The boxes of masse 2 kg and 8 kg are connected by a massless string passing over smooth pulleys. Calculate the time taken by box of mass 8 kg to strike the ground starting from rest. (use g = 10 m/s²)

Solution

Physics sir · 4 steps

The 8 kg box hangs on a movable pulley, so the 2 kg box moves twice as fast. Start from that link.

1. String constraint. One string runs from the ceiling, under the movable pulley, over the fixed pulley and down to the 2 kg box. If the 8 kg box drops by x, both strands around the movable pulley lengthen by x, so the 2 kg box rises by 2x. That gives a₂ = 2a, where a is the acceleration of the 8 kg box.
2. Forces on each box. The tension T is the same along the whole string. The movable pulley has two strands pulling up, so the 8 kg box feels 2T: 80 - 2T = 8a. The 2 kg box feels one strand: T - 20 = 2(2a) = 4a.
3. Solve for a. From the second equation, T = 20 + 4a. Substituting gives 80 - 40 - 8a = 8a, so 16a = 40 and a = 2.5 m s⁻². Here T = 30 N, which is more than 20 N, so the 2 kg box does lift off.

   s = 1/2at² ⇒ 0.2 = 1/2(2.5)t² ⇒ t² = 0.16

4. So. The box starts from rest and falls 20 cm = 0.2 m, so t = 0.4 s. That is option (d).

The idea to keep. With a movable pulley, first write the length constraint to link the accelerations. Then count the strands pulling on each body: two strands on the pulley means 2T.

Answer from the NTA final answer key. Solution by PrepWiser.