.1...

a) Assume that polleys are fritionless and light, what is the acceleration of each object?

b) What if the polleys have 0.5kgm^2 moments of ineition ( I1 = I2 = 0.5kgm^2 ) and 50cm radius ( R1 = R2 = 50cm), what would be the acceleration of each object

Three objects are connected on the table as show the mass are 6kg, 2kg, and 3kg. The talbe has a kinetic friction coefficient of 0.1... Assume that polleys are frictionless and light, what is the acceleration of each object? What if the polleys have 0.5kgm^2 moments of ineition ( I1 = I2 = 0.5kgm^2 ) and 50cm radius ( R1 = R2 = 50cm), what would be the acceleration of each object

Explanation / Answer

The net Gravitational force on the system = ( 6-3 )g = 3 X 9.8 = 29.4 N

Net mass of the system = 6+3+2 = 11 kg

Normal force acting on the lbock on the table = N = 2g N

Force of friction = f = 0.1 X N = 0.2g = 1.96 N

( 29.4 - 1.96 ) = 11 X a

==> a = 2.495 m/s^2

Acceleration of each object = 2.495 m/s^2

Let the left and right pulleys be P1 and P2

T1--> tension between 6kg and P1

T2--> tension between P1 and 2kg

T3--> tension between 2kg and P2

T4--> tension between P2 and 3kg

Angular acceleration of the pulleys = A = a/R [ since we assume there is no slipping ]

So net torque acing on P1 = ( T1-T2 ) X R

net torque on P2 = ( T3 - T4 ) X R

Equation on block of mass 6kg : 6g - T1 = 6a....... (i)

Equation on block of mass 2kg : T2 - T3 - f = 2a.........(ii)

Equation on block of mass 3kg : T4 - 3g = 3a...........(iii)

Equation on P1 : ( T1 - T2 )R = Ia/R.............(iv)

Equation on P2 : ( T3 - T4 )R = Ia /R...........(v)

Substituting (iv) and (v) in the other 3 eqns we have the eqns

6g - T1 = 6a

T1 - T3 - 0.2g = 2a + Ia/r^2

T3 - 3g = 3a + Ia/r^2

Solving these three equation we have a = 2.8g / ( 11 + 2I / R^2 ) = 1.83 m/s^2

Net acceleration if the pulleys are rolling = 1.83 m/s^2

Note : However of if the friction between the pulleys is not sufficient and if the ropes just slip over the pulleys then even if the pulleys have some mass and moment of inertia the acceleration of each object will not change and willr emain the same as in the first case

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