m₁ = mass 1 = 1.4 kg m₂ = mass 2 = 2.7 kgm₃ = mass of the pulley = 0.72 kgR = radius of the pulley = 0.25 mg = accelerayion by gravity = 9.8 m/s²First, you need to ‘choose’ a direction in which the system will accelerate. As m₂ is bigger than m₁, it seems logic to choose m₂ moving down, and m₁ moving up.(Note: If you choose wrong, that’s not a problem. You will end up with a negative value for acceleration so you will know that the system moves in the other direction than the one you chose)The forces acting on m₁ are:* tensile force in the rope, pulling up * force of gravity, pulling downAs m₁ is moving up, you know that the tensile force is bigger than the gravitational force. The net force on m₁ will be:Fnet₁ = T₁ – m₁×gBy Newton’s second law, you can say:T₁ – m₁×g = m₁×aT₁ = m₁×g + m₁×a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . equation 1The forces acting on m₂ are:* tensile force in the rope, pulling up* force of gravity, pulling downAs m₂ is moving down, you know that the gravitational force is bigger than the tensile force. The net force on m₂ will be:Fnet₂ = m₂×g – T₂By Newton’s second law, you can say:m₂×g – T₂ = m₂×aT₂ = m₂×g – m₂×a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . equation 2The moment of inertia of the pulley (assuming it is a solid disk) is:I = m₃×R²/2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . equation 3The equation for torque on the pulley is:T₂×R – T₁×R = I×a/RRearranging this a bit, you get:(T₂ – T₁)×R = I×a/RT₂ – T₁ = I×a/R² . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . equation 4 Substituting equation 3 into equation 4, you get:T₂ – T₁ = (m₃×R²/2)×a/R²T₂ – T₁ = m₃×a/2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . equation 5(note how the radius R cancels out)Now substituting equations 1 and 2 into equation 5, you get:(m₂×g – m₂×a) – (m₁×g + m₁×a) = m₃×a/2m₂×g – m₂×a – m₁×g – m₁×a = m₃×a/2m₂×g – m₁×g = m₃×a/2 + m₂×a + m₁×a(m₂ – m₁)×g = (m₃/2 + m₂ + m₁)×aa = g × (m₂ – m₁) / (m₃/2 + m₂ + m₁)And finally, entering the numbers, you get:a = (9.8 m/s²) × [(2.7 kg) – (1.4 kg)] / [(0.72 kg)/2 + (2.7 kg) + (1.4 kg)]a = 2.8565… m/s²a = 2.9 m/s² (to 2 sf)...
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