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The 4.00 kg block is attached to a vertical rod by means of two strings. When the system rotates about the axis of the rod, the strings are extended as shown in the (Figure 1) and the tension in the upper string is 76.0 N.

Find the number of revolutions per minute at which the lower cord just goes slack.

The 400 kg block is attached to a vertical rod by means of two strings When the system rotates about the axis of the rod the strings are extended as shown in th class=

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Answer:

Let theta be angle of upper string with rod.

This a 3-4-5 triangle where R is distance of block from rod. (.75 - 1 - 1.25)

Fc = m w^2 R    where Fc the centripetal force and w the angular velocity

Fc = T sin theta = T * .75 / 1.25 = .6 T = 45.6 N

w^2 = 45.6 / (4 * .75)

w = 3.90 /sec

RPM = 3.90 / sec * 60 sec / min / (2 * pi) = 37.2 RPM      (2 pi rad / rev)

onyebuchinnaji

The number of revolutions per minute at which the lower cord just goes slack is 45.36 RPM.

The given parameters;

  • mass of the block, m = 4 kg
  • the tension on the upper string, T₁ = 76 N

The following triangle can be formed from the given diagram;

     |

     | 1 m

     |---------x--------(θ)  4.00 kg

The hypotenuse of this triangle = 1.25 m

The height of the triangle = 1 m

The base of the triangle is calculated as;

[tex]x^2 = 1.25^2 - 1^2\\\\x^2 = 0.5625\\\\x = \sqrt{0.5625} \\\\x = 0.75 \ m[/tex]

Resolving the vertical component of the force;

Tension at the upper = Tension at the lower string

[tex]T_1 sin(\theta) = T_2 sin(\theta) + mg\\\\76 \times \frac{1}{1.25} = T_2 \times \frac{1}{1.25} + (4\times 9.8)\\\\76 = T_2 + 39.2\\\\T_2 = 76 - 39.2\\\\T_2 = 36.8 \ N[/tex]

Resolving the horizontal component of the force;

[tex]T_1cos(\theta) + T_2cos(\theta) = F_c\\\\T_1cos(\theta) + T_2cos(\theta) = mr\omega^2 \\\\76\times \frac{0.75}{1.25} \ + \ 36.8\times \frac{0.75}{1.25}= 4\times 0.75\tmes (\frac{2\pi N}{60} )^2\\\\67.68 = 0.0329N^2\\\\N^2 = \frac{67.68}{0.0329} \\\\N^2 = 2057.14\\\\N = \sqrt{2057.14} \\\\N = 45.36 \ rpm[/tex]

Thus, the number of revolutions per minute at which the lower cord just goes slack is 45.36 RPM.

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