A body of mass 0.4 kg is whirled in a vertical circle making 2 rev/sec. If the radius of the circle is 2 m, th

Question

A body of mass 0.4 kg is whirled in a vertical circle making 2 rev/sec. If the radius of the circle is 2 m, then tension in the string when the body is at the top of the circle is:

Accepted Answer

1. **Identify Given Values:**
   - Mass ($m$) = 0.4 kg
   - Frequency ($f$) = 2 rev/sec
   - Radius ($r$) = 2 m
   - Acceleration due to gravity ($g$) $\approx 9.8 	ext{ m/s}^2$

2. **Calculate Angular Velocity ($\omega$):**
   $$ \omega = 2\pi f = 2\pi(2) = 4\pi 	ext{ rad/s} $$
   [Source 43]

3. **Forces at the Top:**
   At the highest point of a vertical circle, both the Tension ($T$) and the Weight ($mg$) act downwards towards the center. The sum of these forces provides the necessary centripetal force ($F_c$).
   $$ T + mg = F_c = m\omega^2 r $$
   [Source 76, 82]

4. **Calculate Centripetal Force:**
   $$ F_c = 0.4 	imes (4\pi)^2 	imes 2 $$
   $$ F_c = 0.8 	imes 16\pi^2 = 12.8\pi^2 $$
   Using $\pi^2 \approx 9.87$:
   $$ F_c \approx 12.8 	imes 9.87 \approx 126.34 	ext{ N} $$

5. **Calculate Tension:**
   Rearranging the force equation for $T$:
   $$ T = m\omega^2 r - mg $$
   $$ T = 126.34 - (0.4 	imes 9.8) $$
   $$ T = 126.34 - 3.92 $$
   $$ T = 122.42 	ext{ N} $$
   This matches Option 4.

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