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A block of mass m1 and a bucket of mass m2 are connected by a rope over a pulley. Find the acceleration of the system and the tension in the rope, neglecting pulley and rope masses.
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Solution: Step 1: Apply Newton's second law to the block and bucket. For the block: m1*g - T = m1*a For the bucket: T - m2*g = m2*a Step 2: Add the equations to eliminate T: m1*g - m2*g = (m1 + m2)*a Step 3: Solve for a: a = (m1 - m2)*g / (m1 + m2) Step 4: Substitute a back into one of the original equations to solve for T. Using the block equation: T = m1*(g - a) Step 5: Substitute a: T = m1*g - m1*(m1 - m2)*g / (m1 + m2) Step 6: Simplify T: T = m1*m2*g / (m1 + m2)
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