Two halls are tied together with a string that passes through a smooth, tapered

Question

Two halls are tied together with a string that passes through a smooth, tapered hole in a tabletop so that the tension in the string is uniform throughout, as shown in Fig.-1. The top of the table is frictionless. Ball 1 on the tabletop has mass m1 = 6.00 kg and Ball 2 hanging under the table has mass m2 = 12.00 kg, the length of the string above the hole is d1 = 0.500 m and the length of the string below the tabletop is d2 = 0.750 m. Clearly, if you hold Ball 1 and then let go. Ball 2 will fall to the ground, dragging Ball 1 into the hole. Before letting go, what could you do to Ball 1 to stop d1 or d2 from changing? You must give a precise, numerical answer. You cannot touch the table. Ball 2 (the one hanging under the table), or the string, or change them in any way, and you have to eventually let go of Ball 1. You cannot place anything on top of Ball 1, or tie it to something, or anything like that, to stop it from moving. So what else can you do? Remember to give a precise, numerical answer. Figure 1 - The balls are tied together with a string, which passes through a smooth, tapered hole so that the tension in the string is uniform throughout. The tabletop is frictionless. m1 = 6.00kg and m2 = 12.00 kg. d1 = 0.500 m and d2 = 0.750 m.

Explanation / Answer

You can rotate m1 through a radius d1 so that

Centripetal force = weight of m2

Thus,

m1 v^2 / d1 = m2 g

Thus, solving for v,

v = sqrt [m2 g d1 / m1]

v = 3.13 m/s   [ANSWER is to rotate m1 with a speed of 3.13 m/s]

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