A 0.51 kg object is tied using a string and spun in a vertical circle of radius {r} m. The tension of the string varies such that it is zero at the top of the circle. If this was spun in free-fall: Determine the magnitude of the tension in the string when the object is at the bottom of the circle. Assume that the speed of the object is constant throughout the circular motion.

Question

BlackTom AI · Accepted Answer

At the top of the circle the centripetal force must be provided entirely by gravity because the string tension is zero. Writing the centripetal force condition at the top gives m v^2 / r = mg, so v^2 = g r. (This uses: centripetal acceleration a_c = v^2 / r, weight = mg, and the sign convention that both act toward the center at the top.)

Step-by-step reasoning and calculations:
1. At top: sum of forces toward center = m v^2 / r.
   Downward forces toward center: mg + T_top. Given T_top = 0, so m v^2 / r = mg.
2. Solve for v^2:
   v^2 = g r.
   (No numerical value for r is needed because it cancels in the next step.)
3. At bottom: centripetal acceleration is upward toward the circle center. Sum of forces toward center:
   T_bottom - mg = m v^2 / r.
4. Substitute v^2 = g r into the bottom equation:
   T_bottom - mg = m (g r) / r = m g.
   Therefore T_bottom = mg + m g = 2 m g.
5. Numerical substitution: m = 0.51 kg, g = 9.81 m/s^2.
   mg = 0.51 × 9.81 = 5.0031 N.
   T_bottom = 2 × 5.0031 = 10.0062 N ≈ 10.01 N.

Why this answer is correct: the tension at the bottom must both support the weight and supply the upward centripetal force equal to m v^2 / r. Because v^2 was fixed by the top condition to equal g r, the required bottom tension becomes 2 m g.

Why the other choices are incorrect:
- a. 0.00 N: tension zero only at the top, not at the bottom.
- c. 9.81 N: this equals mg for a 1 kg mass; for m = 0.51 kg mg ≈ 5.00 N, so 9.81 N is irrelevant.
- d. 5.00 N: this is approximately mg (0.51×9.81 ≈ 5.00 N) but does not include the additional centripetal requirement; the bottom tension must be twice mg.

Final confirmation: the correct magnitude is 10.01 N, matching answer b.

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