A 4 kg block is attached to a vertical rod by means of two strings of equal length. When the system rotates about the axis of the rod, the strings are extended as shown in figure. (a) How many revolut

Question

A 4 kg block is attached to a vertical rod by means of two strings of equal length. When the system rotates about the axis of the rod, the strings are extended as shown in figure. (a) How many revolutions per minute must the system make in order for the tension in the upper string to be 200 N? (b) What is the tension in the lower string then?

TestHub · Accepted Answer

Each string is 5 m and the rod segment between the attachment points is 8 m, so the radius is $r = 3 m$ ,
$cos θ = \frac{3}{5}, sin θ$
$= \frac{4}{5}$ (angle with the rod).
Vertical balance: $T_{1} sin θ = T_{2} sin θ + m g$ ,
i.e. $200 \times \frac{4}{5} = T_{2} \times \frac{4}{5} + 40$ ,
giving $T_{2} = 150 N$ .
Radial: $(T_{1} + T_{2}) cos θ = m r ω^{2}$ ,
i.e. $350 \times \frac{3}{5} = 4 \times 3 \times (2 πn)^{2}$ ,
giving $210 = 48 π^{2} n^{2}$ ,
so $n \approx 0.66 rps \approx 39.6 rpm$ .

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